set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta".
-include "Base.ma".
-
-inductive B: Set \def
-| Abbr: B
-| Abst: B
-| Void: B.
-
-inductive F: Set \def
-| Appl: F
-| Cast: F.
-
-inductive K: Set \def
-| Bind: B \to K
-| Flat: F \to K.
-
-inductive T: Set \def
-| TSort: nat \to T
-| TLRef: nat \to T
-| THead: K \to (T \to (T \to T)).
-
-inductive TList: Set \def
-| TNil: TList
-| TCons: T \to (TList \to TList).
-
-definition THeads: K \to (TList \to (T \to T)) \def let rec THeads (k: K) (us: TList): (T \to T) \def (\lambda (t: T).(match us with [TNil \Rightarrow t | (TCons u ul) \Rightarrow (THead k u (THeads k ul t))])) in THeads.
-
-definition s: K \to (nat \to nat) \def \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow (S i) | (Flat _) \Rightarrow i])).
-
-axiom not_abbr_abst: not (eq B Abbr Abst) .
-
-axiom not_void_abst: not (eq B Void Abst) .
-
-axiom terms_props__bind_dec: \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall (P: Prop).P)))) .
-
-axiom terms_props__flat_dec: \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall (P: Prop).P)))) .
-
-axiom terms_props__kind_dec: \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall (P: Prop).P)))) .
-
-axiom term_dec: \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall (P: Prop).P)))) .
-
-axiom binder_dec: \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) .
-
-axiom abst_dec: \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))))) .
-
-axiom thead_x_y_y: \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to (\forall (P: Prop).P)))) .
-
-axiom s_S: \forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i)))) .
-
-axiom s_plus: \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) (plus (s k i) j)))) .
-
-axiom s_plus_sym: \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j)) (plus i (s k j))))) .
-
-axiom s_minus: \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s k (minus i j)) (minus (s k i) j))))) .
-
-axiom minus_s_s: \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s k j)) (minus i j)))) .
-
-axiom s_le: \forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i) (s k j))))) .
-
-axiom s_lt: \forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i) (s k j))))) .
-
-axiom s_inj: \forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (s k i) (s k j)) \to (eq nat i j)))) .
-
-axiom s_arith0: \forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i)) .
-
-axiom s_arith1: \forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i)) .
-
-definition wadd: ((nat \to nat)) \to (nat \to (nat \to nat)) \def \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n with [O \Rightarrow w | (S m) \Rightarrow (f m)]))).
-
-definition weight_map: ((nat \to nat)) \to (T \to nat) \def let rec weight_map (f: ((nat \to nat))) (t: T): nat \def (match t with [(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O) t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f t0)))])]) in weight_map.
-
-definition weight: T \to nat \def weight_map (\lambda (_: nat).O).
-
-definition tlt: T \to (T \to Prop) \def \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))).
-
-axiom wadd_le: \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to (\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) .
-
-axiom wadd_lt: \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to (\forall (n: nat).(le (wadd f v n) (wadd g w n)))))))) .
-
-axiom wadd_O: \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O) .
-
-axiom weight_le: \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t) (weight_map g t))))) .
-
-axiom weight_eq: \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f t) (weight_map g t))))) .
-
-axiom weight_add_O: \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t) (weight_map (\lambda (_: nat).O) t)) .
-
-axiom weight_add_S: \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O) O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t))) .
-
-axiom tlt_trans: \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to (tlt u t))))) .
-
-axiom tlt_head_sx: \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t)))) .
-
-axiom tlt_head_dx: \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t)))) .
-
-axiom tlt_wf__q_ind: \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P: ((T \to Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P t))))) P n))) \to (\forall (t: T).(P t))) .
-
-axiom tlt_wf_ind: \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t) \to (P v)))) \to (P t)))) \to (\forall (t: T).(P t))) .
+include "LambdaDelta/theory.ma".
+
+definition wadd:
+ ((nat \to nat)) \to (nat \to (nat \to nat))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (w: nat).(\lambda (n: nat).(match n
+with [O \Rightarrow w | (S m) \Rightarrow (f m)]))).
+
+definition weight_map:
+ ((nat \to nat)) \to (T \to nat)
+\def
+ let rec weight_map (f: ((nat \to nat))) (t: T) on t: nat \def (match t with
+[(TSort _) \Rightarrow O | (TLRef n) \Rightarrow (f n) | (THead k u t0)
+\Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr
+\Rightarrow (S (plus (weight_map f u) (weight_map (wadd f (S (weight_map f
+u))) t0))) | Abst \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f
+O) t0))) | Void \Rightarrow (S (plus (weight_map f u) (weight_map (wadd f O)
+t0)))]) | (Flat _) \Rightarrow (S (plus (weight_map f u) (weight_map f
+t0)))])]) in weight_map.
+
+definition weight:
+ T \to nat
+\def
+ weight_map (\lambda (_: nat).O).
+
+definition tlt:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(lt (weight t1) (weight t2))).
+
+theorem wadd_le:
+ \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
+nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((le v w) \to
+(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
+((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
+nat).(\lambda (H0: (le v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(le (wadd f v n0) (wadd g w n0))) H0 (\lambda (n0: nat).(\lambda (_: (le
+(wadd f v n0) (wadd g w n0))).(H n0))) n))))))).
+
+theorem wadd_lt:
+ \forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
+nat).(le (f n) (g n)))) \to (\forall (v: nat).(\forall (w: nat).((lt v w) \to
+(\forall (n: nat).(le (wadd f v n) (wadd g w n))))))))
+\def
+ \lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
+((\forall (n: nat).(le (f n) (g n))))).(\lambda (v: nat).(\lambda (w:
+nat).(\lambda (H0: (lt v w)).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(le (wadd f v n0) (wadd g w n0))) (le_S_n v w (le_S (S v) w H0))
+(\lambda (n0: nat).(\lambda (_: (le (wadd f v n0) (wadd g w n0))).(H n0)))
+n))))))).
+
+theorem wadd_O:
+ \forall (n: nat).(eq nat (wadd (\lambda (_: nat).O) O n) O)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (wadd (\lambda (_:
+nat).O) O n0) O)) (refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat
+(wadd (\lambda (_: nat).O) O n0) O)).(refl_equal nat O))) n).
+
+theorem weight_le:
+ \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t)
+(weight_map g t)))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t0) (weight_map g t0)))))) (\lambda (n: nat).(\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (n:
+nat).(le (f n) (g n))))).(le_n (weight_map g (TSort n))))))) (\lambda (n:
+nat).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H:
+((\forall (n: nat).(le (f n) (g n))))).(H n))))) (\lambda (k: K).(K_ind
+(\lambda (k0: K).(\forall (t0: T).(((\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le
+(weight_map f t0) (weight_map g t0)))))) \to (\forall (t1: T).(((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
+(g n)))) \to (le (weight_map f t1) (weight_map g t1)))))) \to (\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
+(g n)))) \to (le (weight_map f (THead k0 t0 t1)) (weight_map g (THead k0 t0
+t1))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (t0:
+T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t0) (weight_map g t0))))))
+\to (\forall (t1: T).(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t1)
+(weight_map g t1)))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (match b0 with
+[Abbr \Rightarrow (S (plus (weight_map f t0) (weight_map (wadd f (S
+(weight_map f t0))) t1))) | Abst \Rightarrow (S (plus (weight_map f t0)
+(weight_map (wadd f O) t1))) | Void \Rightarrow (S (plus (weight_map f t0)
+(weight_map (wadd f O) t1)))]) (match b0 with [Abbr \Rightarrow (S (plus
+(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1))) | Abst
+\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) | Void
+\Rightarrow (S (plus (weight_map g t0) (weight_map (wadd g O)
+t1)))])))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
+\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
+t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus
+(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1))
+(plus_le_compat (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S
+(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g
+H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0)))
+(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0))
+(le_n_S (weight_map f t0) (weight_map g t0) (H f g H1)) n))))))))))))
+(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f
+t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
+(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
+(f n) (g n))))).(le_S_n (S (plus (weight_map f t0) (weight_map (wadd f O)
+t1))) (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) (le_n_S (S
+(plus (weight_map f t0) (weight_map (wadd f O) t1))) (S (plus (weight_map g
+t0) (weight_map (wadd g O) t1))) (le_n_S (plus (weight_map f t0) (weight_map
+(wadd f O) t1)) (plus (weight_map g t0) (weight_map (wadd g O) t1))
+(plus_le_compat (weight_map f t0) (weight_map g t0) (weight_map (wadd f O)
+t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda
+(n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))))) (\lambda (t0:
+T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f t0)
+(weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g
+n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
+(f n) (g n))))).(le_S_n (S (plus (weight_map f t0) (weight_map (wadd f O)
+t1))) (S (plus (weight_map g t0) (weight_map (wadd g O) t1))) (le_n_S (S
+(plus (weight_map f t0) (weight_map (wadd f O) t1))) (S (plus (weight_map g
+t0) (weight_map (wadd g O) t1))) (le_n_S (plus (weight_map f t0) (weight_map
+(wadd f O) t1)) (plus (weight_map g t0) (weight_map (wadd g O) t1))
+(plus_le_compat (weight_map f t0) (weight_map g t0) (weight_map (wadd f O)
+t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda
+(n: nat).(wadd_le f g H1 O O (le_n O) n)))))))))))))) b)) (\lambda (_:
+F).(\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (n: nat).(le (f n) (g n)))) \to (le (weight_map f
+t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n)
+(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f0:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n:
+nat).(le (f0 n) (g n))))).(lt_le_S (plus (weight_map f0 t0) (weight_map f0
+t1)) (S (plus (weight_map g t0) (weight_map g t1))) (le_lt_n_Sm (plus
+(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g
+t1)) (plus_le_compat (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1)
+(weight_map g t1) (H f0 g H1) (H0 f0 g H1)))))))))))) k)) t).
+
+theorem weight_eq:
+ \forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (n: nat).(eq nat (f n) (g n)))) \to (eq nat (weight_map f
+t) (weight_map g t)))))
+\def
+ \lambda (t: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H: ((\forall (n: nat).(eq nat (f n) (g n))))).(le_antisym
+(weight_map f t) (weight_map g t) (weight_le t f g (\lambda (n:
+nat).(eq_ind_r nat (g n) (\lambda (n0: nat).(le n0 (g n))) (le_n (g n)) (f n)
+(H n)))) (weight_le t g f (\lambda (n: nat).(eq_ind_r nat (g n) (\lambda (n0:
+nat).(le (g n) n0)) (le_n (g n)) (f n) (H n)))))))).
+
+theorem weight_add_O:
+ \forall (t: T).(eq nat (weight_map (wadd (\lambda (_: nat).O) O) t)
+(weight_map (\lambda (_: nat).O) t))
+\def
+ \lambda (t: T).(weight_eq t (wadd (\lambda (_: nat).O) O) (\lambda (_:
+nat).O) (\lambda (n: nat).(wadd_O n))).
+
+theorem weight_add_S:
+ \forall (t: T).(\forall (m: nat).(le (weight_map (wadd (\lambda (_: nat).O)
+O) t) (weight_map (wadd (\lambda (_: nat).O) (S m)) t)))
+\def
+ \lambda (t: T).(\lambda (m: nat).(weight_le t (wadd (\lambda (_: nat).O) O)
+(wadd (\lambda (_: nat).O) (S m)) (\lambda (n: nat).(le_S_n (wadd (\lambda
+(_: nat).O) O n) (wadd (\lambda (_: nat).O) (S m) n) (le_n_S (wadd (\lambda
+(_: nat).O) O n) (wadd (\lambda (_: nat).O) (S m) n) (wadd_le (\lambda (_:
+nat).O) (\lambda (_: nat).O) (\lambda (_: nat).(le_n O)) O (S m) (le_O_n (S
+m)) n)))))).
+
+theorem tlt_trans:
+ \forall (v: T).(\forall (u: T).(\forall (t: T).((tlt u v) \to ((tlt v t) \to
+(tlt u t)))))
+\def
+ \lambda (v: T).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (lt (weight u)
+(weight v))).(\lambda (H0: (lt (weight v) (weight t))).(lt_trans (weight u)
+(weight v) (weight t) H H0))))).
+
+theorem tlt_head_sx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt u (THead k u t))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) (THead
+k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
+(t: T).(lt (weight_map (\lambda (_: nat).O) u) (match b0 with [Abbr
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
+(u: T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda (_: nat).O) u)) (S
+(plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_:
+nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (le_n_S (S (weight_map
+(\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O)
+u))) t))) (le_n_S (weight_map (\lambda (_: nat).O) u) (plus (weight_map
+(\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map
+(\lambda (_: nat).O) u))) t)) (le_plus_l (weight_map (\lambda (_: nat).O) u)
+(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O)
+u))) t))))))) (\lambda (u: T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda
+(_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map
+(wadd (\lambda (_: nat).O) O) t))) (le_n_S (S (weight_map (\lambda (_:
+nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) O) t))) (le_n_S (weight_map (\lambda (_: nat).O) u)
+(plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_:
+nat).O) O) t)) (le_plus_l (weight_map (\lambda (_: nat).O) u) (weight_map
+(wadd (\lambda (_: nat).O) O) t))))))) (\lambda (u: T).(\lambda (t:
+T).(le_S_n (S (weight_map (\lambda (_: nat).O) u)) (S (plus (weight_map
+(\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))
+(le_n_S (S (weight_map (\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))) (le_n_S
+(weight_map (\lambda (_: nat).O) u) (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (wadd (\lambda (_: nat).O) O) t)) (le_plus_l (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t))))))) b))
+(\lambda (_: F).(\lambda (u: T).(\lambda (t: T).(le_S_n (S (weight_map
+(\lambda (_: nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (\lambda (_: nat).O) t))) (le_n_S (S (weight_map (\lambda (_:
+nat).O) u)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda
+(_: nat).O) t))) (le_n_S (weight_map (\lambda (_: nat).O) u) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))
+(le_plus_l (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_:
+nat).O) t)))))))) k).
+
+theorem tlt_head_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(tlt t (THead k u t))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (t: T).(lt
+(weight_map (\lambda (_: nat).O) t) (weight_map (\lambda (_: nat).O) (THead
+k0 u t)))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (u: T).(\forall
+(t: T).(lt (weight_map (\lambda (_: nat).O) t) (match b0 with [Abbr
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) | Abst
+\Rightarrow (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) O) t))) | Void \Rightarrow (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) O) t)))]))))) (\lambda
+(u: T).(\lambda (t: T).(lt_le_trans (weight_map (\lambda (_: nat).O) t) (S
+(weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda (_:
+nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_:
+nat).O) u))) t))) (lt_n_Sn (weight_map (\lambda (_: nat).O) t)) (le_n_S
+(weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u)
+(weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O)
+u))) t)) (le_trans (weight_map (\lambda (_: nat).O) t) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (eq_ind nat (weight_map (wadd
+(\lambda (_: nat).O) O) t) (\lambda (n: nat).(le n (weight_map (wadd (\lambda
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t))) (weight_add_S t
+(weight_map (\lambda (_: nat).O) u)) (weight_map (\lambda (_: nat).O) t)
+(weight_add_O t)) (le_plus_r (weight_map (\lambda (_: nat).O) u) (weight_map
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t)))))))
+(\lambda (u: T).(\lambda (t: T).(eq_ind_r nat (weight_map (\lambda (_:
+nat).O) t) (\lambda (n: nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus
+(weight_map (\lambda (_: nat).O) u) n)))) (le_S_n (S (weight_map (\lambda (_:
+nat).O) t)) (S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (\lambda
+(_: nat).O) t))) (le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)))
+(le_n_S (weight_map (\lambda (_: nat).O) t) (plus (weight_map (\lambda (_:
+nat).O) u) (weight_map (\lambda (_: nat).O) t)) (le_plus_r (weight_map
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t))))) (weight_map
+(wadd (\lambda (_: nat).O) O) t) (weight_add_O t)))) (\lambda (u: T).(\lambda
+(t: T).(eq_ind_r nat (weight_map (\lambda (_: nat).O) t) (\lambda (n:
+nat).(lt (weight_map (\lambda (_: nat).O) t) (S (plus (weight_map (\lambda
+(_: nat).O) u) n)))) (le_S_n (S (weight_map (\lambda (_: nat).O) t)) (S (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)))
+(le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) (le_n_S (weight_map
+(\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map
+(\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u)
+(weight_map (\lambda (_: nat).O) t))))) (weight_map (wadd (\lambda (_:
+nat).O) O) t) (weight_add_O t)))) b)) (\lambda (_: F).(\lambda (u:
+T).(\lambda (t: T).(le_S_n (S (weight_map (\lambda (_: nat).O) t)) (S (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) t)))
+(le_n_S (S (weight_map (\lambda (_: nat).O) t)) (S (plus (weight_map (\lambda
+(_: nat).O) u) (weight_map (\lambda (_: nat).O) t))) (le_n_S (weight_map
+(\lambda (_: nat).O) t) (plus (weight_map (\lambda (_: nat).O) u) (weight_map
+(\lambda (_: nat).O) t)) (le_plus_r (weight_map (\lambda (_: nat).O) u)
+(weight_map (\lambda (_: nat).O) t)))))))) k).
+
+theorem tlt_wf__q_ind:
+ \forall (P: ((T \to Prop))).(((\forall (n: nat).((\lambda (P: ((T \to
+Prop))).(\lambda (n0: nat).(\forall (t: T).((eq nat (weight t) n0) \to (P
+t))))) P n))) \to (\forall (t: T).(P t)))
+\def
+ let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
+T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (t: T).((eq nat (weight t)
+n) \to (P t)))))).(\lambda (t: T).(H (weight t) t (refl_equal nat (weight
+t)))))).
+
+theorem tlt_wf_ind:
+ \forall (P: ((T \to Prop))).(((\forall (t: T).(((\forall (v: T).((tlt v t)
+\to (P v)))) \to (P t)))) \to (\forall (t: T).(P t)))
+\def
+ let Q \def (\lambda (P: ((T \to Prop))).(\lambda (n: nat).(\forall (t:
+T).((eq nat (weight t) n) \to (P t))))) in (\lambda (P: ((T \to
+Prop))).(\lambda (H: ((\forall (t: T).(((\forall (v: T).((lt (weight v)
+(weight t)) \to (P v)))) \to (P t))))).(\lambda (t: T).(tlt_wf__q_ind
+(\lambda (t0: T).(P t0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (t0:
+T).(P t0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (t: T).(P t)) m))))).(\lambda (t0: T).(\lambda (H1: (eq nat
+(weight t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n: nat).(\forall (m:
+nat).((lt m n) \to (\forall (t: T).((eq nat (weight t) m) \to (P t)))))) H0
+(weight t0) H1) in (H t0 (\lambda (v: T).(\lambda (H3: (lt (weight v) (weight
+t0))).(H2 (weight v) H3 v (refl_equal nat (weight v))))))))))))) t)))).
inductive iso: T \to (T \to Prop) \def
| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
| iso_lref: \forall (i1: nat).(\forall (i2: nat).(iso (TLRef i1) (TLRef i2)))
-| iso_head: \forall (k: K).(\forall (v1: T).(\forall (v2: T).(\forall (t1: T).(\forall (t2: T).(iso (THead k v1 t1) (THead k v2 t2)))))).
-
-axiom iso_flats_lref_bind_false: \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))))))) .
-
-axiom iso_flats_flat_bind_false: \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall (v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads (Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))))))))) .
-
-axiom iso_trans: \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 t3) \to (iso t1 t3))))) .
+| iso_head: \forall (k: K).(\forall (v1: T).(\forall (v2: T).(\forall (t1:
+T).(\forall (t2: T).(iso (THead k v1 t1) (THead k v2 t2)))))).
+
+theorem iso_flats_lref_bind_false:
+ \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall
+(t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind
+b) v t)) \to (\forall (P: Prop).P)))))))
+\def
+ \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda
+(t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads
+(Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))
+(\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0
+\def (match H return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (iso t0
+t1)).((eq T t0 (TLRef i)) \to ((eq T t1 (THead (Bind b) v t)) \to P))))) with
+[(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef
+i))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2 \def
+(eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow False])) I (TLRef i) H0) in (False_ind ((eq T (TSort n2)
+(THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H0: (eq T (TLRef i1) (TLRef i))).(\lambda (H1: (eq T (TLRef i2)
+(THead (Bind b) v t))).((let H2 \def (f_equal T nat (\lambda (e: T).(match e
+return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i1 | (TLRef n)
+\Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef i) H0) in
+(eq_ind nat i (\lambda (_: nat).((eq T (TLRef i2) (THead (Bind b) v t)) \to
+P)) (\lambda (H3: (eq T (TLRef i2) (THead (Bind b) v t))).(let H4 \def
+(eq_ind T (TLRef i2) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Bind b) v t) H3) in (False_ind P H4))) i1
+(sym_eq nat i1 i H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda
+(H0: (eq T (THead k v1 t1) (TLRef i))).(\lambda (H1: (eq T (THead k v2 t2)
+(THead (Bind b) v t))).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i)
+H0) in (False_ind ((eq T (THead k v2 t2) (THead (Bind b) v t)) \to P) H2))
+H1)))]) in (H0 (refl_equal T (TLRef i)) (refl_equal T (THead (Bind b) v
+t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads
+(Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P:
+Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef
+i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 return
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (iso t2 t3)).((eq T t2 (THead
+(Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) \to ((eq T t3 (THead (Bind b) v
+t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort
+n1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T
+(TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True
+| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead
+(Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T (TSort
+n2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H1: (eq T (TLRef i1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef
+i))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def
+(eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))
+H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) |
+(iso_head k v1 v2 t2 t3) \Rightarrow (\lambda (H1: (eq T (THead k v1 t2)
+(THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T
+(THead k v2 t3) (THead (Bind b) v t))).((let H3 \def (f_equal T T (\lambda
+(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t2 |
+(TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k v1 t2)
+(THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H4 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t]))
+(THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in
+((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1
+(TLRef i))) H1) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T v1 t0) \to ((eq
+T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead k0 v2 t3) (THead (Bind
+b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_:
+T).((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead (Flat f) v2
+t3) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t2 (THeads (Flat f) t1
+(TLRef i)))).(eq_ind T (THeads (Flat f) t1 (TLRef i)) (\lambda (_: T).((eq T
+(THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T
+(THead (Flat f) v2 t3) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead
+(Flat f) v2 t3) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in
+(False_ind P H9))) t2 (sym_eq T t2 (THeads (Flat f) t1 (TLRef i)) H7))) v1
+(sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f) H5))) H4)) H3)) H2)))]) in (H1
+(refl_equal T (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) (refl_equal
+T (THead (Bind b) v t))))))))) vs)))))).
+
+theorem iso_flats_flat_bind_false:
+ \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall
+(v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads
+(Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P:
+Prop).P)))))))))
+\def
+ \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda
+(v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind
+(\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2))
+(THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead
+(Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0 \def
+(match H return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (iso t0
+t1)).((eq T t0 (THead (Flat f2) v2 t2)) \to ((eq T t1 (THead (Bind b) v t))
+\to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
+(THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v
+t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2
+t2) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2))
+H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead
+(Flat f2) v2 t2))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v
+t))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2
+t2) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H2))
+H1))) | (iso_head k v1 v0 t1 t0) \Rightarrow (\lambda (H0: (eq T (THead k v1
+t1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (THead k v0 t0) (THead
+(Bind b) v t))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1
+| (THead _ _ t) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0)
+in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t
+_) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in ((let H4
+\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in (eq_ind K
+(Flat f2) (\lambda (k0: K).((eq T v1 v2) \to ((eq T t1 t2) \to ((eq T (THead
+k0 v0 t0) (THead (Bind b) v t)) \to P)))) (\lambda (H5: (eq T v1 v2)).(eq_ind
+T v2 (\lambda (_: T).((eq T t1 t2) \to ((eq T (THead (Flat f2) v0 t0) (THead
+(Bind b) v t)) \to P))) (\lambda (H6: (eq T t1 t2)).(eq_ind T t2 (\lambda (_:
+T).((eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P)) (\lambda (H7:
+(eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t))).(let H8 \def (eq_ind T
+(THead (Flat f2) v0 t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H7)
+in (False_ind P H8))) t1 (sym_eq T t1 t2 H6))) v1 (sym_eq T v1 v2 H5))) k
+(sym_eq K k (Flat f2) H4))) H3)) H2)) H1)))]) in (H0 (refl_equal T (THead
+(Flat f2) v2 t2)) (refl_equal T (THead (Bind b) v t)))))) (\lambda (t0:
+T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat
+f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0:
+(iso (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))
+(THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 return
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (iso t3 t4)).((eq T t3 (THead
+(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))) \to ((eq T t4
+(THead (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda
+(H1: (eq T (TSort n1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat
+f2) v2 t2))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3
+\def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1
+(THead (Flat f2) v2 t2))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b)
+v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T
+(TLRef i1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2
+t2))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def
+(eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead
+(Flat f2) v2 t2))) H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t))
+\to P) H3)) H2))) | (iso_head k v1 v0 t3 t4) \Rightarrow (\lambda (H1: (eq T
+(THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2
+t2))))).(\lambda (H2: (eq T (THead k v0 t4) (THead (Bind b) v t))).((let H3
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t)
+\Rightarrow t])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1
+(THead (Flat f2) v2 t2))) H1) in ((let H4 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t3) (THead
+(Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H5
+\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1
+(THead (Flat f2) v2 t2))) H1) in (eq_ind K (Flat f1) (\lambda (k0: K).((eq T
+v1 t0) \to ((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq
+T (THead k0 v0 t4) (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1
+t0)).(eq_ind T t0 (\lambda (_: T).((eq T t3 (THeads (Flat f1) t1 (THead (Flat
+f2) v2 t2))) \to ((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to
+P))) (\lambda (H7: (eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2
+t2)))).(eq_ind T (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (\lambda (_:
+T).((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P)) (\lambda (H8:
+(eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t))).(let H9 \def (eq_ind T
+(THead (Flat f1) v0 t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8)
+in (False_ind P H9))) t3 (sym_eq T t3 (THeads (Flat f1) t1 (THead (Flat f2)
+v2 t2)) H7))) v1 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f1) H5))) H4))
+H3)) H2)))]) in (H1 (refl_equal T (THead (Flat f1) t0 (THeads (Flat f1) t1
+(THead (Flat f2) v2 t2)))) (refl_equal T (THead (Bind b) v t)))))))))
+vs)))))))).
+
+theorem iso_trans:
+ \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
+t3) \to (iso t1 t3)))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
+(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TSort n2) t3)).(let H1 \def (match H0 return (\lambda (t: T).(\lambda (t0:
+T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to (iso
+(TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq T
+(TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def
+(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with
+[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_:
+nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T
+(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
+(iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2)
+\Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T
+(TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in
+(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) |
+(iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1)
+(TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T
+(THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2) t3)
+\to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2))
+(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
+T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 return (\lambda
+(t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef i2)) \to
+((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2) \Rightarrow
+(\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T (TSort n2)
+t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in
+(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) |
+(iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef
+i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat
+(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
+i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind
+T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0
+(sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda
+(H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2)
+t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in
+(False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))])
+in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
+K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
+(match H0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
+t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
+t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
+(THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T
+(TSort n1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2) t5) \to (iso
+(THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0:
+(eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef i2) t5)).((let
+H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T
+(TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_head k0 v0 v3
+t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0) (THead k v2
+t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def (f_equal T T (\lambda
+(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 |
+(TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) (THead k0 v0 t0)
+(THead k v2 t4) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _)
+\Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 v0 t0) (THead k v2
+t4) H0) in (eq_ind K k (\lambda (k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to
+((eq T (THead k1 v3 t4) t5) \to (iso (THead k v1 t3) t5))))) (\lambda (H5:
+(eq T v0 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k
+v3 t4) t5) \to (iso (THead k v1 t3) t5)))) (\lambda (H6: (eq T t0
+t4)).(eq_ind T t4 (\lambda (_: T).((eq T (THead k v3 t4) t5) \to (iso (THead
+k v1 t3) t5))) (\lambda (H7: (eq T (THead k v3 t4) t5)).(eq_ind T (THead k v3
+t4) (\lambda (t: T).(iso (THead k v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7))
+t0 (sym_eq T t0 t4 H6))) v0 (sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4)))
+H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k v2 t4)) (refl_equal T
+t5)))))))))) t1 t2 H))).
inductive C: Set \def
| CSort: nat \to C
| CHead: C \to (K \to (T \to C)).
-definition r: K \to (nat \to nat) \def \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i | (Flat _) \Rightarrow (S i)])).
-
-definition clen: C \to nat \def let rec clen (c: C): nat \def (match c with [(CSort _) \Rightarrow O | (CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen.
-
-axiom r_S: \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i)))) .
-
-axiom r_plus: \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) (plus (r k i) j)))) .
-
-axiom r_plus_sym: \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j)) (plus i (r k j))))) .
-
-axiom r_minus: \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat (minus (r k i) (S n)) (r k (minus i (S n))))))) .
-
-axiom r_dis: \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P))) .
-
-axiom s_r: \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i))) .
-
-axiom r_arith0: \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i))) .
-
-axiom r_arith1: \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S i)) (S j)) (minus (r k i) j)))) .
-
-definition tweight: T \to nat \def let rec tweight (t: T): nat \def (match t with [(TSort _) \Rightarrow (S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus (tweight u) (tweight t0)))]) in tweight.
-
-definition cweight: C \to nat \def let rec cweight (c: C): nat \def (match c with [(CSort _) \Rightarrow O | (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight.
-
-definition clt: C \to (C \to Prop) \def \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
-
-definition cle: C \to (C \to Prop) \def \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
-
-axiom tweight_lt: \forall (t: T).(lt O (tweight t)) .
-
-axiom clt_cong: \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: T).(clt (CHead c k t) (CHead d k t)))))) .
-
-axiom clt_head: \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u)))) .
-
-axiom clt_wf__q_ind: \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P: ((C \to Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P c))))) P n))) \to (\forall (c: C).(P c))) .
-
-axiom clt_wf_ind: \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c) \to (P d)))) \to (P c)))) \to (\forall (c: C).(P c))) .
-
-definition CTail: K \to (T \to (C \to C)) \def let rec CTail (k: K) (t: T) (c: C): C \def (match c with [(CSort n) \Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k t d) h u)]) in CTail.
-
-axiom chead_ctail: \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d)))))))) .
-
-axiom clt_thead: \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c)))) .
-
-axiom c_tail_ind: \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to (((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t c))))))) \to (\forall (c: C).(P c)))) .
-
-definition fweight: C \to (T \to nat) \def \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
-
-definition flt: C \to (T \to (C \to (T \to Prop))) \def \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt (fweight c1 t1) (fweight c2 t2))))).
-
-axiom flt_thead_sx: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c (THead k u t))))) .
-
-axiom flt_thead_dx: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c (THead k u t))))) .
-
-axiom flt_shift: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c k u) t c (THead k u t))))) .
-
-axiom flt_arith0: \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t (CHead c k t) (TLRef i))))) .
-
-axiom flt_arith1: \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle (CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i: nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i))))))))) .
-
-axiom flt_arith2: \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1 t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) .
-
-axiom flt_wf__q_ind: \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P: ((C \to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq nat (fweight c t) n0) \to (P c t)))))) P n))) \to (\forall (c: C).(\forall (t: T).(P c t)))) .
-
-axiom flt_wf_ind: \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2: T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1))))) \to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t)))) .
-
-definition lref_map: ((nat \to nat)) \to (nat \to (T \to T)) \def let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T): T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map.
-
-definition lift: nat \to (nat \to (T \to T)) \def \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x: nat).(plus x h)) i t))).
-
-definition lifts: nat \to (nat \to (TList \to TList)) \def let rec lifts (h: nat) (d: nat) (ts: TList): TList \def (match ts with [TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts h d ts0))]) in lifts.
-
-axiom lift_sort: \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort n)) (TSort n)))) .
-
-axiom lift_lref_lt: \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T (lift h d (TLRef n)) (TLRef n))))) .
-
-axiom lift_lref_ge: \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T (lift h d (TLRef n)) (TLRef (plus n h)))))) .
-
-axiom lift_head: \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) t))))))) .
-
-axiom lift_bind: \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) (lift h (S d) t))))))) .
-
-axiom lift_flat: \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) (lift h d t))))))) .
-
-axiom lift_gen_sort: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T (TSort n) (lift h d t)) \to (eq T t (TSort n)))))) .
-
-axiom lift_gen_lref: \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le (plus d h) i) (eq T t (TLRef (minus i h))))))))) .
-
-axiom lift_gen_lref_lt: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n))))))) .
-
-axiom lift_gen_lref_false: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n (plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall (P: Prop).P))))))) .
-
-axiom lift_gen_lref_ge: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall (t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n))))))) .
-
-axiom lift_gen_head: \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))))) .
-
-axiom lift_gen_bind: \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))))))) .
-
-axiom lift_gen_flat: \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))))))) .
-
-axiom thead_x_lift_y_y: \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P)))))) .
-
-axiom lift_r: \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) .
-
-axiom lift_lref_gt: \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef (pred n))) (TLRef n)))) .
-
-axiom lift_inj: \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d x) (lift h d t)) \to (eq T x t))))) .
-
-axiom lift_gen_lift: \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1 t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2))))))))))) .
-
-axiom lift_free: \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e (lift h d t)) (lift (plus k h) d t)))))))) .
-
-axiom lift_d: \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t)) (lift k e (lift h d t)))))))) .
-
-axiom lift_weight_map: \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat (weight_map f (lift h d t)) (weight_map f t)))))) .
-
-axiom lift_weight: \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d t)) (weight t)))) .
-
-axiom lift_weight_add: \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t)) (weight_map g (lift (S h) d t))))))))))) .
-
-axiom lift_weight_add_O: \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) O t)))))) .
-
-axiom lift_tlt_dx: \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).(tlt t (THead k u (lift h d t))))))) .
+definition r:
+ K \to (nat \to nat)
+\def
+ \lambda (k: K).(\lambda (i: nat).(match k with [(Bind _) \Rightarrow i |
+(Flat _) \Rightarrow (S i)])).
+
+definition clen:
+ C \to nat
+\def
+ let rec clen (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O |
+(CHead c0 k _) \Rightarrow (s k (clen c0))]) in clen.
+
+theorem r_S:
+ \forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
+i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
+f) i))))) k).
+
+theorem r_plus:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
+(plus (r k i) j))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
+(Flat f) i) j))))) k).
+
+theorem r_plus_sym:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
+(plus i (r k j)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
+
+theorem r_minus:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
+(minus (r k i) (S n)) (r k (minus i (S n)))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
+K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
+n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
+F).(minus_x_Sy i n H)) k)))).
+
+theorem r_dis:
+ \forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
+\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
+nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
+(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
+((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
+((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
+nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
+((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
+((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
+(i: nat).(refl_equal nat (S i)))))))) k).
+
+theorem s_r:
+ \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
+\def
+ \lambda (k: K).(match k return (\lambda (k0: K).(\forall (i: nat).(eq nat (s
+k0 (r k0 i)) (S i)))) with [(Bind _) \Rightarrow (\lambda (i:
+nat).(refl_equal nat (S i))) | (Flat _) \Rightarrow (\lambda (i:
+nat).(refl_equal nat (S i)))]).
+
+theorem r_arith0:
+ \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
+\def
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
+nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
+nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
+(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
+
+theorem r_arith1:
+ \forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
+i)) (S j)) (minus (r k i) j))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
+(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
+(minus (r k i) j)) (r k (S i)) (r_S k i)))).
+
+definition tweight:
+ T \to nat
+\def
+ let rec tweight (t: T) on t: nat \def (match t with [(TSort _) \Rightarrow
+(S O) | (TLRef _) \Rightarrow (S O) | (THead _ u t0) \Rightarrow (S (plus
+(tweight u) (tweight t0)))]) in tweight.
+
+definition cweight:
+ C \to nat
+\def
+ let rec cweight (c: C) on c: nat \def (match c with [(CSort _) \Rightarrow O
+| (CHead c0 _ t) \Rightarrow (plus (cweight c0) (tweight t))]) in cweight.
+
+definition clt:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(lt (cweight c1) (cweight c2))).
+
+definition cle:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(le (cweight c1) (cweight c2))).
+
+theorem tweight_lt:
+ \forall (t: T).(lt O (tweight t))
+\def
+ \lambda (t: T).(match t return (\lambda (t0: T).(lt O (tweight t0))) with
+[(TSort _) \Rightarrow (le_n (S O)) | (TLRef _) \Rightarrow (le_n (S O)) |
+(THead _ t0 t1) \Rightarrow (le_S_n (S O) (S (plus (tweight t0) (tweight
+t1))) (le_n_S (S O) (S (plus (tweight t0) (tweight t1))) (le_n_S O (plus
+(tweight t0) (tweight t1)) (le_O_n (plus (tweight t0) (tweight t1))))))]).
+
+theorem clt_cong:
+ \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
+T).(clt (CHead c k t) (CHead d k t))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
+d))).(\lambda (_: K).(\lambda (t: T).(lt_le_S (plus (cweight c) (tweight t))
+(plus (cweight d) (tweight t)) (plus_lt_compat_r (cweight c) (cweight d)
+(tweight t) H)))))).
+
+theorem clt_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
+c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) (lt_le_S (plus
+(cweight c) O) (plus (cweight c) (tweight u)) (plus_le_lt_compat (cweight c)
+(cweight c) O (tweight u) (le_n (cweight c)) (tweight_lt u))) (cweight c)
+(plus_n_O (cweight c))))).
+
+theorem clt_wf__q_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P: ((C \to
+Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P
+c))))) P n))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
+n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
+c)))))).
+
+theorem clt_wf_ind:
+ \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
+\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
+\def
+ let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
+C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
+Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
+(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
+(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
+C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (c: C).(P c)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
+(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n: nat).(\forall
+(m: nat).((lt m n) \to (\forall (c: C).((eq nat (cweight c) m) \to (P c))))))
+H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt (cweight d)
+(cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight d)))))))))))))
+c)))).
+
+definition CTail:
+ K \to (T \to (C \to C))
+\def
+ let rec CTail (k: K) (t: T) (c: C) on c: C \def (match c with [(CSort n)
+\Rightarrow (CHead (CSort n) k t) | (CHead d h u) \Rightarrow (CHead (CTail k
+t d) h u)]) in CTail.
+
+theorem chead_ctail:
+ \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
+K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
+(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
+K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
+(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
+(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
+H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
+(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
+c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
+C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
+c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
+(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
+k t) H1))))) H0))))))))) c).
+
+theorem clt_thead:
+ \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
+c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
+C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
+T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
+
+theorem c_tail_ind:
+ \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
+(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t
+c))))))) \to (\forall (c: C).(P c))))
+\def
+ \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
+n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
+T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
+c0)) (\lambda (c0: C).(match c0 return (\lambda (c1: C).(((\forall (d:
+C).((clt d c1) \to (P d)))) \to (P c1))) with [(CSort n) \Rightarrow (\lambda
+(_: ((\forall (d: C).((clt d (CSort n)) \to (P d))))).(H n)) | (CHead c1 k t)
+\Rightarrow (\lambda (H1: ((\forall (d: C).((clt d (CHead c1 k t)) \to (P
+d))))).(let H_x \def (chead_ctail c1 t k) in (let H2 \def H_x in (ex_3_ind K
+C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
+(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H3: (eq C (CHead c1 k t) (CTail x0 x2
+x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H4 \def
+(eq_ind C (CHead c1 k t) (\lambda (c: C).(\forall (d: C).((clt d c) \to (P
+d)))) H1 (CTail x0 x2 x1) H3) in (H0 x1 (H4 x1 (clt_thead x0 x2 x1)) x0 x2))
+(CHead c1 k t) H3))))) H2))))])) c)))).
+
+definition fweight:
+ C \to (T \to nat)
+\def
+ \lambda (c: C).(\lambda (t: T).(plus (cweight c) (tweight t))).
+
+definition flt:
+ C \to (T \to (C \to (T \to Prop)))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (c2: C).(\lambda (t2: T).(lt
+(fweight c1 t1) (fweight c2 t2))))).
+
+theorem flt_thead_sx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_le_S
+(plus (cweight c) (tweight u)) (plus (cweight c) (S (plus (tweight u)
+(tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight u) (S
+(plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (tweight u)
+(plus (tweight u) (tweight t)) (le_plus_l (tweight u) (tweight t)))))))).
+
+theorem flt_thead_dx:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c
+(THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_le_S
+(plus (cweight c) (tweight t)) (plus (cweight c) (S (plus (tweight u)
+(tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight t) (S
+(plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (tweight t)
+(plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t)))))))).
+
+theorem flt_shift:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
+k u) t c (THead k u t)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
+(S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
+(plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
+(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
+(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
+c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
+t))) (plus_assoc (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
+(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
+(tweight t))))))).
+
+theorem flt_arith0:
+ \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
+(CHead c k t) (TLRef i)))))
+\def
+ \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: nat).(le_S_n (S
+(plus (cweight c) (tweight t))) (plus (plus (cweight c) (tweight t)) (S O))
+(lt_le_S (S (plus (cweight c) (tweight t))) (S (plus (plus (cweight c)
+(tweight t)) (S O))) (lt_n_S (plus (cweight c) (tweight t)) (plus (plus
+(cweight c) (tweight t)) (S O)) (lt_x_plus_x_Sy (plus (cweight c) (tweight
+t)) O))))))).
+
+theorem flt_arith1:
+ \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
+(CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i:
+nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
+\def
+ \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
+(H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
+K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
+(tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
+(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
+nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
+(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
+(tweight t2)) (S O)) (plus_comm (plus (cweight c2) (tweight t2)) (S
+O))))))))))).
+
+theorem flt_arith2:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
+t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt
+c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
+(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
+(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
+(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
+t2)) (S O)) H (le_S_n (plus (cweight c2) (S O)) (plus (plus (cweight c2)
+(tweight t2)) (S O)) (lt_le_S (plus (cweight c2) (S O)) (S (plus (plus
+(cweight c2) (tweight t2)) (S O))) (le_lt_n_Sm (plus (cweight c2) (S O))
+(plus (plus (cweight c2) (tweight t2)) (S O)) (plus_le_compat (cweight c2)
+(plus (cweight c2) (tweight t2)) (S O) (S O) (le_plus_l (cweight c2) (tweight
+t2)) (le_n (S O)))))))))))))).
+
+theorem flt_wf__q_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P: ((C
+\to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
+nat (fweight c t) n0) \to (P c t)))))) P n))) \to (\forall (c: C).(\forall
+(t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
+C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
+C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
+
+theorem flt_wf_ind:
+ \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
+\to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
+\def
+ let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
+(c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
+(P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
+T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
+\to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda
+(n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m:
+nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda
+(H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n:
+nat).(\forall (m: nat).((lt m n) \to (\forall (c: C).(\forall (t: T).((eq nat
+(fweight c t) m) \to (P c t))))))) H0 (fweight c0 t0) H1) in (H c0 t0
+(\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2
+(fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c
+t))))).
+
+definition lref_map:
+ ((nat \to nat)) \to (nat \to (T \to T))
+\def
+ let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t
+with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
+(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u
+t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in
+lref_map.
+
+definition lift:
+ nat \to (nat \to (T \to T))
+\def
+ \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x:
+nat).(plus x h)) i t))).
+
+definition lifts:
+ nat \to (nat \to (TList \to TList))
+\def
+ let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts
+h d ts0))]) in lifts.
+
+theorem lift_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
+n)) (TSort n))))
+\def
+ \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort
+n)))).
+
+theorem lift_lref_lt:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T
+(lift h d (TLRef n)) (TLRef n)))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n
+d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true
+\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T
+(TLRef n)) (blt n d) (sym_equal bool (blt n d) true (lt_blt d n H)))))).
+
+theorem lift_lref_ge:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T
+(lift h d (TLRef n)) (TLRef (plus n h))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d
+n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true
+\Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h))))
+(refl_equal T (TLRef (plus n h))) (blt n d) (sym_equal bool (blt n d) false
+(le_bge d n H)))))).
+
+theorem lift_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d)
+t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
+
+theorem lift_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u)
+(lift h (S d) t)))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
+
+theorem lift_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u)
+(lift h d t)))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
+
+theorem lift_gen_sort:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T
+(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind
+(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))
+(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort
+n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H:
+(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort
+n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0))
+(\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in
+(let H2 \def (match H1 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T
+t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal \Rightarrow
+(\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T (TSort n)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))])
+in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d n0)).(let H1 \def
+(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TSort n) t)) H (TLRef
+(plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def (match H1 return
+(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 h))) \to
+(eq T (TLRef n0) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq
+T (TSort n) (TLRef (plus n0 h)))).(let H2 \def (eq_ind T (TSort n) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True
+| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
+(plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))]) in (H2
+(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort
+n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq
+T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0
+t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq
+T (TSort n) t)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k
+t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t: T).(\lambda (_: (eq
+? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T
+(THead k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H2: (eq
+T (TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def
+(eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H2) in
+(False_ind (eq T (THead k t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T
+(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))).
+
+theorem lift_gen_lref:
+ \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T
+(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le
+(plus d h) i) (eq T t (TLRef (minus i h)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(\forall (h:
+nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
+(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i
+h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda
+(i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def
+(eq_ind T (lift h d (TSort n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TSort
+n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n)
+H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i))) (land (le
+(plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1)))))))) (\lambda (n:
+nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H: (eq T
+(TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or (land (lt i d) (eq T (TLRef
+n) (TLRef i))) (land (le (plus d h) i) (eq T (TLRef n) (TLRef (minus i h)))))
+(\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda
+(t: T).(eq T (TLRef i) t)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2
+\def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _)
+\Rightarrow i])) (TLRef i) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0:
+nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le (plus d h)
+n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (or_introl (land (lt n d) (eq T
+(TLRef n) (TLRef n))) (land (le (plus d h) n) (eq T (TLRef n) (TLRef (minus n
+h)))) (conj (lt n d) (eq T (TLRef n) (TLRef n)) H0 (refl_equal T (TLRef n))))
+i H2)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n))
+(\lambda (t: T).(eq T (TLRef i) t)) H (TLRef (plus n h)) (lift_lref_ge n h d
+H0)) in (let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda
+(_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead
+_ _ _) \Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat
+(plus n h) (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef
+n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h))))))
+(eq_ind_r nat n (\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef
+n) (TLRef (plus n h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n)
+(TLRef n0))))) (or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef
+(plus n h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n)))
+(conj (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n)) (plus_le_compat d
+n h h H0 (le_n h)) (refl_equal T (TLRef n)))) (minus (plus n h) h)
+(minus_plus_r n h)) i H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
+(_: ((\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i)
+(lift h d t0)) \to (or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d
+h) i) (eq T t0 (TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_:
+((\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift
+h d t1)) \to (or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i)
+(eq T t1 (TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (i: nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0
+t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq
+T (TLRef i) t)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k
+t0 t1 h d)) in (let H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d
+t0) (lift h (s k d) t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k
+t0 t1) (TLRef i))) (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef
+(minus i h))))) H3)))))))))))) t).
+
+theorem lift_gen_lref_lt:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall
+(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n
+d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0))
+\to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n)
+(lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0:
+nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq
+T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift
+h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0)
+(lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda
+(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t:
+T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in
+(let H3 \def (match H2 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T
+t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal
+\Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 h)))).(let H3 \def
+(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _)
+\Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n0 h)
+(\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def (eq_ind nat n
+(\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 (eq T (TLRef
+n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus n0 h) d
+(le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3
+(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef
+n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq
+T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0
+t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq
+T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k
+t0 t1 h d)) in (let H4 \def (match H3 return (\lambda (t: T).(\lambda (_: (eq
+? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T
+(THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq
+T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def
+(eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in
+(False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T
+(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
+
+theorem lift_gen_lref_false:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
+(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall
+(P: Prop).P)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
+n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0:
+T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0:
+nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P:
+Prop).(let H2 \def (match H1 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow
+(\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T
+(TLRef n) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (lift h d (TSort n0)) H2) in (False_ind P H3)))]) in (H2
+(refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda (H1:
+(eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d P
+(\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0))
+(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2))
+in (let H4 \def (match H3 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3:
+(eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e:
+T).(match e return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0)
+H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0
+(\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat
+n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d
+n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T
+(TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4
+\def (match H3 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
+(TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq
+T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e:
+T).(match e return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
+(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
+(plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def
+(eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in
+(le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq
+nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h)))))))))))
+(\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d
+t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T
+(TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T
+(TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3
+(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let
+H5 \def (match H4 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
+(THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal
+\Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k
+d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d
+t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T
+(THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
+
+theorem lift_gen_lref_ge:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
+(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
+n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h
+d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T
+(TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 return
+(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort n0))) \to
+(eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq
+T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def (eq_ind T (TLRef
+(plus n h)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (lift h d (TSort n0)) H1) in (False_ind (eq T (TSort
+n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort n0)))))))
+(\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TLRef
+n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0
+d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef
+(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def
+(match H2 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef
+n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda
+(H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T nat
+(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
+nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
+plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec
+plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O
+\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef
+(plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T
+(TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n
+d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H
+(lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in
+(H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def
+(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t))
+H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 h)))
+\to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H2:
+(eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T nat
+(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m:
+nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in
+plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec
+plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O
+\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef
+(plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_:
+nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h
+n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3
+(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef
+n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d
+t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift
+h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1))
+(\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift
+h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 return
+(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift h d t0)
+(lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift
+h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h))
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq
+T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d
+t0) (lift h (s k d) t1)))))))))))) t))))).
+
+theorem lift_gen_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z)))))))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
+(lift h d (TSort n)))).(let H0 \def (match H return (\lambda (t0: T).(\lambda
+(_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k
+u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead k u t) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d
+(TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H1)))])
+in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef
+n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
+(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0:
+(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
+(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def
+(match H1 return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef
+n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal
+\Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def
+(eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0:
+(le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T
+(THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2
+\def (match H1 return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0
+(TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with
+[refl_equal \Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef (plus n
+h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h))
+H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
+(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2
+(refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k0: K).(\lambda (t0:
+T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1:
+T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0
+t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t0:
+T).(eq T (THead k u t) t0)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
+(lift_head k0 t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t2:
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k0 (lift h d t0) (lift h (s k0
+d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0
+t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with
+[refl_equal \Rightarrow (\lambda (H2: (eq T (THead k u t) (THead k0 (lift h d
+t0) (lift h (s k0 d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match
+e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
+\Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k u t) (THead k0 (lift
+h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k u t) (THead k0
+(lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K
+k0 (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1))
+\to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T
+u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h
+(s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0
+t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z)))))))
+(\lambda (H7: (eq T t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1)
+(\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0
+t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0)
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d)
+z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0
+t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0)
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1)
+(lift h (s k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T
+(lift h d t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s
+k0 d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4))
+H3)))]) in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d)
+t1)))))))))))))) x)))).
+
+theorem lift_gen_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (S d) z)))))))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u
+t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
+(THead (Bind b) u t) (lift h d (TSort n)))).(let H0 \def (match H return
+(\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n)))
+\to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Bind
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal
+\Rightarrow (\lambda (H0: (eq T (THead (Bind b) u t) (lift h d (TSort
+n)))).(let H1 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n))
+H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n)
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H1)))]) in
+(H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d
+(TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))
+(\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda
+(t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef n) (lift_lref_lt n h d H0))
+in (let H2 \def (match H1 return (\lambda (t0: T).(\lambda (_: (eq ? ?
+t0)).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq
+T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d)
+z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Bind b) u
+t) (TLRef n))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
+H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in
+(H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind
+T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H
+(TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return
+(\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef (plus n h))) \to
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal
+\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n
+h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h))
+H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n)
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in
+(H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u
+t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d)
+z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq T (THead (Bind b) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T
+(THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T
+(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0))
+H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in
+(let H3 \def (match H2 return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq
+T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (S d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2:
+(eq T (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let
+H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d)
+t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t _) \Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0)
+(lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match
+e return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _)
+\Rightarrow (Bind b) | (THead k _ _) \Rightarrow k])) (THead (Bind b) u t)
+(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Bind b)
+(\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) (\lambda (H6: (eq T u
+(lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s
+(Bind b) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d)
+z))))))) (\lambda (H7: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h
+(s (Bind b) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z:
+T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (ex3_2_intro T T (\lambda
+(y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) t1) (lift h (S d) z))))
+t0 t1 (refl_equal T (THead (Bind b) t0 t1)) (refl_equal T (lift h d t0))
+(refl_equal T (lift h (S d) t1))) t (sym_eq T t (lift h (s (Bind b) d) t1)
+H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) H4)) H3)))]) in (H3
+(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))).
+
+theorem lift_gen_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u
+t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))))) (\lambda
+(n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat
+f) u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (t0:
+T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow
+(\lambda (H0: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H1 \def
+(eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z))))) H1)))]) in (H0 (refl_equal T
+(lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TLRef n)))).(lt_le_e
+n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat
+f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h d z))))) (\lambda (H0: (lt n d)).(let
+H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f)
+u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1
+return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow
+(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T
+(THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T (TLRef n))))))
+(\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda
+(t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n
+h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).(\lambda (_: (eq ?
+? t0)).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda
+(z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d
+z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Flat f) u
+t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead (Flat f) u t) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z:
+T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d
+z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T
+(THead (Flat f) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda
+(z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d
+z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq T (THead (Flat f) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h d z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T
+(THead (Flat f) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T
+(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0))
+H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in
+(let H3 \def (match H2 return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq
+T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq
+T (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t)
+\Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d)
+t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t _) \Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0)
+(lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match
+e return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat f) | (TLRef _)
+\Rightarrow (Flat f) | (THead k _ _) \Rightarrow k])) (THead (Flat f) u t)
+(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Flat f)
+(\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat
+f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u
+(lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s
+(Flat f) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))
+(\lambda (H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat
+f) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
+t (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T
+(THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T
+(lift h (s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat
+f) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t
+(sym_eq T t (lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0)
+H6))) k H5)) H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h
+(s k d) t1)))))))))))))) x)))).
+
+theorem thead_x_lift_y_y:
+ \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
+(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
+\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
+(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
+(TSort n))) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
+nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
+(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
+(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
+h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
+h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
+t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t]))
+(THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H4 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t:
+T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
+(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0)
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t
+with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
+(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u
+t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in
+lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow
+(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t) \Rightarrow t])) (THead k
+v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in (\lambda (_: (eq T v
+t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k (\lambda (k:
+K).(\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v
+(lift h d t1)) t1) \to (\forall (P: Prop).P)))))) H0 k0 H6) in (let H8 \def
+(eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t: T).(eq T t t1)) H4 (THead
+k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift
+h d t0) h (s k0 d) H8 P)))))) H3)) H2)))))))))))) t)).
+
+theorem lift_r:
+ \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
+t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
+(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
+(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
+n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
+(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
+t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
+(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
+t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_equal T (THead k t0 t1)
+(THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d
+t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead
+k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d
+t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d))
+(sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1))
+(lift_head k t0 t1 O d)))))))) t).
+
+theorem lift_lref_gt:
+ \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
+(pred n))) (TLRef n))))
+\def
+ \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
+(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
+(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
+(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
+(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_comm (S O) (pred n)))
+(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
+(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
+(S_pred n d H))))))).
+
+theorem lift_inj:
+ \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
+(lift h d x) (lift h d t)) \to (eq T x t)))))
+\def
+ \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
+t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
+(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
+H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
+T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
+(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
+n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
+d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
+h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
+(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
+t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
+T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
+(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to
+(eq T t0 t)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
+nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
+t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: T).(\forall
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0
+t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
+(lift h d (THead (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h d t1))) H1
+(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
+(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
+h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
+T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
+(sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead
+(Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1)
+(THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0
+(refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1
+h (S d) H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0)
+t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall
+(t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d
+t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to
+(eq T t0 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d
+t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) (\lambda (t:
+T).(eq T t (lift h d t1))) H1 (THead (Flat f) (lift h d t) (lift h d t0))
+(lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq
+T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d
+t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift
+h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift
+h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d
+x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat
+f) t t0) t2)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0)
+(sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead
+(Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat
+f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T
+t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d
+t0) t1 h d H2)))))))))))) k)) x).
+
+theorem lift_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
+((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
+t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
+d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
+x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
+(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
+(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
+(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
+d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
+(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
+(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
+(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
+(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
+h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
+d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
+(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
+(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
+(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
+n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
+(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
+(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
+(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
+(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
+(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
+T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
+(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
+n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
+t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
+h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
+(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
+(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (plus_lt_compat_r n d2 h1 H3) x
+H2))) (\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2
+t2)))) (\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2
+h1) (plus n h1) (le_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S
+(plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat d2 n h1 h1
+H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt
+(plus n h1) n0)) (lt_le_S (plus n h1) (plus (plus d2 h2) h1)
+(plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2)
+(plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1
+d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4:
+(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n:
+nat).(eq T (TLRef n) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1)
+h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1
+(le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2))
+(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda
+(t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq
+T (TLRef (minus (plus n h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(TLRef n) (lift h2 d2 t2))) (TLRef (minus n h2)) (eq_ind_r nat (plus (minus n
+h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n
+h2))))) (eq_ind_r T (TLRef (plus (minus n h2) h1)) (\lambda (t: T).(eq T
+(TLRef (plus (minus n h2) h1)) t)) (refl_equal T (TLRef (plus (minus n h2)
+h1))) (lift h1 d1 (TLRef (minus n h2))) (lift_lref_ge (minus n h2) h1 d1
+(le_trans d1 d2 (minus n h2) H (le_minus d2 n h2 H4)))) (minus (plus n h1)
+h2) (le_minus_plus h2 n (le_trans_plus_r d2 h2 n H4) h1)) (eq_ind_r nat (plus
+(minus n h2) h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef
+(minus n0 h2))))) (eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2)
+h2)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T
+TLRef (plus (minus n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2)
+(f_equal2 nat nat nat plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2
+h2 (sym_eq nat (minus (plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r
+(minus n h2) h2)) (refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus
+(minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2
+(le_minus d2 (plus (minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2
+(le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n
+(le_trans_plus_r d2 h2 n H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus
+(plus n h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
+(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
+h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
+(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
+T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
+nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
+t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
+(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
+x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
+t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
+t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
+(Bind b) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H3 (THead
+(Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2
+h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
+(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
+b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
+(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
+H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
+t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
+T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
+t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
+h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
+x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
+d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
+(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
+h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
+(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
+T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
+t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
+h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
+d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
+(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
+t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
+(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
+H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S
+(S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm d1 d2 H1)))) H11)))) t H9) x0 H8)))) (H
+x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S
+d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift
+h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind
+T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus
+d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t
+t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead
+(Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift
+h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0)
+(lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0
+x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2))))
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0)
+(lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T
+(THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3:
+T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2
+d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T
+(lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat
+f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T
+(THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))
+(lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1
+d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1
+x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift
+h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1))
+(eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2:
+T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T
+(THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f)
+x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1
+H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f
+(lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2)))))))))))))
+t1).
+
+theorem lift_free:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
+(lift h d t)) (lift (plus k h) d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
+nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
+(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
+n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
+(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
+(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
+h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
+(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
+h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
+n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
+(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
+n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
+(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
+k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
+H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
+n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
+(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
+(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
+(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
+nat T TLRef (plus (plus n h) k) (plus n (plus k h))
+(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
+(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
+(plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1
+(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0)))))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
+(lift h d t1)) (lift (plus k h) d t1)))))))))).(\lambda (h: nat).(\lambda
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
+h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
+d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
+t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
+h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
+t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
+t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
+e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
+d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
+(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
+(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
+k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
+(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
+d))))))))))))) t).
+
+theorem lift_d:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
+(lift k e (lift h d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
+d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
+(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
+(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
+(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
+T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
+T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
+T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
+e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
+(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
+nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
+(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
+(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
+n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
+(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
+e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
+h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
+(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
+d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
+n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
+t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
+nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
+(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
+(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
+(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
+(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
+(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
+(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
+(lift_lref_lt (plus n k) h (plus d k) (lt_le_S (plus n k) (plus d k)
+(plus_lt_compat_r n d k H1))))) (\lambda (H1: (le d n)).(eq_ind_r T (TLRef
+(plus (plus n k) h)) (\lambda (t0: T).(eq T t0 (lift k e (lift h d (TLRef
+n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus
+(plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef (plus (plus n h) k))
+(\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) (f_equal nat T TLRef
+(plus (plus n k) h) (plus (plus n h) k) (sym_eq nat (plus (plus n h) k) (plus
+(plus n k) h) (plus_permute_2_in_3 n h k))) (lift k e (TLRef (plus n h)))
+(lift_lref_ge (plus n h) k e (le_S_n e (plus n h) (lt_le_S e (S (plus n h))
+(le_lt_n_Sm e (plus n h) (le_plus_trans e n h H0)))))) (lift h d (TLRef n))
+(lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge
+(plus n k) h (plus d k) (le_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S
+(plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1
+(le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n))
+(lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
+(H: ((\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e:
+nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t0)) (lift k e (lift h
+d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
+d) (lift k e t1)) (lift k e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e
+d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
+T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
+(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
+k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
+(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
+(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
+(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
+k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
+(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
+k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
+(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
+k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
+(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
+t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
+t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
+(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0
+d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
+(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
+(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
+(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
+d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
+
+theorem lift_weight_map:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t)) (weight_map f t))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
+(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0)))))))
+(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f
+(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat
+(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0)))
+(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq
+nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda
+(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_S_n d (plus n h)
+(le_n_S d (plus n h) (le_plus_trans d n h H0)))) (f n) (H n H0)) (lift h d
+(TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t0)) (weight_map f t0)))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t1)) (weight_map f t1)))))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (f: ((nat \to nat))).(\lambda (H1: ((\forall
+(m: nat).((le d m) \to (eq nat (f m) O))))).(K_ind (\lambda (k0: K).(eq nat
+(weight_map f (lift h d (THead k0 t0 t1))) (weight_map f (THead k0 t0 t1))))
+(\lambda (b: B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b)
+d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f (THead (Bind
+b) t0 t1)))) (B_ind (\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow
+(S (plus (weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f
+(lift h d t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map
+f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void
+\Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O)
+(lift h (S d) t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map
+f t0) (weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S
+(plus (weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S
+(plus (weight_map f t0) (weight_map (wadd f O) t1)))]))) (eq_ind_r nat
+(weight_map f t0) (\lambda (n: nat).(eq nat (S (plus n (weight_map (wadd f (S
+n)) (lift h (S d) t1)))) (S (plus (weight_map f t0) (weight_map (wadd f (S
+(weight_map f t0))) t1))))) (eq_ind_r nat (weight_map (wadd f (S (weight_map
+f t0))) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))))
+(refl_equal nat (S (plus (weight_map f t0) (weight_map (wadd f (S (weight_map
+f t0))) t1)))) (weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1))
+(H0 h (S d) (wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2:
+(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n:
+nat).(le d n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x:
+nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m
+H3)))) (le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1))
+(eq_ind_r nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus
+(weight_map f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd
+f O) t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0))
+(weight_map (wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O)
+t1)) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f
+t0) (weight_map (wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1)
+(refl_equal nat (weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h
+(S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (wadd f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S
+x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat
+(wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat
+(weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map f
+(lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) t1)))))
+(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O)
+t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat
+nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map (wadd f
+O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat (weight_map
+(wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd
+f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n:
+nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O)
+(\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d
+x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4)
+m H3)))) (le_gen_S d m H2)))))) b) (lift h d (THead (Bind b) t0 t1))
+(lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat
+f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat
+(weight_map f t2) (weight_map f (THead (Flat f0) t0 t1)))) (f_equal nat nat S
+(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus
+(weight_map f t0) (weight_map f t1)) (f_equal2 nat nat nat plus (weight_map f
+(lift h d t0)) (weight_map f t0) (weight_map f (lift h d t1)) (weight_map f
+t1) (H h d f H1) (H0 h d f H1))) (lift h d (THead (Flat f0) t0 t1))
+(lift_head (Flat f0) t0 t1 h d))) k)))))))))) t).
+
+theorem lift_weight:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d
+t)) (weight t))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d
+(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat
+O)))))).
+
+theorem lift_weight_add:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
+(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
+(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h:
+nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat
+(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))
+\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
+t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m:
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d)
+w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n))))))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m
+d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
+((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d
+(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d
+(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n))
+(weight_map g t0))) (sym_equal nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef
+n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d
+H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
+(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f
+(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda
+(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_equal nat (g (S (plus n h)))
+(f (plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h))
+(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift
+h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda
+(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to
+(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d
+m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0))
+(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0:
+((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
+m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g
+(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift
+(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m:
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d)
+w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
+t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b:
+B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1))
+(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead
+(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h)
+(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b)
+(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind
+(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus
+(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d
+t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h
+d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S
+(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
+t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h)
+d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h)
+(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0))
+(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
+t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
+(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift
+(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map
+f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S
+(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S
+(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2
+H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S
+(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d
+t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m
+O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift
+(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat
+nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0))
+(sym_equal nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0))
+(H h d f g H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq
+nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda (m0:
+nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S
+(weight_map g (lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0)))
+m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x
+d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g
+(lift (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x
+H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4:
+(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n:
+nat).(le d n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m))
+(\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d
+x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g n) (wadd f (S
+(weight_map f (lift h d t0))) n))) (H3 x H6) m H5)))) (le_gen_S d m H4)))))))
+(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O)
+(lift h (S d) t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd
+g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h
+d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d)
+t1)) (weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h
+(S d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda
+(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n)
+(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0:
+nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda
+(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O
+m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda
+(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n)
+(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m:
+nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
+n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x:
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5))))
+(le_gen_S d m H4))))))) (f_equal nat nat S (plus (weight_map f (lift h d t0))
+(weight_map (wadd f O) (lift h (S d) t1))) (plus (weight_map g (lift (S h) d
+t0)) (weight_map (wadd g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat
+plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0))
+(weight_map (wadd f O) (lift h (S d) t1)) (weight_map (wadd g O) (lift (S h)
+(S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) (wadd g O) (\lambda
+(m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) (ex2 nat (\lambda
+(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O
+m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n:
+nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m H5)) (\lambda
+(H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m
+d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0:
+nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: nat).(\lambda
+(H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda
+(n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) H5))
+(lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S
+x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g
+n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) b) (lift (S h) d
+(THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 (S h) d)) (lift h d (THead
+(Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r
+T (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2:
+T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead (Flat f0) t0
+t1))))) (eq_ind_r T (THead (Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat
+f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Flat f0) (lift h d
+t0) (lift h (s (Flat f0) d) t1))) (weight_map g t2))) (f_equal nat nat S
+(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus
+(weight_map g (lift (S h) d t0)) (weight_map g (lift (S h) d t1))) (f_equal2
+nat nat nat plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
+t0)) (weight_map f (lift h d t1)) (weight_map g (lift (S h) d t1)) (H h d f g
+H1 H2 H3) (H0 h d f g H1 H2 H3))) (lift (S h) d (THead (Flat f0) t0 t1))
+(lift_head (Flat f0) t0 t1 (S h) d)) (lift h d (THead (Flat f0) t0 t1))
+(lift_head (Flat f0) t0 t1 h d))) k))))))))))))) t)).
+
+theorem lift_weight_add_O:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to
+nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h)
+O t))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to
+nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m:
+nat).(\lambda (H: (lt m O)).(let H0 \def (match H return (\lambda (n:
+nat).(\lambda (_: (le ? n)).((eq nat n O) \to (eq nat (wadd f w m) (f m)))))
+with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) O)).(let H1 \def (eq_ind
+nat (S m) (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
+(wadd f w m) (f m)) H1))) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S
+m0) O)).((let H2 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H1) in (False_ind ((le (S m) m0) \to (eq nat (wadd f w m) (f m))) H2))
+H0))]) in (H0 (refl_equal nat O))))) (plus_n_O (wadd f w O)) (\lambda (m:
+nat).(\lambda (_: (le O m)).(refl_equal nat (f m)))))))).
+
+theorem lift_tlt_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(tlt t (THead k u (lift h d t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight
+(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t)
+(lift_weight t h d)))))).
inductive PList: Set \def
| PNil: PList
| PCons: nat \to (nat \to (PList \to PList)).
-definition PConsTail: PList \to (nat \to (nat \to PList)) \def let rec PConsTail (hds: PList): (nat \to (nat \to PList)) \def (\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow (PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 h0 d0))]))) in PConsTail.
-
-definition trans: PList \to (nat \to nat) \def let rec trans (hds: PList): (nat \to nat) \def (\lambda (i: nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false \Rightarrow (plus j h)]))])) in trans.
-
-definition Ss: PList \to PList \def let rec Ss (hds: PList): PList \def (match hds with [PNil \Rightarrow PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss.
-
-definition lift1: PList \to (T \to T) \def let rec lift1 (hds: PList): (T \to T) \def (\lambda (t: T).(match hds with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0 t))])) in lift1.
-
-definition lifts1: PList \to (TList \to TList) \def let rec lifts1 (hds: PList) (ts: TList): TList \def (match ts with [TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t) (lifts1 hds ts0))]) in lifts1.
-
-axiom lift1_lref: \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef (trans hds i)))) .
-
-axiom lift1_bind: \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T (lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss hds) t)))))) .
-
-axiom lift1_flat: \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T (lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds t)))))) .
-
-axiom lift1_cons_tail: \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t)))))) .
-
-axiom lifts1_flat: \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts: TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds ts) (lift1 hds t)))))) .
-
-axiom lifts1_nil: \forall (ts: TList).(eq TList (lifts1 PNil ts) ts) .
-
-axiom lifts1_cons: \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts)))))) .
-
-axiom lift1_xhg: \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t)) (lift (S O) O (lift1 hds t)))) .
-
-axiom lifts1_xhg: \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts (S O) O ts)) (lifts (S O) O (lifts1 hds ts)))) .
+definition PConsTail:
+ PList \to (nat \to (nat \to PList))
+\def
+ let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def
+(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow
+(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0
+h0 d0))]))) in PConsTail.
+
+definition trans:
+ PList \to (nat \to nat)
+\def
+ let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i:
+nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let
+j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false
+\Rightarrow (plus j h)]))])) in trans.
+
+definition Ss:
+ PList \to PList
+\def
+ let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow
+PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss.
+
+definition lift1:
+ PList \to (T \to T)
+\def
+ let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds
+with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0
+t))])) in lift1.
+
+definition lifts1:
+ PList \to (TList \to TList)
+\def
+ let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t)
+(lifts1 hds ts0))]) in lifts1.
+
+theorem lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef
+(trans hds i))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T
+(TLRef (trans PNil i)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef
+(trans p i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda
+(t: T).(eq T (lift h d t) (TLRef (match (blt (trans p i) d) with [true
+\Rightarrow (trans p i) | false \Rightarrow (plus (trans p i) h)]))))
+(refl_equal T (TLRef (match (blt (trans p i) d) with [true \Rightarrow (trans
+p i) | false \Rightarrow (plus (trans p i) h)]))) (lift1 p (TLRef i)) (H
+i))))))) hds).
+
+theorem lift1_bind:
+ \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss
+hds) t))))))
+\def
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal
+T (THead (Bind b) (lift1 PNil u) (lift1 (Ss PNil) t))))) (\lambda (h:
+nat).(\lambda (d: nat).(\lambda (p: PList).(\lambda (H: ((\forall (u:
+T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t))))))).(\lambda (u: T).(\lambda (t: T).(eq_ind_r
+T (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)) (\lambda (t0: T).(eq T (lift
+h d t0) (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d) (lift1 (Ss p)
+t))))) (eq_ind_r T (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d)
+(lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 (THead (Bind b) (lift h d (lift1
+p u)) (lift h (S d) (lift1 (Ss p) t))))) (refl_equal T (THead (Bind b) (lift
+h d (lift1 p u)) (lift h (S d) (lift1 (Ss p) t)))) (lift h d (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t))) (lift_bind b (lift1 p u) (lift1 (Ss p) t) h
+d)) (lift1 p (THead (Bind b) u t)) (H u t)))))))) hds)).
+
+theorem lift1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds
+t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f)
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T
+(THead (Flat f) (lift1 PNil u) (lift1 PNil t))))) (\lambda (h: nat).(\lambda
+(d: nat).(\lambda (p: PList).(\lambda (H: ((\forall (u: T).(\forall (t:
+T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p
+t))))))).(\lambda (u: T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p
+u) (lift1 p t)) (\lambda (t0: T).(eq T (lift h d t0) (THead (Flat f) (lift h
+d (lift1 p u)) (lift h d (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift h
+d (lift1 p u)) (lift h d (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat
+f) (lift h d (lift1 p u)) (lift h d (lift1 p t))))) (refl_equal T (THead
+(Flat f) (lift h d (lift1 p u)) (lift h d (lift1 p t)))) (lift h d (THead
+(Flat f) (lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) h d))
+(lift1 p (THead (Flat f) u t)) (H u t)))))))) hds)).
+
+theorem lift1_cons_tail:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq
+T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t)
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p
+h d) t) H))))) hds)))).
+
+theorem lifts1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts:
+TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds
+ts) (lift1 hds t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts:
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f)
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f)
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1)
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0
+(THeads (Flat f) t1 t)))))) ts)))).
+
+theorem lifts1_nil:
+ \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
+\def
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t)
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1
+PNil t0) H)))) ts).
+
+theorem lifts1_cons:
+ \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts:
+TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t)
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1:
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0)
+H)))) ts)))).
+
+theorem lift1_xhg:
+ \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t))
+(lift (S O) O (lift1 hds t))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T
+(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t:
+T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p)
+(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T
+(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S
+O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n:
+nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d
+(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda
+(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift
+(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1
+p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S
+d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
+
+theorem lifts1_xhg:
+ \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts
+(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
+\def
+ \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq
+TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t))))
+(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq
+TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds
+t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList
+(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1
+hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O
+(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1
+hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds
+t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
+(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds)
+(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
inductive cnt: T \to Prop \def
| cnt_sort: \forall (n: nat).(cnt (TSort n))
-| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt (THead k v t))))).
-
-axiom cnt_lift: \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i d t))))) .
+| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt
+(THead k v t))))).
+
+theorem cnt_lift:
+ \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i
+d t)))))
+\def
+ \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i:
+nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i:
+nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0))
+(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0:
+T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d:
+nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0))
+(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k
+(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)).
inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
| drop_refl: \forall (c: C).(drop O O c c)
-| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u) e))))))
-| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d) (CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
-
-axiom drop_gen_sort: \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O)))))) .
-
-axiom drop_gen_refl: \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e))) .
-
-axiom drop_gen_drop: \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x)))))) .
-
-axiom drop_gen_skip_r: \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall (d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))))))) .
-
-axiom drop_gen_skip_l: \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall (d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k d) c e)))))))))) .
-
-axiom drop_skip_bind: \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) (lift h d u)) (CHead e (Bind b) u)))))))) .
-
-axiom drop_skip_flat: \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h (S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat f) (lift h (S d) u)) (CHead e (Flat f) u)))))))) .
-
-axiom drop_S: \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) .
-
-axiom drop_ctail: \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1) (CTail k u c2)))))))) .
-
-axiom drop_mono: \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2))))))) .
-
-axiom drop_conf_lt: \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall (c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) .
-
-axiom drop_conf_ge: \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) i) \to (drop (minus i h) O e a))))))))) .
-
-axiom drop_conf_rev: \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1))))))))) .
-
-axiom drop_trans_le: \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 e2))))))))))) .
-
-axiom drop_trans_ge: \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 e2) \to ((le d i) \to (drop (plus i h) O c1 e2))))))))) .
+| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e:
+C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u)
+e))))))
+| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c:
+C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d)
+(CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
+
+theorem drop_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop
+h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x:
+C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c:
+C).(drop h d c x)) (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))
+(\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n))
+\to (and3 (eq C c0 (CSort n)) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C
+c0 (CSort n)) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort
+n)) (eq nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O)
+(refl_equal nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c:
+C).(\lambda (e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c
+(CSort n)) \to (and3 (eq C e (CSort n)) (eq nat (r k h0) O) (eq nat O
+O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4
+\def (eq_ind C (CHead c k u) (\lambda (ee: C).(match ee return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort n) H3) in (False_ind (and3 (eq C e (CSort n)) (eq nat (S h0)
+O) (eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda
+(d0: nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c
+e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e (CSort n)) (eq nat h0
+O) (eq nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k
+(lift h0 (r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0
+(r k d0) u)) (\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n)
+H3) in (False_ind (and3 (eq C (CHead e k u) (CSort n)) (eq nat h0 O) (eq nat
+(S d0) O)) H4))))))))))) h d y x H0))) H))))).
+
+theorem drop_gen_refl:
+ \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
+\def
+ \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O
+(\lambda (n: nat).(drop n O x e)) (eq C x e) (\lambda (y: nat).(\lambda (H0:
+(drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y n x e)) ((eq nat y
+O) \to (eq C x e)) (\lambda (y0: nat).(\lambda (H1: (drop y y0 x
+e)).(drop_ind (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c: C).(\lambda
+(c0: C).((eq nat n0 O) \to ((eq nat n O) \to (eq C c c0))))))) (\lambda (c:
+C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq nat O O)).(refl_equal C c))))
+(\lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda
+(_: (drop (r k h) O c e0)).(\lambda (_: (((eq nat O O) \to ((eq nat (r k h)
+O) \to (eq C c e0))))).(\lambda (u: T).(\lambda (_: (eq nat O O)).(\lambda
+(H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S h) (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H5) in (False_ind (eq C (CHead c k u) e0)
+H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c e0)).(\lambda (H3:
+(((eq nat (r k d) O) \to ((eq nat h O) \to (eq C c e0))))).(\lambda (u:
+T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat h O)).(let H6 \def
+(f_equal nat nat (\lambda (e1: nat).e1) h O H5) in (let H7 \def (eq_ind nat h
+(\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat n O) \to (eq C c e0)))) H3
+O H6) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n (r k d) c e0))
+H2 O H6) in (eq_ind_r nat O (\lambda (n: nat).(eq C (CHead c k (lift n (r k
+d) u)) (CHead e0 k u))) (let H9 \def (eq_ind nat (S d) (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H4) in (False_ind (eq C (CHead c k (lift O (r k d)
+u)) (CHead e0 k u)) H9)) h H6)))))))))))))) y y0 x e H1))) H0))) H))).
+
+theorem drop_gen_drop:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
+nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
+nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u)
+(\lambda (c0: C).(drop (S h) O c0 x)) (drop (r k h) O c x) (\lambda (y:
+C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda (n: nat).(drop
+(S h) n y x)) ((eq C y (CHead c k u)) \to (drop (r k h) O c x)) (\lambda (y0:
+nat).(\lambda (H1: (drop (S h) y0 y x)).(insert_eq nat (S h) (\lambda (n:
+nat).(drop n y0 y x)) ((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r
+k h) O c x))) (\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq
+nat n (S h)) \to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k
+h) O c c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda
+(_: (eq nat O O)).(\lambda (_: (eq C c0 (CHead c k u))).(let H6 \def (match
+H3 return (\lambda (n: nat).(\lambda (_: (eq ? ? n)).((eq nat n (S h)) \to
+(drop (r k h) O c c0)))) with [refl_equal \Rightarrow (\lambda (H2: (eq nat O
+(S h))).(let H3 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda
+(_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S h)
+H2) in (False_ind (drop (r k h) O c c0) H3)))]) in (H6 (refl_equal nat (S
+h)))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e:
+C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (_: (((eq nat (r k0 h0) (S
+h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c
+e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S h))).(\lambda (_:
+(eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H8
+\def (match H5 return (\lambda (n: nat).(\lambda (_: (eq ? ? n)).((eq nat n
+(S h)) \to (drop (r k h) O c e)))) with [refl_equal \Rightarrow (\lambda (H4:
+(eq nat (S h0) (S h))).(let H5 \def (f_equal nat nat (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow h0 | (S n)
+\Rightarrow n])) (S h0) (S h) H4) in (eq_ind nat h (\lambda (_: nat).(drop (r
+k h) O c e)) (let H6 \def (match H7 return (\lambda (c0: C).(\lambda (_: (eq
+? ? c0)).((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))) with
+[refl_equal \Rightarrow (\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c k
+u))).(let H6 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead
+c0 k0 u0) (CHead c k u) H4) in ((let H7 \def (f_equal C K (\lambda (e0:
+C).(match e0 return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 |
+(CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H8
+\def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0)
+(CHead c k u) H4) in (eq_ind C c (\lambda (_: C).((eq K k0 k) \to ((eq T u0
+u) \to (drop (r k h) O c e)))) (\lambda (H9: (eq K k0 k)).(eq_ind K k
+(\lambda (_: K).((eq T u0 u) \to (drop (r k h) O c e))) (\lambda (H10: (eq T
+u0 u)).(eq_ind T u (\lambda (_: T).(drop (r k h) O c e)) (eq_ind nat h0
+(\lambda (n: nat).(drop (r k n) O c e)) (eq_ind C c0 (\lambda (c: C).(drop (r
+k h0) O c e)) (eq_ind K k0 (\lambda (k: K).(drop (r k h0) O c0 e)) H3 k H9) c
+H8) h H5) u0 (sym_eq T u0 u H10))) k0 (sym_eq K k0 k H9))) c0 (sym_eq C c0 c
+H8))) H7)) H6)))]) in (H6 (refl_equal C (CHead c k u)))) h0 (sym_eq nat h0 h
+H5))))]) in (H8 (refl_equal nat (S h)))))))))))))) (\lambda (k0: K).(\lambda
+(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (_:
+(drop h0 (r k0 d) c0 e)).(\lambda (_: (((eq nat h0 (S h)) \to ((eq nat (r k0
+d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))))).(\lambda
+(u0: T).(\lambda (_: (eq nat h0 (S h))).(\lambda (H6: (eq nat (S d)
+O)).(\lambda (_: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c k
+u))).(let H8 \def (match H6 return (\lambda (n: nat).(\lambda (_: (eq ? ?
+n)).((eq nat n O) \to (drop (r k h) O c (CHead e k0 u0))))) with [refl_equal
+\Rightarrow (\lambda (H4: (eq nat (S d) O)).(let H5 \def (eq_ind nat (S d)
+(\lambda (e0: nat).(match e0 return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (r
+k h) O c (CHead e k0 u0)) H5)))]) in (H8 (refl_equal nat O)))))))))))))) y1
+y0 y x H2))) H1))) H0))) H)))))).
+
+theorem drop_gen_skip_r:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda
+(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k
+d) e c)))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k u))).(let H0
+\def (match H return (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0:
+C).(\lambda (c1: C).(\lambda (_: (drop n n0 c0 c1)).((eq nat n h) \to ((eq
+nat n0 (S d)) \to ((eq C c0 x) \to ((eq C c1 (CHead c k u)) \to (ex2 C
+(\lambda (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e:
+C).(drop h (r k d) e c)))))))))))) with [(drop_refl c0) \Rightarrow (\lambda
+(H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0
+x)).(\lambda (H3: (eq C c0 (CHead c k u))).(eq_ind nat O (\lambda (n:
+nat).((eq nat O (S d)) \to ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2
+C (\lambda (e: C).(eq C x (CHead e k (lift n (r k d) u)))) (\lambda (e:
+C).(drop n (r k d) e c))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def
+(eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with
+[O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind
+((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x
+(CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c)))))
+H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) \Rightarrow (\lambda
+(H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C
+(CHead c0 k0 u0) x)).(\lambda (H4: (eq C e (CHead c k u))).(eq_ind nat (S h0)
+(\lambda (n: nat).((eq nat O (S d)) \to ((eq C (CHead c0 k0 u0) x) \to ((eq C
+e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq
+C x (CHead e0 k (lift n (r k d) u)))) (\lambda (e0: C).(drop n (r k d) e0
+c)))))))) (\lambda (H5: (eq nat O (S d))).(let H6 \def (eq_ind nat O (\lambda
+(e0: nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow True |
+(S _) \Rightarrow False])) I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0)
+x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C
+(\lambda (e0: C).(eq C x (CHead e0 k (lift (S h0) (r k d) u)))) (\lambda (e0:
+C).(drop (S h0) (r k d) e0 c)))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip
+k0 h0 d0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2:
+(eq nat (S d0) (S d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0)
+u0)) x)).(\lambda (H4: (eq C (CHead e k0 u0) (CHead c k u))).(eq_ind nat h
+(\lambda (n: nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r
+k0 d0) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop n (r k0
+d0) c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d)
+u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))))) (\lambda (H5: (eq nat (S
+d0) (S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return
+(\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0)
+(S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r
+k0 n) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 n)
+c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u))))
+(\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H7: (eq C (CHead c0 k0
+(lift h (r k0 d) u0)) x)).(eq_ind C (CHead c0 k0 (lift h (r k0 d) u0))
+(\lambda (c1: C).((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 d)
+c0 e) \to (ex2 C (\lambda (e0: C).(eq C c1 (CHead e0 k (lift h (r k d) u))))
+(\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H8: (eq C (CHead e k0
+u0) (CHead c k u))).(let H9 \def (f_equal C T (\lambda (e0: C).(match e0
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead e k0 u0) (CHead c k u) H8) in ((let H10 \def (f_equal
+C K (\lambda (e0: C).(match e0 return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e k0 u0) (CHead c k u)
+H8) in ((let H11 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c]))
+(CHead e k0 u0) (CHead c k u) H8) in (eq_ind C c (\lambda (c1: C).((eq K k0
+k) \to ((eq T u0 u) \to ((drop h (r k0 d) c0 c1) \to (ex2 C (\lambda (e0:
+C).(eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead e0 k (lift h (r k d) u))))
+(\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H12: (eq K k0
+k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u) \to ((drop h (r k1 d) c0 c) \to
+(ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 (lift h (r k1 d) u0)) (CHead e0 k
+(lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda
+(H13: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((drop h (r k d) c0 c) \to
+(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) t)) (CHead e0 k
+(lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))) (\lambda
+(H14: (drop h (r k d) c0 c)).(let H15 \def (eq_ind T u0 (\lambda (t: T).(eq C
+(CHead c0 k0 (lift h (r k0 d) t)) x)) H7 u H13) in (let H16 \def (eq_ind K k0
+(\lambda (k: K).(eq C (CHead c0 k (lift h (r k d) u)) x)) H15 k H12) in (let
+H17 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) c0 (CHead c k u))) H
+(CHead c0 k (lift h (r k d) u)) H16) in (ex_intro2 C (\lambda (e0: C).(eq C
+(CHead c0 k (lift h (r k d) u)) (CHead e0 k (lift h (r k d) u)))) (\lambda
+(e0: C).(drop h (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift h (r k d)
+u))) H14))))) u0 (sym_eq T u0 u H13))) k0 (sym_eq K k0 k H12))) e (sym_eq C e
+c H11))) H10)) H9))) x H7)) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h
+H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d))
+(refl_equal C x) (refl_equal C (CHead c k u)))))))))).
+
+theorem drop_gen_skip_l:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T
+(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r k d) c e))))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) x)).(let H0
+\def (match H return (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0:
+C).(\lambda (c1: C).(\lambda (_: (drop n n0 c0 c1)).((eq nat n h) \to ((eq
+nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c1 x) \to (ex3_2 C T
+(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r k d) c e))))))))))))) with [(drop_refl c0) \Rightarrow (\lambda
+(H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0
+(CHead c k u))).(\lambda (H3: (eq C c0 x)).(eq_ind nat O (\lambda (n:
+nat).((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to
+(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda
+(_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda
+(_: T).(drop n (r k d) c e)))))))) (\lambda (H4: (eq nat O (S d))).(let H5
+\def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop)
+with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in
+(False_ind ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to (ex3_2 C T (\lambda
+(e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift O (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop O (r k
+d) c e)))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0)
+\Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S
+d))).(\lambda (H3: (eq C (CHead c0 k0 u0) (CHead c k u))).(\lambda (H4: (eq C
+e x)).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C
+(CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e)
+\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop n (r k d) c e0))))))))) (\lambda (H5: (eq nat O (S
+d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 return (\lambda
+(_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d)
+H5) in (False_ind ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to
+((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq
+C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (S h0) (r
+k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (S h0) (r k d) c e0)))))))
+H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0) \Rightarrow
+(\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S d))).(\lambda
+(H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(\lambda (H4:
+(eq C (CHead e k0 u0) x)).(eq_ind nat h (\lambda (n: nat).((eq nat (S d0) (S
+d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) (CHead c k u)) \to ((eq C
+(CHead e k0 u0) x) \to ((drop n (r k0 d0) c0 e) \to (ex3_2 C T (\lambda (e0:
+C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
+k d) c e0))))))))) (\lambda (H5: (eq nat (S d0) (S d))).(let H6 \def (f_equal
+nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O
+\Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H5) in (eq_ind nat d
+(\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r k0 n) u0)) (CHead c k u))
+\to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 n) c0 e) \to (ex3_2 C T
+(\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda
+(_: T).(drop h (r k d) c e0)))))))) (\lambda (H7: (eq C (CHead c0 k0 (lift h
+(r k0 d) u0)) (CHead c k u))).(let H8 \def (f_equal C T (\lambda (e0:
+C).(match e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x h)) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t]))
+(CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in ((let H9 \def
+(f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 (lift
+h (r k0 d) u0)) (CHead c k u) H7) in ((let H10 \def (f_equal C C (\lambda
+(e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c _ _) \Rightarrow c])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k
+u) H7) in (eq_ind C c (\lambda (c1: C).((eq K k0 k) \to ((eq T (lift h (r k0
+d) u0) u) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 d) c1 e) \to
+(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H11: (eq K k0
+k)).(eq_ind K k (\lambda (k1: K).((eq T (lift h (r k1 d) u0) u) \to ((eq C
+(CHead e k1 u0) x) \to ((drop h (r k1 d) c e) \to (ex3_2 C T (\lambda (e0:
+C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
+k d) c e0)))))))) (\lambda (H12: (eq T (lift h (r k d) u0) u)).(eq_ind T
+(lift h (r k d) u0) (\lambda (t: T).((eq C (CHead e k u0) x) \to ((drop h (r
+k d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k
+v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d) v)))) (\lambda
+(e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))) (\lambda (H13: (eq C
+(CHead e k u0) x)).(eq_ind C (CHead e k u0) (\lambda (c1: C).((drop h (r k d)
+c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C c1 (CHead e0 k
+v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0) (lift h (r k
+d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))) (\lambda
+(H14: (drop h (r k d) c e)).(let H15 \def (eq_ind_r T u (\lambda (t: T).(drop
+h (S d) (CHead c k t) x)) H (lift h (r k d) u0) H12) in (let H16 \def
+(eq_ind_r C x (\lambda (c0: C).(drop h (S d) (CHead c k (lift h (r k d) u0))
+c0)) H15 (CHead e k u0) H13) in (ex3_2_intro C T (\lambda (e0: C).(\lambda
+(v: T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T (lift h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda
+(_: T).(drop h (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal
+T (lift h (r k d) u0)) H14)))) x H13)) u H12)) k0 (sym_eq K k0 k H11))) c0
+(sym_eq C c0 c H10))) H9)) H8))) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0
+h H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d))
+(refl_equal C (CHead c k u)) (refl_equal C x))))))))).
+
+theorem drop_skip_bind:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
+d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
+(lift h d u)) (CHead e (Bind b) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b)
+d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e
+(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
+
+theorem drop_skip_flat:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
+(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat
+f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat
+f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
+e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S
+d))))))))).
+
+theorem drop_S:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
+nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b)
+u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O
+O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort
+n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O
+(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead
+e (Bind b) u) (\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n)
+H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1)))) (drop_gen_sort n h
+O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e:
+C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to
+(drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e:
+C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O
+(CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e)))
+(\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) u))).(let H1 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t)
+(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0))
+in ((let H2 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead
+c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b)
+u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow
+t])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead
+e (Bind b) u) H0)) in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0
+e)).(eq_ind C c0 (\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r
+K (Bind b) (\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop
+(Bind b) O c0 c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n:
+nat).(\lambda (_: (((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop
+(S n) O (CHead c0 k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
+(CHead e (Bind b) u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n))
+(\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0
+(CHead e (Bind b) u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+theorem drop_ctail:
+ \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop
+h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1)
+(CTail k u c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u:
+T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
+c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h
+O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0:
+(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n))
+(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u
+(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O
+(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2
+H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2:
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k
+u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n
+(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u
+(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop
+n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O
+(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t)
+(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
+(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3
+H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to
+(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail
+k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0:
+K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O
+(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n:
+nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to
+(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail
+k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t)
+c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
+(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e)))
+(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2:
+(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h n (CHead c2 k t)
+c) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t))
+(CTail k0 u c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k x1)
+(\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
+(let H5 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead
+c2 k t) (CHead x0 k x1)) \to (\forall (k0: K).(\forall (u: T).(drop h n
+(CTail k0 u (CHead c2 k t)) (CTail k0 u (CHead x0 k x1)))))))) H4 (lift h (r
+k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(drop h (S
+n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) (drop_skip k h
+n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) x1) t H2)) c3
+H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) c1).
+
+theorem drop_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h
+d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0
+x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2:
+C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq
+nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort
+n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
+x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1
+(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r
+C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h
+(\lambda (n: nat).(eq nat n O)) H2 O H5) in (let H8 \def (eq_ind nat d
+(\lambda (n: nat).(eq nat n O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda
+(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1))))
+(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda
+(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1
+x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t)
+x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2))))))
+(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1)
+\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2)))))
+(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1:
+(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C
+x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t)))
+(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2
+(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_:
+(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t)
+x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
+x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O
+(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n
+H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq
+C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t)
+x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t)
+x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0:
+C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4:
+(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0
+x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
+C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7:
+(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r
+C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1
+(\lambda (c: C).(\forall (h: nat).((drop h n (CHead c0 k t) c) \to (\forall
+(x2: C).((drop h n (CHead c0 k t) x2) \to (eq C c x2)))))) H0 (CHead x4 k x5)
+H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 (CHead x0 k
+x3))) (let H10 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n
+(CHead c0 k t) (CHead x4 k x5)) \to (\forall (x2: C).((drop h n (CHead c0 k
+t) x2) \to (eq C (CHead x4 k x5) x2)))))) H9 (lift h (r k n) x5) H7) in (let
+H11 \def (eq_ind T t (\lambda (t: T).(eq T t (lift h (r k n) x3))) H4 (lift h
+(r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (t: T).(\forall (h0:
+nat).((drop h0 n (CHead c0 k (lift h (r k n) t)) (CHead x4 k t)) \to (\forall
+(x2: C).((drop h0 n (CHead c0 k (lift h (r k n) t)) x2) \to (eq C (CHead x4 k
+t) x2)))))) H10 x3 (lift_inj x5 x3 h (r k n) H11)) in (eq_ind_r T x3 (\lambda
+(t0: T).(eq C (CHead x4 k t0) (CHead x0 k x3))) (sym_equal C (CHead x0 k x3)
+(CHead x4 k x3) (sym_equal C (CHead x4 k x3) (CHead x0 k x3) (sym_equal C
+(CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C CHead x0 x4 k k x3 x3 (H x0
+(r k n) h H5 x4 H8) (refl_equal K k) (refl_equal T x3))))) x5 (lift_inj x5 x3
+h (r k n) H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k
+H1))))))) (drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c).
+
+theorem drop_conf_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall
+(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h (r k d) c0 e0)))))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u:
+T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
+(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u:
+T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k
+u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop
+h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c: C).(drop h (S
+(plus O d)) c e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in
+(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda
+(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d))
+x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k
+x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
+h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T
+(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda
+(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda
+(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k
+d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0
+e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall
+(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d))
+c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda
+(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0)
+O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S
+i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind
+(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort
+n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d))
+O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq
+T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
+H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1)))))))) (\lambda (c1:
+C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to (\forall (e:
+C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0) d)) c1 e) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
+(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda (k0:
+K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 t)
+(CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
+d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b)
+(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1
+x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b)
+c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T
+u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O
+(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7:
+(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k
+x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0
+x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h
+(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
+d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f)
+(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S
+i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
+c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1)
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d)
+x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h
+(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead
+x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r
+k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1)
+H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e
+H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2)))))))))
+k0)))) c)))))) i)).
+
+theorem drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
+(plus d h) i) \to (drop (minus i h) O e a)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c:
+C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
+a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
+d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
+(c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1
+return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop
+(minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h)
+O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in
+(eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r
+nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq
+nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h
+O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in
+(let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in
+(eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a
+e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow
+(\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m) \to (drop
+(minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O)))))))))))) (\lambda
+(i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
+(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a:
+C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d
+h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n:
+nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2:
+(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d
+O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda
+(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n))
+(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6:
+(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O
+O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O
+H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0)))
+H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e
+a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a))
+(eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n)
+c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) H11)) a
+H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d
+e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le
+(plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k:
+K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a)
+\to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0
+k0 t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
+a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O
+(CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le
+(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b)
+t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a))))
+(\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O
+h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e)
+\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda
+(H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus O O) (S
+i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O)
+O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t)
+e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda
+(_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to
+(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0
+(Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a c0
+(drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e t
+h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: nat).(\lambda
+(_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to
+(drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0
+(Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T
+(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind b) v)))) (\lambda
+(_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop (minus (S i0) h) O
+e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind
+b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H8:
+(drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda
+(c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda
+(n: nat).(drop n O (CHead x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0
+h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n
+(plus d0 h) i0 H5)) x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0
+h i0 (le_S_n (plus d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0
+(Bind b) H4)))))) d H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t)
+e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h
+n (CHead c0 (Flat f) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S
+i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda
+(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0
+(Flat f) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e
+a)))) (\lambda (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le
+(plus O O) (S i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop
+(minus (S i0) O) O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f)
+c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda
+(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O
+h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0)
+O (CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0
+(drop_gen_drop (Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e
+t h0 H6) H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0
+(CHead c0 (Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0)
+h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda
+(H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda
+(v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T
+t (lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
+(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t
+(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0)
+h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n:
+nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0)
+H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n
+(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop
+n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1)
+(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0
+h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2
+H3))))))))) k)))) c))))) i).
+
+theorem drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
+(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1:
+C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
+\def
+ \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2:
+C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2)
+\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1
+e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1
+e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let
+H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl
+e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1:
+C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda
+(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2:
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O
+c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind
+(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2:
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n:
+nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda
+(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2
+(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1:
+(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O
+O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n)
+H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda
+(c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2
+H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O
+e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C
+(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1
+e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H:
+(drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i:
+nat).(\lambda (H0: (drop i O c2 e3)).((match k return (\lambda (k0: K).((drop
+(r k0 j0) O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2))
+(\lambda (c1: C).(drop i (S j0) c1 (CHead e2 k0 t)))))) with [(Bind b)
+\Rightarrow (\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj
+e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop j0
+O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop
+(S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))))
+(\lambda (x: C).(\lambda (H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x
+e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
+C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) (CHead x (Bind b) (lift i (r
+(Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 (lift i (r (Bind b) j0) t))
+(drop_skip (Bind b) i j0 x e2 H4 t))))) H2)))) | (Flat f) \Rightarrow
+(\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def (IHe1 e3 H1 c2 i
+H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (S j0) O c1 c2))
+(\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0)
+O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))))
+(\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: (drop i (S
+j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
+C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) (lift i (r
+(Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat f) j0) t))
+(drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))]) (drop_gen_drop k e2 e3 t j0
+H))))))))))) e1)))) j).
+
+theorem drop_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
+(minus d i) e1 e2)))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
+(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
+(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
+e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
+nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
+c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
+(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
+(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
+(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
+(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
+nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
+C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
+C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
+(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
+(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
+C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
+C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
+O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
+O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let H2 \def
+(match H return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to
+(ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h
+(minus O (S i0)) e1 e2)))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S
+i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda
+(e1: C).(drop h (minus O (S i0)) e1 e2))) H3))) | (le_S m H2) \Rightarrow
+(\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H3) in (False_ind ((le (S i0) m) \to (ex2 C
+(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S
+i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat O)))))))))) (\lambda (d0:
+nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2:
+C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
+C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S
+d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
+nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to
+(ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus
+(S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda
+(H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq
+nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
+(e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort
+n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def
+(eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in
+(and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda
+(e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0)
+(S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S
+i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2
+C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h
+(minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H7) in (False_ind (ex2 C (\lambda (e1: C).(drop (S
+i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort
+n)))) H9)) e2 H6)))) (drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h
+(S d0) c2 H0)))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall (c3:
+C).(\forall (h: nat).((drop h (S d0) c2 c3) \to (\forall (e2: C).((drop (S
+i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2)))))))))).(\lambda (k: K).(K_ind
+(\lambda (k0: K).(\forall (t: T).(\forall (c3: C).(\forall (h: nat).((drop h
+(S d0) (CHead c2 k0 t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to
+(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 k0 t) e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2)))))))))) (\lambda (b: B).(\lambda (t:
+T).(\lambda (c3: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2
+(Bind b) t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3
+e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Bind
+b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0)
+v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r (Bind b) d0) c2 e))) (ex2 C
+(\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) t) e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H2: (eq C c3 (CHead x0 (Bind b) x1))).(\lambda (H3: (eq T t
+(lift h (r (Bind b) d0) x1))).(\lambda (H4: (drop h (r (Bind b) d0) c2
+x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H1
+(CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h (r (Bind b) d0) x1)
+(\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b)
+t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))) (ex2_ind C
+(\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop h (minus d0 i0) e1
+e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) (lift h (r
+(Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda (H7: (drop h
+(minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
+(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
+d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r (Bind b) d0)
+x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 (drop_gen_drop (Bind b) x0
+e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 t h d0 (Bind b) H0)))))))))
+(\lambda (f: F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(\lambda
+(H0: (drop h (S d0) (CHead c2 (Flat f) t) c3)).(\lambda (e2: C).(\lambda (H1:
+(drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C
+c3 (CHead e (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r
+(Flat f) d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r (Flat f) d0) c2
+e))) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) t) e1))
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 (Flat f) x1))).(\lambda
+(H3: (eq T t (lift h (r (Flat f) d0) x1))).(\lambda (H4: (drop h (r (Flat f)
+d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2))
+H1 (CHead x0 (Flat f) x1) H2) in (eq_ind_r T (lift h (r (Flat f) d0) x1)
+(\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
+t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))) (ex2_ind C
+(\lambda (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S
+d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
+(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
+i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda
+(H7: (drop h (minus (S d0) (S i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop
+(S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6
+(lift h (r (Flat f) d0) x1)) H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f)
+x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 t h d0 (Flat f)
+H0))))))))) k)))) c1))))) d)))) i).
+
+theorem drop_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
+e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
+C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
+c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
+O c1 c)) (let H2 \def (match H1 return (\lambda (n: nat).(\lambda (_: (le ?
+n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n \Rightarrow
+(\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop (plus O h)
+O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O)) H1 O H0) in
+(let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H O H0) in
+H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2: (eq nat
+(S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1 c2)) H1)) H0))]) in
+(H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 H0)))))))))) (\lambda (i0:
+nat).(\lambda (IHi: ((\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2
+e2) \to ((le d i0) \to (drop (plus i0 h) O c1 e2))))))))))).(\lambda (c1:
+C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to ((le d
+(S i0)) \to (drop (plus (S i0) h) O c e2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
+c2)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c2 e2)).(\lambda (H1: (le
+d (S i0))).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq nat d O) (drop (S
+(plus i0 h)) O (CSort n) e2) (\lambda (H2: (eq C c2 (CSort n))).(\lambda (H3:
+(eq nat h O)).(\lambda (H4: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
+nat).(drop (S (plus i0 n0)) O (CSort n) e2)) (let H5 \def (eq_ind nat d
+(\lambda (n: nat).(le n (S i0))) H1 O H4) in (let H6 \def (eq_ind C c2
+(\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n) H2) in (and3_ind (eq C e2
+(CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S (plus i0 O)) O (CSort n)
+e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8: (eq nat (S i0)
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(drop (S
+(plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0) (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort
+n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3))))
+(drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall
+(c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall
+(e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O
+c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t)
+c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop
+(S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind
+(\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop
+(S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t)
+e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda
+(H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def
+(eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t)
+(drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n:
+nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k
+c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda
+(_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3
+e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t)
+e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2:
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind
+nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2))
+(drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n)))
+(\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n))
+(\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k
+c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus
+i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h))
+(\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k
+t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to
+(drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda
+(H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop
+(S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda
+(e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r
+k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3:
+(eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h d0 (CHead c2 k
+t) c) \to (\forall (e2: C).((drop (S i0) O c e2) \to ((le d0 (S i0)) \to
+(drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) IHd (CHead x0 k x1) H2) in
+(let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0
+k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t: T).(\forall (h:
+nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall (e2: C).((drop
+(S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O
+(CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r T (lift h
+(r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k t0) e2))
+(drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r k0 d0) c2
+x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2 e2))))
+(\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda (H9:
+(drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9 (le_S_n
+(r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r (Flat f)
+d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r (Flat f)
+d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0)
+x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1))))
+i).
inductive drop1: PList \to (C \to (C \to Prop)) \def
| drop1_nil: \forall (c: C).(drop1 PNil c c)
-| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
-
-definition ctrans: PList \to (nat \to (T \to T)) \def let rec ctrans (hds: PList): (nat \to (T \to T)) \def (\lambda (i: nat).(\lambda (t: T).(match hds with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (let j \def (trans hds0 i) in (let u \def (ctrans hds0 i t) in (match (blt j d) with [true \Rightarrow (lift h (minus d (S j)) u) | false \Rightarrow u])))]))) in ctrans.
-
-axiom drop1_skip_bind: \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) (lift1 hds u)) (CHead e (Bind b) u))))))) .
-
-axiom drop1_cons_tail: \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to (drop1 (PConsTail hds h d) c1 c3)))))))) .
-
-axiom lift1_free: \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds (lift (S i) O t)) (lift (S (trans hds i)) O (ctrans hds i t))))) .
+| drop1_cons: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c1 c2) \to (\forall (c3: C).(\forall (hds: PList).((drop1 hds
+c2 c3) \to (drop1 (PCons h d hds) c1 c3)))))))).
+
+definition ctrans:
+ PList \to (nat \to (T \to T))
+\def
+ let rec ctrans (hds: PList) on hds: (nat \to (T \to T)) \def (\lambda (i:
+nat).(\lambda (t: T).(match hds with [PNil \Rightarrow t | (PCons h d hds0)
+\Rightarrow (let j \def (trans hds0 i) in (let u \def (ctrans hds0 i t) in
+(match (blt j d) with [true \Rightarrow (lift h (minus d (S j)) u) | false
+\Rightarrow u])))]))) in ctrans.
+
+theorem drop1_skip_bind:
+ \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
+C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
+(lift1 hds u)) (CHead e (Bind b) u)))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p:
+PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p)
+(CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H0 \def (match H
+return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_:
+(drop1 p c0 c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to
+(drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))))) with
+[(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H1:
+(eq C c0 c)).(\lambda (H2: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C
+c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))) (\lambda
+(H3: (eq C c e)).(eq_ind C e (\lambda (c: C).(drop1 PNil (CHead c (Bind b) u)
+(CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e H3)))
+c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds H1)
+\Rightarrow (\lambda (H2: (eq PList (PCons h d hds) PNil)).(\lambda (H3: (eq
+C c1 c)).(\lambda (H4: (eq C c3 e)).((let H5 \def (eq_ind PList (PCons h d
+hds) (\lambda (e0: PList).(match e0 return (\lambda (_: PList).Prop) with
+[PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H2) in
+(False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1
+hds c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))
+H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal C c)
+(refl_equal C e)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to
+(drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b)
+u))))))).(\lambda (c: C).(\lambda (u: T).(\lambda (H0: (drop1 (PCons n n0 p)
+c e)).(let H1 \def (match H0 return (\lambda (p0: PList).(\lambda (c0:
+C).(\lambda (c1: C).(\lambda (_: (drop1 p0 c0 c1)).((eq PList p0 (PCons n n0
+p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 (PCons n (S n0) (Ss p))
+(CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))) with
+[(drop1_nil c0) \Rightarrow (\lambda (H1: (eq PList PNil (PCons n n0
+p))).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).((let H4 \def
+(eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_:
+PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False]))
+I (PCons n n0 p) H1) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (drop1
+(PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e
+(Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2)
+\Rightarrow (\lambda (H3: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda
+(H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (f_equal PList
+PList (\lambda (e0: PList).(match e0 return (\lambda (_: PList).PList) with
+[PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons
+n n0 p) H3) in ((let H7 \def (f_equal PList nat (\lambda (e0: PList).(match
+e0 return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n _)
+\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in ((let H8 \def (f_equal
+PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: PList).nat) with
+[PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n
+n0 p) H3) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList
+hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1
+hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0
+(lift1 p u))) (CHead e (Bind b) u))))))))) (\lambda (H9: (eq nat d
+n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to
+((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons
+n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b)
+u)))))))) (\lambda (H10: (eq PList hds p)).(eq_ind PList p (\lambda (p0:
+PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2
+c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p
+u))) (CHead e (Bind b) u))))))) (\lambda (H11: (eq C c1 c)).(eq_ind C c
+(\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to
+(drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u)))
+(CHead e (Bind b) u)))))) (\lambda (H12: (eq C c3 e)).(eq_ind C e (\lambda
+(c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (drop1 (PCons n (S n0) (Ss
+p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))
+(\lambda (H13: (drop n n0 c c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons
+(CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n
+(S n0) (drop_skip_bind n n0 c c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss
+p) (H c2 u H14)))) c3 (sym_eq C c3 e H12))) c1 (sym_eq C c1 c H11))) hds
+(sym_eq PList hds p H10))) d (sym_eq nat d n0 H9))) h (sym_eq nat h n H8)))
+H7)) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList (PCons n n0 p))
+(refl_equal C c) (refl_equal C e)))))))))) hds))).
+
+theorem drop1_cons_tail:
+ \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
+h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to
+(drop1 (PConsTail hds h d) c1 c3))))))))
+\def
+ \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda
+(p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
+c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H1 \def (match
+H0 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_:
+(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to
+(drop1 (PCons h d PNil) c1 c3)))))))) with [(drop1_nil c) \Rightarrow
+(\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3:
+(eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2) \to (drop1 (PCons h
+d PNil) c1 c3))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2 (\lambda (c0:
+C).(drop1 (PCons h d PNil) c0 c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil
+c3)) c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c0 c4
+h0 d0 H1 c5 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h0 d0 hds)
+PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5 c2)).((let H6 \def
+(eq_ind PList (PCons h0 d0 hds) (\lambda (e: PList).(match e return (\lambda
+(_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow
+True])) I PNil H3) in (False_ind ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop
+h0 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons h d PNil) c1 c3)))))
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1)
+(refl_equal C c2))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H0: ((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail
+p h d) c1 c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1
+c2)).(let H2 \def (match H1 return (\lambda (p0: PList).(\lambda (c:
+C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n n0
+p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons n n0 (PConsTail p h
+d)) c1 c3)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList
+PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c
+c2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return
+(\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
+\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq
+C c c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))) H5)) H3 H4)))) |
+(drop1_cons c0 c4 h0 d0 H2 c5 hds H3) \Rightarrow (\lambda (H4: (eq PList
+(PCons h0 d0 hds) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6:
+(eq C c5 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e
+return (\lambda (_: PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p)
+\Rightarrow p])) (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H8 \def
+(f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_:
+PList).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons
+h0 d0 hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda
+(e: PList).(match e return (\lambda (_: PList).nat) with [PNil \Rightarrow h0
+| (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0 p) H4) in
+(eq_ind nat n (\lambda (n1: nat).((eq nat d0 n0) \to ((eq PList hds p) \to
+((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n1 d0 c0 c4) \to ((drop1 hds c4
+c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) (\lambda (H10:
+(eq nat d0 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq
+C c0 c1) \to ((eq C c5 c2) \to ((drop n n1 c0 c4) \to ((drop1 hds c4 c5) \to
+(drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq PList
+hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c5 c2)
+\to ((drop n n0 c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0
+(PConsTail p h d)) c1 c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1
+(\lambda (c: C).((eq C c5 c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to
+(drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5
+c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to
+(drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1
+c4)).(\lambda (H15: (drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail
+p h d) (H0 c4 H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds
+(sym_eq PList hds p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n
+H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
+(refl_equal C c1) (refl_equal C c2))))))))) hds)))))).
+
+theorem lift1_free:
+ \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds
+(lift (S i) O t)) (lift (S (trans hds i)) O (ctrans hds i t)))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i:
+nat).(\forall (t: T).(eq T (lift1 p (lift (S i) O t)) (lift (S (trans p i)) O
+(ctrans p i t)))))) (\lambda (i: nat).(\lambda (t: T).(refl_equal T (lift (S
+i) O t)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0:
+PList).(\lambda (H: ((\forall (i: nat).(\forall (t: T).(eq T (lift1 hds0
+(lift (S i) O t)) (lift (S (trans hds0 i)) O (ctrans hds0 i t))))))).(\lambda
+(i: nat).(\lambda (t: T).(eq_ind_r T (lift (S (trans hds0 i)) O (ctrans hds0
+i t)) (\lambda (t0: T).(eq T (lift h d t0) (lift (S (match (blt (trans hds0
+i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans
+hds0 i) h)])) O (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h
+(minus d (S (trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans
+hds0 i t)])))) (xinduction bool (blt (trans hds0 i) d) (\lambda (b: bool).(eq
+T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (match b
+with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0
+i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans hds0 i t)])))) (\lambda
+(x_x: bool).(bool_ind (\lambda (b: bool).((eq bool (blt (trans hds0 i) d) b)
+\to (eq T (lift h d (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S
+(match b with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus
+(trans hds0 i) h)])) O (match b with [true \Rightarrow (lift h (minus d (S
+(trans hds0 i))) (ctrans hds0 i t)) | false \Rightarrow (ctrans hds0 i
+t)]))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(eq_ind_r nat
+(plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (\lambda (n: nat).(eq
+T (lift h n (lift (S (trans hds0 i)) O (ctrans hds0 i t))) (lift (S (trans
+hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t)))))
+(eq_ind_r T (lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i t))) (\lambda (t0: T).(eq T t0 (lift (S (trans hds0 i)) O
+(lift h (minus d (S (trans hds0 i))) (ctrans hds0 i t))))) (refl_equal T
+(lift (S (trans hds0 i)) O (lift h (minus d (S (trans hds0 i))) (ctrans hds0
+i t)))) (lift h (plus (S (trans hds0 i)) (minus d (S (trans hds0 i)))) (lift
+(S (trans hds0 i)) O (ctrans hds0 i t))) (lift_d (ctrans hds0 i t) h (S
+(trans hds0 i)) (minus d (S (trans hds0 i))) O (le_O_n (minus d (S (trans
+hds0 i)))))) d (le_plus_minus (S (trans hds0 i)) d (bge_le (S (trans hds0 i))
+d (le_bge (S (trans hds0 i)) d (lt_le_S (trans hds0 i) d (blt_lt d (trans
+hds0 i) H0))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d)
+false)).(eq_ind_r T (lift (plus h (S (trans hds0 i))) O (ctrans hds0 i t))
+(\lambda (t0: T).(eq T t0 (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i
+t)))) (eq_ind nat (S (plus h (trans hds0 i))) (\lambda (n: nat).(eq T (lift n
+O (ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t))))
+(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O
+(ctrans hds0 i t)) (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t))))
+(refl_equal T (lift (S (plus (trans hds0 i) h)) O (ctrans hds0 i t))) (plus h
+(trans hds0 i)) (plus_comm h (trans hds0 i))) (plus h (S (trans hds0 i)))
+(plus_n_Sm h (trans hds0 i))) (lift h d (lift (S (trans hds0 i)) O (ctrans
+hds0 i t))) (lift_free (ctrans hds0 i t) (S (trans hds0 i)) h O d (eq_ind nat
+(S (plus O (trans hds0 i))) (\lambda (n: nat).(le d n)) (eq_ind_r nat (plus
+(trans hds0 i) O) (\lambda (n: nat).(le d (S n))) (le_S d (plus (trans hds0
+i) O) (le_plus_trans d (trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus
+O (trans hds0 i)) (plus_comm O (trans hds0 i))) (plus O (S (trans hds0 i)))
+(plus_n_Sm O (trans hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O
+t)) (H i t)))))))) hds).
inductive clear: C \to (C \to Prop) \def
-| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e (Bind b) u) (CHead e (Bind b) u))))
-| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f: F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))).
+| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e
+(Bind b) u) (CHead e (Bind b) u))))
+| clear_flat: \forall (e: C).(\forall (c: C).((clear e c) \to (\forall (f:
+F).(\forall (u: T).(clear (CHead e (Flat f) u) c))))).
inductive getl (h:nat) (c1:C) (c2:C): Prop \def
-| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h c1 c2))).
-
-definition cimp: C \to (C \to Prop) \def \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))).
-
-axiom clear_gen_sort: \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: Prop).P))) .
-
-axiom clear_gen_bind: \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear (CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u)))))) .
-
-axiom clear_gen_flat: \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear (CHead e (Flat f) u) x) \to (clear e x))))) .
-
-axiom clear_gen_flat_r: \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x (CHead e (Flat f) u)) \to (\forall (P: Prop).P))))) .
-
-axiom clear_gen_all: \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u)))))))) .
-
-axiom drop_clear: \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))))))) .
-
-axiom drop_clear_O: \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c e2)))))))) .
-
-axiom drop_clear_S: \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))) .
-
-axiom clear_clear: \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2))) .
-
-axiom clear_mono: \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c c2) \to (eq C c1 c2))))) .
-
-axiom clear_trans: \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c c2) \to (clear c1 c2))))) .
-
-axiom clear_ctail: \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c1) (CHead (CTail k u1 c2) (Bind b) u2)))))))) .
-
-axiom getl_gen_all: \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))))) .
-
-axiom getl_gen_sort: \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to (\forall (P: Prop).P)))) .
-
-axiom getl_gen_O: \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x))) .
-
-axiom getl_gen_S: \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x)))))) .
-
-axiom getl_refl: \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u) (CHead c (Bind b) u)))) .
-
-axiom clear_getl_trans: \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl i c1 c3)))))) .
-
-axiom getl_clear_trans: \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to (\forall (c3: C).((clear c2 c3) \to (getl i c1 c3)))))) .
-
-axiom getl_head: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e)))))) .
-
-axiom getl_flat: \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall (f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e)))))) .
-
-axiom getl_drop: \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e)))))) .
-
-axiom getl_clear_bind: \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c e2)))))))) .
-
-axiom getl_ctail: \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u))))))))) .
-
-axiom getl_ctail_clen: \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t)))))) .
-
-axiom getl_dec: \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c d) \to (\forall (P: Prop).P))))) .
-
-axiom clear_cle: \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1))) .
-
-axiom getl_flt: \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i))))))) .
-
-axiom getl_gen_flat: \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d)))))) .
-
-axiom getl_gen_bind: \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j e d))))))))) .
-
-axiom getl_gen_tail: \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall (c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))))))) .
-
-axiom cimp_flat_sx: \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) c))) .
-
-axiom cimp_flat_dx: \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) v)))) .
-
-axiom cimp_bind: \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall (v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v)))))) .
-
-axiom getl_mono: \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to (\forall (x2: C).((getl h c x2) \to (eq C x1 x2)))))) .
-
-axiom getl_clear_conf: \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl i c2 c3)))))) .
-
-axiom getl_drop_conf_lt: \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0))))))))))))) .
-
-axiom getl_drop_conf_ge: \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) i) \to (getl (minus i h) e a))))))))) .
-
-axiom getl_conf_ge_drop: \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1 c2) \to (drop i O c2 e)))))))) .
-
-axiom getl_conf_le: \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall (e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e a)))))))) .
-
-axiom getl_drop_conf_rev: \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to (\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) (\lambda (c1: C).(drop (S i) j c1 e1))))))))))) .
-
-axiom drop_getl_trans_lt: \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2: C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d (S i)) e1 e2))))))))))))) .
-
-axiom drop_getl_trans_le: \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))))))))))) .
-
-axiom drop_getl_trans_ge: \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to ((le d i) \to (getl (plus i h) c1 e2))))))))) .
-
-axiom getl_drop_trans: \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i h)) O c1 e2))))))) .
-
-axiom getl_trans: \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 e2))))))) .
-
-axiom drop1_getl_trans: \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (ctrans hds i v))))))))))))) .
-
-axiom cimp_getl_conf: \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall (d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) \to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))))))))))) .
+| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h
+c1 c2))).
+
+definition cimp:
+ C \to (C \to Prop)
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\forall (b: B).(\forall (d1: C).(\forall
+(w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C
+(\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w)))))))))).
+
+theorem clear_gen_sort:
+ \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda
+(P: Prop).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0:
+C).(\lambda (_: (clear c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to P)))))
+with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u)
+(CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def
+(eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u) x) \to P)
+H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq C (CHead e
+(Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C
+(CHead e (Flat f) u) (\lambda (e0: C).(match e0 return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H1) in (False_ind ((eq C c x) \to ((clear e c) \to P)) H3)) H2
+H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x)))))).
+
+theorem clear_gen_bind:
+ \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
+(CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
+(clear (CHead e (Bind b) u) x)).(let H0 \def (match H return (\lambda (c:
+C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e (Bind b) u))
+\to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u))))))) with [(clear_bind b0
+e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind
+b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) x)).((let H2 \def
+(f_equal C T (\lambda (e1: C).(match e1 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind
+b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def (f_equal C B (\lambda (e1:
+C).(match e1 return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 |
+(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead e0 (Bind b0) u0) (CHead e
+(Bind b) u) H0) in ((let H4 \def (f_equal C C (\lambda (e1: C).(match e1
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _)
+\Rightarrow c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in (eq_ind
+C e (\lambda (c: C).((eq B b0 b) \to ((eq T u0 u) \to ((eq C (CHead c (Bind
+b0) u0) x) \to (eq C x (CHead e (Bind b) u)))))) (\lambda (H5: (eq B b0
+b)).(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq C (CHead e (Bind b1)
+u0) x) \to (eq C x (CHead e (Bind b) u))))) (\lambda (H6: (eq T u0
+u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e (Bind b) t) x) \to (eq C x
+(CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead e (Bind b) u) x)).(eq_ind
+C (CHead e (Bind b) u) (\lambda (c: C).(eq C c (CHead e (Bind b) u)))
+(refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq T u0 u H6))) b0 (sym_eq
+B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1))) | (clear_flat e0 c H0 f
+u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b)
+u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C (CHead e0 (Flat f) u0)
+(\lambda (e1: C).(match e1 return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(CHead e (Bind b) u) H1) in (False_ind ((eq C c x) \to ((clear e0 c) \to (eq
+C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e
+(Bind b) u)) (refl_equal C x))))))).
+
+theorem clear_gen_flat:
+ \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear
+(CHead e (Flat f) u) x) \to (clear e x)))))
+\def
+ \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H:
+(clear (CHead e (Flat f) u) x)).(let H0 \def (match H return (\lambda (c:
+C).(\lambda (c0: C).(\lambda (_: (clear c c0)).((eq C c (CHead e (Flat f) u))
+\to ((eq C c0 x) \to (clear e x)))))) with [(clear_bind b e0 u0) \Rightarrow
+(\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(\lambda
+(H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C (CHead e0 (Bind
+b) u0) (\lambda (e1: C).(match e1 return (\lambda (_: C).Prop) with [(CSort
+_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(CHead e (Flat f) u) H0) in (False_ind ((eq C (CHead e0 (Bind b) u0) x) \to
+(clear e x)) H2)) H1))) | (clear_flat e0 c H0 f0 u0) \Rightarrow (\lambda
+(H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(\lambda (H2: (eq C
+c x)).((let H3 \def (f_equal C T (\lambda (e1: C).(match e1 return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t]))
+(CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in ((let H4 \def (f_equal C
+F (\lambda (e1: C).(match e1 return (\lambda (_: C).F) with [(CSort _)
+\Rightarrow f0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).F)
+with [(Bind _) \Rightarrow f0 | (Flat f) \Rightarrow f])])) (CHead e0 (Flat
+f0) u0) (CHead e (Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e1:
+C).(match e1 return (\lambda (_: C).C) with [(CSort _) \Rightarrow e0 |
+(CHead c _ _) \Rightarrow c])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u)
+H1) in (eq_ind C e (\lambda (c0: C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C
+c x) \to ((clear c0 c) \to (clear e x)))))) (\lambda (H6: (eq F f0
+f)).(eq_ind F f (\lambda (_: F).((eq T u0 u) \to ((eq C c x) \to ((clear e c)
+\to (clear e x))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (_:
+T).((eq C c x) \to ((clear e c) \to (clear e x)))) (\lambda (H8: (eq C c
+x)).(eq_ind C x (\lambda (c0: C).((clear e c0) \to (clear e x))) (\lambda
+(H9: (clear e x)).H9) c (sym_eq C c x H8))) u0 (sym_eq T u0 u H7))) f0
+(sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4)) H3)) H2 H0)))]) in (H0
+(refl_equal C (CHead e (Flat f) u)) (refl_equal C x))))))).
+
+theorem clear_gen_flat_r:
+ \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x
+(CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (f: F).(\lambda (x: C).(\lambda (e: C).(\lambda (u: T).(\lambda (H:
+(clear x (CHead e (Flat f) u))).(\lambda (P: Prop).(insert_eq C (CHead e
+(Flat f) u) (\lambda (c: C).(clear x c)) P (\lambda (y: C).(\lambda (H0:
+(clear x y)).(clear_ind (\lambda (_: C).(\lambda (c0: C).((eq C c0 (CHead e
+(Flat f) u)) \to P))) (\lambda (b: B).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(let H2
+\def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (ee: C).(match ee return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H1) in
+(False_ind P H2)))))) (\lambda (e0: C).(\lambda (c: C).(\lambda (H1: (clear
+e0 c)).(\lambda (H2: (((eq C c (CHead e (Flat f) u)) \to P))).(\lambda (_:
+F).(\lambda (_: T).(\lambda (H3: (eq C c (CHead e (Flat f) u))).(let H4 \def
+(eq_ind C c (\lambda (c: C).((eq C c (CHead e (Flat f) u)) \to P)) H2 (CHead
+e (Flat f) u) H3) in (let H5 \def (eq_ind C c (\lambda (c: C).(clear e0 c))
+H1 (CHead e (Flat f) u) H3) in (H4 (refl_equal C (CHead e (Flat f)
+u)))))))))))) x y H0))) H)))))).
+
+theorem clear_gen_all:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (ex_3 B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u: T).(eq C c2 (CHead e (Bind b) u))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (clear c1 c2)).(clear_ind
+(\lambda (_: C).(\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u: T).(eq C c0 (CHead e (Bind b) u)))))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (u: T).(ex_3_intro B C T (\lambda (b0:
+B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead e (Bind b) u) (CHead e0
+(Bind b0) u0))))) b e u (refl_equal C (CHead e (Bind b) u)))))) (\lambda (e:
+C).(\lambda (c: C).(\lambda (H0: (clear e c)).(\lambda (H1: (ex_3 B C T
+(\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c (CHead e (Bind b)
+u))))))).(\lambda (_: F).(\lambda (_: T).(let H2 \def H1 in (ex_3_ind B C T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c (CHead e0 (Bind b)
+u0))))) (ex_3 B C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C c
+(CHead e0 (Bind b) u0)))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (H3: (eq C c (CHead x1 (Bind x0) x2))).(let H4 \def (eq_ind C c
+(\lambda (c: C).(clear e c)) H0 (CHead x1 (Bind x0) x2) H3) in (eq_ind_r C
+(CHead x1 (Bind x0) x2) (\lambda (c0: C).(ex_3 B C T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u0: T).(eq C c0 (CHead e0 (Bind b) u0))))))) (ex_3_intro B
+C T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(eq C (CHead x1 (Bind
+x0) x2) (CHead e0 (Bind b) u0))))) x0 x1 x2 (refl_equal C (CHead x1 (Bind x0)
+x2))) c H3)))))) H2)))))))) c1 c2 H))).
+
+theorem drop_clear:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to
+(ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead
+e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
+c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i:
+nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
+(e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind
+(eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v)))))
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda
+(_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat
+O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda
+(v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e:
+C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort n (S i) O c2
+H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (i:
+nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
+(e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k
+t) c2)).((match k return (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3
+B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t)
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
+T).(drop i O e c2))))))) with [(Bind b) \Rightarrow (\lambda (H1: (drop (r
+(Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e:
+C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v)))))
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t
+(clear_bind b c t) H1)) | (Flat f) \Rightarrow (\lambda (H1: (drop (r (Flat
+f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda
+(_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T
+(\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t)
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
+T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O
+x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v:
+T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_:
+B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c
+(CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))]) (drop_gen_drop k c c2 t i
+H0))))))))) c1).
+
+theorem drop_clear_O:
+ \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c
+(CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1
+e2) \to (drop (S i) O c e2))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
+C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
+C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2))))))))
+(\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort
+n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_:
+(drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O
+(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
+C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
+C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0
+e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u:
+T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).((match k return
+(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S
+i) O (CHead c0 k0 t) e2))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear
+(CHead c0 (Bind b0) t) (CHead e1 (Bind b) u))).(let H3 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) u) (CHead
+c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let
+H4 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda
+(_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead
+e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind
+b) u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
+t])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
+(CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq
+C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: C).(drop i O c e2)) H1 c0
+H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2))
+(drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3))) | (Flat f)
+\Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b)
+u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind
+b) u) t H2) e2 i H1) t))]) H0))))))))))) c)).
+
+theorem drop_clear_S:
+ \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop
+h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear
+x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1
+(Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
+\def
+ \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
+C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda
+(c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b)
+u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1:
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
+C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
+C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k
+t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear
+(CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1
+(CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C
+(\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1:
+C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift
+h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k
+(lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead
+c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) ((match k
+return (\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to
+((drop h (r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h
+(r k0 d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2)))))) with [(Bind b0) \Rightarrow (\lambda (H4: (clear (CHead c (Bind b0)
+t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let
+H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b)
+u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in
+((let H7 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])]))
+(CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2
+(Bind b) u) t H4)) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind
+b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda (H9: (eq B b b0)).(\lambda
+(H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(ex2 C (\lambda (c1:
+C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b)
+(lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) (eq_ind_r C c (\lambda
+(c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind
+b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda (c1: C).(drop h d c1
+c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda (c1: C).(clear (CHead
+x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b1) (lift h d t))))
+(\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda (c1: C).(clear
+(CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b0) (lift h d
+t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x (lift h d t)) H5)
+b H9) c2 H10) u H8)))) H7)) H6)))) | (Flat f) \Rightarrow (\lambda (H4:
+(clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r
+(Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u (clear_gen_flat f c (CHead
+c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: C).(clear x (CHead c1 (Bind
+b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)) (ex2 C (\lambda (c1:
+C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b)
+(lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) (\lambda (x0: C).(\lambda
+(H7: (clear x (CHead x0 (Bind b) (lift h d u)))).(\lambda (H8: (drop h d x0
+c2)).(ex_intro2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat
+f) d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
+c2)) x0 (clear_flat x (CHead x0 (Bind b) (lift h d u)) H7 f (lift h (r (Flat
+f) d) t)) H8)))) H6))))]) H1 H3) x1 H2)))) (drop_gen_skip_r c x1 t h d k
+H0)))))))))))))) x2).
+
+theorem clear_clear:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
+(clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear
+(CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2
+c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear
+(CHead c k t) c2)).((match k return (\lambda (k0: K).((clear (CHead c k0 t)
+c2) \to (clear c2 c2))) with [(Bind b) \Rightarrow (\lambda (H1: (clear
+(CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0:
+C).(clear c0 c0)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H1))) |
+(Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2
+(clear_gen_flat f c c2 t H1)))]) H0))))))) c1).
+
+theorem clear_mono:
+ \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c
+c2) \to (eq C c1 c2)))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n:
+nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2:
+C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1
+c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to
+(\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t)
+c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).((match k
+return (\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0
+t) c2) \to (eq C c1 c2)))) with [(Bind b) \Rightarrow (\lambda (H2: (clear
+(CHead c0 (Bind b) t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t)
+c2)).(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c1 c3))
+(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c3 (CHead c0 (Bind
+b) t))) (refl_equal C (CHead c0 (Bind b) t)) c1 (clear_gen_bind b c0 c1 t
+H2)) c2 (clear_gen_bind b c0 c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2:
+(clear (CHead c0 (Flat f) t) c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t)
+c2)).(H c1 (clear_gen_flat f c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t
+H3))))]) H0 H1))))))))) c).
+
+theorem clear_trans:
+ \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c
+c2) \to (clear c1 c2)))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2:
+C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n)
+c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to
+(\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t)
+c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).((match k return (\lambda
+(k0: K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) with
+[(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3
+\def (eq_ind C c0 (\lambda (c: C).(clear c c2)) H1 (CHead c (Bind b) t)
+(clear_gen_bind b c c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda
+(c3: C).(clear (CHead c (Bind b) t) c3)) (clear_bind b c t) c2
+(clear_gen_bind b c c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear
+(CHead c (Flat f) t) c0)).(clear_flat c c2 (H c0 (clear_gen_flat f c c0 t H2)
+c2 H1) f t))]) H0))))))))) c1).
+
+theorem clear_ctail:
+ \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1
+(CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k
+u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
+K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b)
+u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H:
+(clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1:
+T).(match k return (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead
+(CTail k0 u1 c2) (Bind b) u2))) with [(Bind b0) \Rightarrow (clear_gen_sort
+(CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Bind b0) u1) (CHead
+(CTail (Bind b0) u1 c2) (Bind b) u2))) | (Flat f) \Rightarrow (clear_gen_sort
+(CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Flat f) u1) (CHead (CTail
+(Flat f) u1 c2) (Bind b) u2)))]))))))) (\lambda (c: C).(\lambda (H: ((\forall
+(c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k:
+K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b)
+u2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2:
+T).(\lambda (H0: (clear (CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0:
+K).(\lambda (u1: T).((match k return (\lambda (k1: K).((clear (CHead c k1 t)
+(CHead c2 (Bind b) u2)) \to (clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail
+k0 u1 c2) (Bind b) u2)))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear
+(CHead c (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead
+c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3
+\def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda
+(_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead
+c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b)
+u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow
+t])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead
+c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2
+c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead (CTail k0 u1 c) (Bind b0) t)
+(CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r C c (\lambda (c0: C).(clear
+(CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c0) (Bind b) t)))
+(eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 u1 c) (Bind b1) t)
+(CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail k0 u1 c) t) b0 H5)
+c2 H6) u2 H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear
+(CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 c)
+(CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead c2
+(Bind b) u2) t H1) k0 u1) f t))]) H0)))))))))) c1)).
+
+theorem getl_gen_all:
+ \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2
+C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (getl i c1
+c2)).(let H0 \def (match H return (\lambda (_: (getl ? ? ?)).(ex2 C (\lambda
+(e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2)))) with [(getl_intro e
+H0 H1) \Rightarrow (ex_intro2 C (\lambda (e0: C).(drop i O c1 e0)) (\lambda
+(e0: C).(clear e0 c2)) e H0 H1)]) in H0)))).
+
+theorem getl_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (x: C).((getl h (CSort n) x) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (x: C).(\lambda (H: (getl h
+(CSort n) x)).(\lambda (P: Prop).(let H0 \def (getl_gen_all (CSort n) x h H)
+in (ex2_ind C (\lambda (e: C).(drop h O (CSort n) e)) (\lambda (e: C).(clear
+e x)) P (\lambda (x0: C).(\lambda (H1: (drop h O (CSort n) x0)).(\lambda (H2:
+(clear x0 x)).(and3_ind (eq C x0 (CSort n)) (eq nat h O) (eq nat O O) P
+(\lambda (H3: (eq C x0 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
+(eq nat O O)).(let H6 \def (eq_ind C x0 (\lambda (c: C).(clear c x)) H2
+(CSort n) H3) in (clear_gen_sort x n H6 P))))) (drop_gen_sort n h O x0
+H1))))) H0)))))).
+
+theorem getl_gen_O:
+ \forall (e: C).(\forall (x: C).((getl O e x) \to (clear e x)))
+\def
+ \lambda (e: C).(\lambda (x: C).(\lambda (H: (getl O e x)).(let H0 \def
+(getl_gen_all e x O H) in (ex2_ind C (\lambda (e0: C).(drop O O e e0))
+(\lambda (e0: C).(clear e0 x)) (clear e x) (\lambda (x0: C).(\lambda (H1:
+(drop O O e x0)).(\lambda (H2: (clear x0 x)).(let H3 \def (eq_ind_r C x0
+(\lambda (c: C).(clear c x)) H2 e (drop_gen_refl e x0 H1)) in H3)))) H0)))).
+
+theorem getl_gen_S:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
+nat).((getl (S h) (CHead c k u) x) \to (getl (r k h) c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
+nat).(\lambda (H: (getl (S h) (CHead c k u) x)).(let H0 \def (getl_gen_all
+(CHead c k u) x (S h) H) in (ex2_ind C (\lambda (e: C).(drop (S h) O (CHead c
+k u) e)) (\lambda (e: C).(clear e x)) (getl (r k h) c x) (\lambda (x0:
+C).(\lambda (H1: (drop (S h) O (CHead c k u) x0)).(\lambda (H2: (clear x0
+x)).(getl_intro (r k h) c x x0 (drop_gen_drop k c x0 u h H1) H2)))) H0))))))).
+
+theorem getl_refl:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u)
+(CHead c (Bind b) u))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(getl_intro O (CHead c (Bind
+b) u) (CHead c (Bind b) u) (CHead c (Bind b) u) (drop_refl (CHead c (Bind b)
+u)) (clear_bind b c u)))).
+
+theorem clear_getl_trans:
+ \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to
+(\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3:
+C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1
+c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2
+c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1
+(drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda
+(n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to
+(\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2:
+C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall
+(c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda
+(c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda
+(_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1
+c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3)
+\to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t)
+c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).((match k
+return (\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead
+c k0 t)) \to (getl (S n) c1 c3)))) with [(Bind b) \Rightarrow (\lambda (H3:
+(getl (S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind
+b) t))).(let H5 \def (getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c
+c3 t n H3)) in (ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e:
+C).(clear e c3)) (getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c
+x)).(\lambda (H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1
+c t H4 x n H6) H7)))) H5)))) | (Flat f) \Rightarrow (\lambda (_: (getl (S n)
+(CHead c (Flat f) t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f)
+t))).(clear_gen_flat_r f c1 c t H4 (getl (S n) c1 c3))))]) H1 H2)))))))))
+c2)))) i).
+
+theorem getl_clear_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to
+(\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (getl i c1
+c2)).(\lambda (c3: C).(\lambda (H0: (clear c2 c3)).(let H1 \def (getl_gen_all
+c1 c2 i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e:
+C).(clear e c2)) (getl i c1 c3) (\lambda (x: C).(\lambda (H2: (drop i O c1
+x)).(\lambda (H3: (clear x c2)).(let H4 \def (clear_gen_all x c2 H3) in
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c2
+(CHead e (Bind b) u))))) (getl i c1 c3) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H5: (eq C c2 (CHead x1 (Bind x0) x2))).(let H6
+\def (eq_ind C c2 (\lambda (c: C).(clear x c)) H3 (CHead x1 (Bind x0) x2) H5)
+in (let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c3)) H0 (CHead x1 (Bind
+x0) x2) H5) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl i c1
+c)) (getl_intro i c1 (CHead x1 (Bind x0) x2) x H2 H6) c3 (clear_gen_bind x0
+x1 c3 x2 H7)))))))) H4))))) H1))))))).
+
+theorem getl_head:
+ \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e: C).((getl (r k
+h) c e) \to (\forall (u: T).(getl (S h) (CHead c k u) e))))))
+\def
+ \lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (getl (r k h) c e)).(\lambda (u: T).(let H0 \def (getl_gen_all c e (r k
+h) H) in (ex2_ind C (\lambda (e0: C).(drop (r k h) O c e0)) (\lambda (e0:
+C).(clear e0 e)) (getl (S h) (CHead c k u) e) (\lambda (x: C).(\lambda (H1:
+(drop (r k h) O c x)).(\lambda (H2: (clear x e)).(getl_intro (S h) (CHead c k
+u) e x (drop_drop k h c x H1 u) H2)))) H0))))))).
+
+theorem getl_flat:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall
+(f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c
+e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in
+(ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e))
+(getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c
+x)).(\lambda (H2: (clear x e)).((match h return (\lambda (n: nat).((drop n O
+c x) \to (getl n (CHead c (Flat f) u) e))) with [O \Rightarrow (\lambda (H3:
+(drop O O c x)).(let H4 \def (eq_ind_r C x (\lambda (c: C).(clear c e)) H2 c
+(drop_gen_refl c x H3)) in (getl_intro O (CHead c (Flat f) u) e (CHead c
+(Flat f) u) (drop_refl (CHead c (Flat f) u)) (clear_flat c e H4 f u)))) | (S
+n) \Rightarrow (\lambda (H3: (drop (S n) O c x)).(getl_intro (S n) (CHead c
+(Flat f) u) e x (drop_drop (Flat f) n c x H3 u) H2))]) H1)))) H0))))))).
+
+theorem getl_drop:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
+nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) (CHead e (Bind b)
+u))).(getl_gen_sort n h (CHead e (Bind b) u) H (drop (S h) O (CSort n)
+e))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
+T).(\forall (h: nat).((getl h c0 (CHead e (Bind b) u)) \to (drop (S h) O c0
+e))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c0 k t)
+(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0:
+(getl O (CHead c0 k t) (CHead e (Bind b) u))).(K_ind (\lambda (k0: K).((clear
+(CHead c0 k0 t) (CHead e (Bind b) u)) \to (drop (S O) O (CHead c0 k0 t) e)))
+(\lambda (b0: B).(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind
+b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c]))
+(CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e
+(Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t)
+(clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 \def (f_equal C
+T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind b) u) (CHead c0
+(Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda
+(H5: (eq B b b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r C c0 (\lambda (c1:
+C).(drop (S O) O (CHead c0 (Bind b0) t) c1)) (eq_ind B b (\lambda (b1:
+B).(drop (S O) O (CHead c0 (Bind b1) t) c0)) (drop_drop (Bind b) O c0 c0
+(drop_refl c0) t) b0 H5) e H6)))) H3)) H2)))) (\lambda (f: F).(\lambda (H1:
+(clear (CHead c0 (Flat f) t) (CHead e (Bind b) u))).(drop_clear_O b (CHead c0
+(Flat f) t) e u (clear_flat c0 (CHead e (Bind b) u) (clear_gen_flat f c0
+(CHead e (Bind b) u) t H1) f t) e O (drop_refl e)))) k (getl_gen_O (CHead c0
+k t) (CHead e (Bind b) u) H0))) (\lambda (n: nat).(\lambda (_: (((getl n
+(CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t)
+e)))).(\lambda (H1: (getl (S n) (CHead c0 k t) (CHead e (Bind b)
+u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
+nat).(drop n0 O c0 e)) (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t
+n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
+
+theorem getl_clear_bind:
+ \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c
+(CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2)
+\to (getl (S n) c e2))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
+C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2:
+C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda
+(n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n)
+(CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_:
+(getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0)
+(CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
+C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2:
+C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda
+(k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear
+(CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n:
+nat).(\lambda (H1: (getl n e1 e2)).((match k return (\lambda (k0: K).((clear
+(CHead c0 k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2)))
+with [(Bind b0) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b0) t)
+(CHead e1 (Bind b) v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _)
+\Rightarrow c])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind
+b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e:
+C).(match e return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead
+_ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0
+(Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead e1 (Bind b)
+v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2))
+in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def
+(eq_ind C e1 (\lambda (c: C).(getl n c e2)) H1 c0 H7) in (eq_ind B b (\lambda
+(b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) n c0 e2
+H8 t) b0 H6))))) H4)) H3))) | (Flat f) \Rightarrow (\lambda (H2: (clear
+(CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v
+(clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))])
+H0))))))))))) c)).
+
+theorem getl_ctail:
+ \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v:
+T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind b) u))).(\lambda (k: K).(\lambda
+(v: T).(let H0 \def (getl_gen_all c (CHead d (Bind b) u) i H) in (ex2_ind C
+(\lambda (e: C).(drop i O c e)) (\lambda (e: C).(clear e (CHead d (Bind b)
+u))) (getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)) (\lambda (x:
+C).(\lambda (H1: (drop i O c x)).(\lambda (H2: (clear x (CHead d (Bind b)
+u))).(getl_intro i (CTail k v c) (CHead (CTail k v d) (Bind b) u) (CTail k v
+x) (drop_ctail c x O i H1 k v) (clear_ctail b x d u H2 k v))))) H0))))))))).
+
+theorem getl_ctail_clen:
+ \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n:
+nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
+\def
+ \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex
+nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n)
+(Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O
+(CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b
+(CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl
+(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k:
+K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl
+(clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat
+(\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0)
+(CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen
+c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(match k return
+(\lambda (k0: K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead
+(CTail (Bind b) t c0) k0 t0) (CHead (CSort n) (Bind b) t))))) with [(Bind b0)
+\Rightarrow (ex_intro nat (\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail
+(Bind b) t c0) (Bind b0) t0) (CHead (CSort n) (Bind b) t))) x (getl_head
+(Bind b0) (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) H1
+t0)) | (Flat f) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (clen c0)
+(CHead (CTail (Bind b) t c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x
+(getl_flat (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f
+t0))]))) H0)))))) c))).
+
+theorem getl_dec:
+ \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl i c d) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))
+(\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P)))
+(\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P:
+Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall
+(i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to
+(\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i:
+nat).(match i return (\lambda (n: nat).(or (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl n (CHead c0 k t) d) \to (\forall (P:
+Prop).P))))) with [O \Rightarrow (match k return (\lambda (k0: K).(or (ex_3 C
+B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 k0 t)
+(CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 k0 t) d) \to
+(\forall (P: Prop).P))))) with [(Bind b) \Rightarrow (or_introl (ex_3 C B T
+(\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b)
+t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 (Bind b) t)
+d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda
+(b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e (Bind b0)
+v))))) c0 b t (getl_refl b c0 t))) | (Flat f) \Rightarrow (let H_x \def (H O)
+in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e
+(Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to
+(\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda
+(b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b)
+v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl
+O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O
+(CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0:
+C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind
+x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d:
+C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat
+f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2)
+O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to
+(\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b:
+B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v))))))
+(\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))
+(\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P:
+Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t
+(getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))]) | (S n) \Rightarrow
+(let H_x \def (H (r k n)) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda
+(e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b)
+v)))))) (\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or
+(ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n)
+(CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead
+c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda
+(e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b)
+v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl (r k n) c0 (CHead e (Bind b) v))))) (or (ex_3 C B T (\lambda (e:
+C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind
+b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P:
+Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2:
+(getl (r k n) c0 (CHead x0 (Bind x1) x2))).(or_introl (ex_3 C B T (\lambda
+(e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e
+(Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall
+(P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v:
+T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k
+n c0 (CHead x0 (Bind x1) x2) H2 t))))))) H1)) (\lambda (H1: ((\forall (d:
+C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T
+(\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t)
+(CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to
+(\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H2: (getl (S n) (CHead c0 k
+t) d)).(\lambda (P: Prop).(H1 d (getl_gen_S k c0 d t n H2) P))))))
+H0)))])))))) c).
+
+theorem clear_cle:
+ \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to
+(le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda
+(H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O)))))
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight
+c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
+C).(\lambda (H0: (clear (CHead c k t) c2)).((match k return (\lambda (k0:
+K).((clear (CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight
+t))))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b) t)
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus
+(cweight c) (tweight t)))) (le_n (plus (cweight c) (tweight t))) c2
+(clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow (\lambda (H1: (clear
+(CHead c (Flat f) t) c2)).(le_S_n (cweight c2) (plus (cweight c) (tweight t))
+(le_n_S (cweight c2) (plus (cweight c) (tweight t)) (le_plus_trans (cweight
+c2) (cweight c) (tweight t) (H c2 (clear_gen_flat f c c2 t H1))))))])
+H0))))))) c1).
+
+theorem getl_flt:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to
+(flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b)
+u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef
+i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u:
+T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef
+i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (i: nat).(match i return (\lambda (n: nat).((getl n (CHead c0 k
+t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) with [O
+\Rightarrow (\lambda (H0: (getl O (CHead c0 k t) (CHead e (Bind b)
+u))).((match k return (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e (Bind
+b) u)) \to (flt e u (CHead c0 k0 t) (TLRef O)))) with [(Bind b0) \Rightarrow
+(\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind b) u))).(let H2
+\def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind b)
+u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1))
+in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+b])])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0
+(CHead e (Bind b) u) t H1)) in ((let H4 \def (f_equal C T (\lambda (e0:
+C).(match e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead
+_ _ t) \Rightarrow t])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t)
+(clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b
+b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0
+(CHead c0 (Bind b0) t) (TLRef O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t
+(CHead c0 (Bind b0) t) (TLRef O))) (eq_ind B b (\lambda (b1: B).(flt c0 t
+(CHead c0 (Bind b1) t) (TLRef O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6)
+u H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c0
+(Flat f) t) (CHead e (Bind b) u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0
+(CHead e (Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat
+f) t O))]) (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) | (S n)
+\Rightarrow (\lambda (H0: (getl (S n) (CHead c0 k t) (CHead e (Bind b)
+u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t n
+H0)) in (flt_arith2 e c0 u (r k n) H_y k t (S n))))])))))))) c)).
+
+theorem getl_gen_flat:
+ \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
+\def
+ \lambda (f: F).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Flat f) v) d) \to (getl n
+e d))) (\lambda (H: (getl O (CHead e (Flat f) v) d)).(getl_intro O e d e
+(drop_refl e) (clear_gen_flat f e d v (getl_gen_O (CHead e (Flat f) v) d
+H)))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead e (Flat f) v) d) \to
+(getl n e d)))).(\lambda (H0: (getl (S n) (CHead e (Flat f) v)
+d)).(getl_gen_S (Flat f) e d v n H0)))) i))))).
+
+theorem getl_gen_bind:
+ \forall (b: B).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i (CHead e (Bind b) v) d) \to (or (land (eq nat i O) (eq C d
+(CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda
+(j: nat).(getl j e d)))))))))
+\def
+ \lambda (b: B).(\lambda (e: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(nat_ind (\lambda (n: nat).((getl n (CHead e (Bind b) v) d) \to (or
+(land (eq nat n O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j:
+nat).(eq nat n (S j))) (\lambda (j: nat).(getl j e d)))))) (\lambda (H: (getl
+O (CHead e (Bind b) v) d)).(eq_ind_r C (CHead e (Bind b) v) (\lambda (c:
+C).(or (land (eq nat O O) (eq C c (CHead e (Bind b) v))) (ex2 nat (\lambda
+(j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e c))))) (or_introl
+(land (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind b) v))) (ex2 nat
+(\lambda (j: nat).(eq nat O (S j))) (\lambda (j: nat).(getl j e (CHead e
+(Bind b) v)))) (conj (eq nat O O) (eq C (CHead e (Bind b) v) (CHead e (Bind
+b) v)) (refl_equal nat O) (refl_equal C (CHead e (Bind b) v)))) d
+(clear_gen_bind b e d v (getl_gen_O (CHead e (Bind b) v) d H)))) (\lambda (n:
+nat).(\lambda (_: (((getl n (CHead e (Bind b) v) d) \to (or (land (eq nat n
+O) (eq C d (CHead e (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat n (S
+j))) (\lambda (j: nat).(getl j e d))))))).(\lambda (H0: (getl (S n) (CHead e
+(Bind b) v) d)).(or_intror (land (eq nat (S n) O) (eq C d (CHead e (Bind b)
+v))) (ex2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j: nat).(getl
+j e d))) (ex_intro2 nat (\lambda (j: nat).(eq nat (S n) (S j))) (\lambda (j:
+nat).(getl j e d)) n (refl_equal nat (S n)) (getl_gen_S (Bind b) e d v n
+H0)))))) i))))).
+
+theorem getl_gen_tail:
+ \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall
+(c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2
+(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
+\def
+ \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i
+(CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C
+c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
+n)))))))) (\lambda (n: nat).(\lambda (i: nat).(match i return (\lambda (n0:
+nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n)
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort
+n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2))
+(\lambda (n1: nat).(eq C c2 (CSort n1))))))) with [O \Rightarrow (\lambda (H:
+(getl O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).((match k return
+(\lambda (k0: K).((clear (CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to
+(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O
+(CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O))
+(\lambda (_: nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2))
+(\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [(Bind b0) \Rightarrow
+(\lambda (H0: (clear (CHead (CSort n) (Bind b0) u1) (CHead c2 (Bind b)
+u2))).(let H1 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead
+c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n)
+(CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e:
+C).(match e return (\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead
+_ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead
+(CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2)
+u1 H0)) in ((let H3 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow
+t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0
+(CSort n) (CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b
+b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c:
+C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e:
+C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
+nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq
+T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda
+(t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e)))
+(\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda
+(_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda
+(_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))))))
+(eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n)
+(CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1)
+u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind
+b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C
+(CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n)
+(CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0)
+u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind
+b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C
+(CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O))
+(\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1))
+(\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat O)
+(refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b H4)
+u2 H3) c2 H5)))) H2)) H1))) | (Flat f) \Rightarrow (\lambda (H0: (clear
+(CHead (CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead
+c2 (Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0)
+(or (ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e:
+C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
+nat O O)) (\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T
+u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))]) (getl_gen_O (CHead
+(CSort n) k u1) (CHead c2 (Bind b) u2) H))) | (S n0) \Rightarrow (\lambda (H:
+(getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n
+(r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k (CSort n) (CHead c2 (Bind b)
+u2) u1 n0 H) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda
+(e: C).(getl (S n0) (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1)))))))]))) (\lambda
+(c: C).(\lambda (H: ((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2
+(Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e)))
+(\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0:
+K).(\lambda (t: T).(\lambda (i: nat).(match i return (\lambda (n: nat).((getl
+n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda
+(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t)
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (clen (CHead c
+k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2))
+(\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [O \Rightarrow (\lambda
+(H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).((match k0
+return (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 (Bind
+b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
+C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))
+with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Bind
+b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead
+c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind
+b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in
+((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])]))
+(CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0
+(CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 \def (f_equal C T
+(\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead
+(CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind
+b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k
+u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2
+(CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n: nat).(eq C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C
+(\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
+(Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O
+(s (Bind b1) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def
+(eq_ind C c2 (\lambda (c0: C).(\forall (i: nat).((getl i (CTail k u1 c)
+(CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1
+e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1
+c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e:
+C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2)
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b)
+(clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
+u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e:
+C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c
+(Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s
+(Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_:
+nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n))))
+(ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda
+(e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C
+(CTail k u1 c)) (getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2))) |
+(Flat f) \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t)
+(CHead c2 (Bind b) u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c)
+(CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl (CTail k u1 c))
+(clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t H1))) in (or_ind
+(ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c
+(CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (clen c)))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k
+u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))))
+(ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_:
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq
+C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
+e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda
+(e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b)
+u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
+C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda
+(_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))
+(\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O
+c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or
+(ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O
+(CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq
+nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda
+(_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl
+(ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e:
+C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda
+(_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x)
+(CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1
+e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x
+(refl_equal C (CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t)))
+c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c)))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O
+(clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1
+u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C
+c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n: nat).(eq C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O
+(clen c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1
+u2)).(\lambda (H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0:
+C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl
+O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n))))))
+(eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0)
+(CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind
+b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda
+(n: nat).(eq C (CSort x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1:
+K).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e:
+C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda
+(_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind
+b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0)
+(CSort n)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail
+(Bind b) u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind
+b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c))))
+(\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1))
+(\lambda (n: nat).(eq C (CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_:
+nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind
+b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0)
+(CSort n))) x0 H4 (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C
+(CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) H2)))]) (getl_gen_O (CHead (CTail
+k u1 c) k0 t) (CHead c2 (Bind b) u2) H0))) | (S n) \Rightarrow (\lambda (H0:
+(getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x
+\def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n
+H0)) in (let H1 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k
+u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind
+b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort
+n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))
+(\lambda (H2: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
+C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq
+C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b)
+u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))
+(\lambda (x: C).(\lambda (H3: (eq C c2 (CTail k u1 x))).(\lambda (H4: (getl
+(r k0 n) c (CHead x (Bind b) u2))).(let H5 \def (eq_ind C c2 (\lambda (c0:
+C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0
+(CTail k u1 c) (CHead c2 (Bind b) u2) t n H0) (CTail k u1 x) H3) in (eq_ind_r
+C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail
+k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))))
+(ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_:
+nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
+nat).(eq C c0 (CSort n0)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail
+k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e
+(Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c))))
+(\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda
+(n0: nat).(eq C (CTail k u1 x) (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq
+C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t)
+(CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_head k0 n c
+(CHead x (Bind b) u2) H4 t))) c2 H3))))) H2)) (\lambda (H2: (ex4 nat (\lambda
+(_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort
+n))))).(ex4_ind nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda
+(_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0:
+nat).(eq C c2 (CSort n0))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1
+e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4
+nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K
+k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2
+(CSort n0))))) (\lambda (x0: nat).(\lambda (H3: (eq nat (r k0 n) (clen
+c))).(\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq T u1 u2)).(\lambda
+(H6: (eq C c2 (CSort x0))).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(getl
+(r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1
+c) (CHead c2 (Bind b) u2) t n H0) (CSort x0) H6) in (eq_ind_r C (CSort x0)
+(\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e)))
+(\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat
+(\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k
+(Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort
+n0)))))) (let H8 \def (eq_ind_r T u2 (\lambda (t: T).(getl (r k0 n) (CTail k
+u1 c) (CHead (CSort x0) (Bind b) t))) H7 u1 H5) in (eq_ind T u1 (\lambda (t0:
+T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b)))
+(\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
+n0)))))) (let H9 \def (eq_ind K k (\lambda (k: K).(getl (r k0 n) (CTail k u1
+c) (CHead (CSort x0) (Bind b) u1))) H8 (Bind b) H4) in (eq_ind_r K (Bind b)
+(\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1
+e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4
+nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K
+k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort
+x0) (CSort n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C
+(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort
+n1)))))) (eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e:
+C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n)
+(CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S
+n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T
+u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C
+(\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e:
+C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_:
+nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b)))
+(\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort
+n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_:
+nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0:
+nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K
+(Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r
+k0 n)) (clen c) H3) k H4)) u2 H5)) c2 H6))))))) H2)) H1))))])))))) c1)))))).
+
+theorem cimp_flat_sx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v)
+c)))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f)
+v) (CHead d1 (Bind b) w))).((match h return (\lambda (n: nat).((getl n (CHead
+c (Flat f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c
+(CHead d2 (Bind b) w)))))) with [O \Rightarrow (\lambda (H0: (getl O (CHead c
+(Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2: C).(getl O c
+(CHead d2 (Bind b) w))) d1 (getl_intro O c (CHead d1 (Bind b) w) c (drop_refl
+c) (clear_gen_flat f c (CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f)
+v) (CHead d1 (Bind b) w) H0))))) | (S n) \Rightarrow (\lambda (H0: (getl (S
+n) (CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2:
+C).(getl (S n) c (CHead d2 (Bind b) w))) d1 (getl_gen_S (Flat f) c (CHead d1
+(Bind b) w) v n H0)))]) H)))))))).
+
+theorem cimp_flat_dx:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f)
+v))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
+C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind
+b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2
+(Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
+
+theorem cimp_bind:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
+(v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
+(b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w:
+T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1
+(Bind b0) w))).((match h return (\lambda (n: nat).((getl n (CHead c1 (Bind b)
+v) (CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind
+b) v) (CHead d2 (Bind b0) w)))))) with [O \Rightarrow (\lambda (H1: (getl O
+(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) w) (CHead
+c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
+(CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H3 \def (f_equal
+C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow b0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind
+b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v
+(getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0)
+w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v
+(getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda
+(H5: (eq B b0 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t:
+T).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0)
+t))))) (eq_ind_r B b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead
+c2 (Bind b) v) (CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl
+O (CHead c2 (Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5)
+w H4)))) H3)) H2))) | (S n) \Rightarrow (\lambda (H1: (getl (S n) (CHead c1
+(Bind b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) n)
+(getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v n H1)) in (let H2 \def H_x
+in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) n) c2 (CHead d2 (Bind b0)
+w))) (ex C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) (CHead d2 (Bind
+b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) n) c2 (CHead x
+(Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v)
+(CHead d2 (Bind b0) w))) x (getl_head (Bind b) n c2 (CHead x (Bind b0) w) H3
+v)))) H2))))]) H0)))))))))).
+
+theorem getl_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to
+(\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
+\def
+ \lambda (c: C).(\lambda (x1: C).(\lambda (h: nat).(\lambda (H: (getl h c
+x1)).(\lambda (x2: C).(\lambda (H0: (getl h c x2)).(let H1 \def (getl_gen_all
+c x2 h H0) in (ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e:
+C).(clear e x2)) (eq C x1 x2) (\lambda (x: C).(\lambda (H2: (drop h O c
+x)).(\lambda (H3: (clear x x2)).(let H4 \def (getl_gen_all c x1 h H) in
+(ex2_ind C (\lambda (e: C).(drop h O c e)) (\lambda (e: C).(clear e x1)) (eq
+C x1 x2) (\lambda (x0: C).(\lambda (H5: (drop h O c x0)).(\lambda (H6: (clear
+x0 x1)).(let H7 \def (eq_ind C x (\lambda (c0: C).(drop h O c c0)) H2 x0
+(drop_mono c x O h H2 x0 H5)) in (let H8 \def (eq_ind_r C x0 (\lambda (c0:
+C).(drop h O c c0)) H7 x (drop_mono c x O h H2 x0 H5)) in (let H9 \def
+(eq_ind_r C x0 (\lambda (c: C).(clear c x1)) H6 x (drop_mono c x O h H2 x0
+H5)) in (clear_mono x x1 H9 x2 H3))))))) H4))))) H1))))))).
+
+theorem getl_clear_conf:
+ \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to
+(\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3:
+C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2
+c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1
+c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c:
+C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in
+(ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3
+(CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3
+\def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead
+x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c:
+C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3
+(getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall
+(c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2)
+\to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall
+(c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n)
+c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n)
+(CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0)
+c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2:
+C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda
+(c2: C).(\lambda (H2: (clear (CHead c k t) c2)).((match k return (\lambda
+(k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to
+(getl (S n) c2 c3)))) with [(Bind b) \Rightarrow (\lambda (H3: (getl (S n)
+(CHead c (Bind b) t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t)
+c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3))
+(getl_head (Bind b) n c c3 (getl_gen_S (Bind b) c c3 t n H3) t) c2
+(clear_gen_bind b c c2 t H4)))) | (Flat f) \Rightarrow (\lambda (H3: (getl (S
+n) (CHead c (Flat f) t) c3)).(\lambda (H4: (clear (CHead c (Flat f) t)
+c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n H3) c2 (clear_gen_flat f c c2 t
+H4))))]) H1 H2))))))))) c1)))) i).
+
+theorem getl_drop_conf_lt:
+ \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c0 e0)))))))))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1:
+C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d))
+c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i
+(CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i
+(CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda
+(c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i:
+nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t)
+(CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def
+(getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C
+(\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0
+(CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x:
+C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead
+c1 (Bind b) u))).((match x return (\lambda (c2: C).((drop i O (CHead c0 k t)
+c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O (CHead c0
+k t) (CSort n))).(\lambda (H6: (clear (CSort n) (CHead c1 (Bind b)
+u))).(clear_gen_sort (CHead c1 (Bind b) u) n H6 (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h d c1 e0))))))) | (CHead c2 k0 t0) \Rightarrow (\lambda (H5: (drop i O
+(CHead c0 k t) (CHead c2 k0 t0))).(\lambda (H6: (clear (CHead c2 k0 t0)
+(CHead c1 (Bind b) u))).((match k0 return (\lambda (k1: K).((drop i O (CHead
+c0 k t) (CHead c2 k1 t0)) \to ((clear (CHead c2 k1 t0) (CHead c1 (Bind b) u))
+\to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [(Bind b0) \Rightarrow
+(\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))).(\lambda
+(H8: (clear (CHead c2 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind b)
+u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0
+H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0)
+(clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def
+(f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind b)
+u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0
+H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 c2)).(let H14
+\def (eq_ind_r T t0 (\lambda (t0: T).(drop i O (CHead c0 k t) (CHead c2 (Bind
+b0) t0))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b: B).(drop i O
+(CHead c0 k t) (CHead c2 (Bind b) u))) H14 b H12) in (let H16 \def (eq_ind_r
+C c2 (\lambda (c: C).(drop i O (CHead c0 k t) (CHead c (Bind b) u))) H15 c1
+H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r
+(Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e (CHead e0
+(Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r (Bind b) d) c1
+e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (H17: (eq T u (lift h (r (Bind b) d) x0))).(\lambda (H18: (drop i
+O e (CHead x1 (Bind b) x0))).(\lambda (H19: (drop h (r (Bind b) d) c1
+x1)).(eq_ind_r T (lift h (r (Bind b) d) x0) (\lambda (t1: T).(ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x0) (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))) x0 x1 (refl_equal T (lift h d x0))
+(getl_intro i e (CHead x1 (Bind b) x0) (CHead x1 (Bind b) x0) H18 (clear_bind
+b x1 x0)) H19) u H17)))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H16 e
+h d H1)))))))) H10)) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (drop i O
+(CHead c0 k t) (CHead c2 (Flat f) t0))).(\lambda (H8: (clear (CHead c2 (Flat
+f) t0) (CHead c1 (Bind b) u))).((match i return (\lambda (n: nat).((drop h (S
+(plus n d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead c2 (Flat
+f) t0)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [O \Rightarrow
+(\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) e)).(\lambda (H10: (drop
+O O (CHead c0 k t) (CHead c2 (Flat f) t0))).(let H11 \def (f_equal C C
+(\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t) (CHead c2
+(Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in
+((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead
+c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat
+f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow
+t])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t)
+(CHead c2 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat f))).(\lambda
+(H15: (eq C c0 c2)).(let H16 \def (eq_ind_r C c2 (\lambda (c: C).(clear c
+(CHead c1 (Bind b) u))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) c0
+H15) in (let H17 \def (eq_ind K k (\lambda (k: K).(drop h (S (plus O d))
+(CHead c0 k t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0:
+C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda
+(v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H18: (eq C e (CHead x0 (Flat f) x1))).(\lambda (H19: (eq T t
+(lift h (r (Flat f) (plus O d)) x1))).(\lambda (H20: (drop h (r (Flat f)
+(plus O d)) c0 x0)).(let H21 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r (Flat f) (plus O d)) x1) H19) in (eq_ind_r C (CHead x0 (Flat f) x1)
+(\lambda (c3: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d
+v)))) (\lambda (v: T).(\lambda (e0: C).(getl O c3 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H22 \def (H c1 u O
+(getl_intro O c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x0 h d H20) in
+(ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl O x0 (CHead e0 (Bind b) v)))) (\lambda
+(_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H23: (eq T u (lift h d x2))).(\lambda (H24: (getl O x0 (CHead x3
+(Bind b) x2))).(\lambda (H25: (drop h d c1 x3)).(let H26 \def (eq_ind T u
+(\lambda (t: T).(clear c0 (CHead c1 (Bind b) t))) H16 (lift h d x2) H23) in
+(eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift
+h d x2)) (getl_flat x0 (CHead x3 (Bind b) x2) O H24 f x1) H25) u H23)))))))
+H22)) e H18))))))) (drop_gen_skip_l c0 e t h (plus O d) (Flat f) H17)))))))
+H12)) H11)))) | (S n) \Rightarrow (\lambda (H9: (drop h (S (plus (S n) d))
+(CHead c0 k t) e)).(\lambda (H10: (drop (S n) O (CHead c0 k t) (CHead c2
+(Flat f) t0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead
+e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S n)
+d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S n) d)) c0
+e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl (S n) e (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H11: (eq C e (CHead x0 k x1))).(\lambda (H12:
+(eq T t (lift h (r k (plus (S n) d)) x1))).(\lambda (H13: (drop h (r k (plus
+(S n) d)) c0 x0)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h
+(r k (plus (S n) d)) x1) H12) in (eq_ind_r C (CHead x0 k x1) (\lambda (c3:
+C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v))))
+(\lambda (v: T).(\lambda (e0: C).(getl (S n) c3 (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H15 \def (eq_ind
+nat (r k (plus (S n) d)) (\lambda (n: nat).(drop h n c0 x0)) H13 (plus (r k
+(S n)) d) (r_plus k (S n) d)) in (let H16 \def (eq_ind nat (r k (S n))
+(\lambda (n: nat).(drop h (plus n d) c0 x0)) H15 (S (r k n)) (r_S k n)) in
+(let H17 \def (H c1 u (r k n) (getl_intro (r k n) c0 (CHead c1 (Bind b) u)
+(CHead c2 (Flat f) t0) (drop_gen_drop k c0 (CHead c2 (Flat f) t0) t n H10)
+(clear_flat c2 (CHead c1 (Bind b) u) (clear_gen_flat f c2 (CHead c1 (Bind b)
+u) t0 H8) f t0)) x0 h d H16) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k n) x0
+(CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda
+(v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2:
+T).(\lambda (x3: C).(\lambda (H18: (eq T u (lift h d x2))).(\lambda (H19:
+(getl (r k n) x0 (CHead x3 (Bind b) x2))).(\lambda (H20: (drop h d c1
+x3)).(let H21 \def (eq_ind T u (\lambda (t: T).(clear c2 (CHead c1 (Bind b)
+t))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) (lift h d x2) H18) in
+(eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift
+h d x2)) (getl_head k n x0 (CHead x3 (Bind b) x2) H19 x1) H20) u H18)))))))
+H17)))) e H11))))))) (drop_gen_skip_l c0 e t h (plus (S n) d) k H9))))]) H1
+H7)))]) H5 H6)))]) H3 H4)))) H2)))))))))))))) c)).
+
+theorem getl_drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d
+h) i) \to (getl (minus i h) e a)))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
+d c e)).(\lambda (H1: (le (plus d h) i)).(let H2 \def (getl_gen_all c a i H)
+in (ex2_ind C (\lambda (e0: C).(drop i O c e0)) (\lambda (e0: C).(clear e0
+a)) (getl (minus i h) e a) (\lambda (x: C).(\lambda (H3: (drop i O c
+x)).(\lambda (H4: (clear x a)).(getl_intro (minus i h) e a x (drop_conf_ge i
+x c H3 e h d H0 H1) H4)))) H2)))))))))).
+
+theorem getl_conf_ge_drop:
+ \forall (b: B).(\forall (c1: C).(\forall (e: C).(\forall (u: T).(\forall (i:
+nat).((getl i c1 (CHead e (Bind b) u)) \to (\forall (c2: C).((drop (S O) i c1
+c2) \to (drop i O c2 e))))))))
+\def
+ \lambda (b: B).(\lambda (c1: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda
+(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O))
+(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e
+u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
+i))) (le_n (S i)) (plus i (S O)) (plus_comm i (S O)))) i (minus_Sx_SO i)) in
+H3)))))))).
+
+theorem getl_conf_le:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
+(e: C).(\forall (h: nat).((getl h c e) \to ((le h i) \to (getl (minus i h) e
+a))))))))
+\def
+ \lambda (i: nat).(\lambda (a: C).(\lambda (c: C).(\lambda (H: (getl i c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (H0: (getl h c e)).(\lambda
+(H1: (le h i)).(let H2 \def (getl_gen_all c e h H0) in (ex2_ind C (\lambda
+(e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl (minus i h) e
+a) (\lambda (x: C).(\lambda (H3: (drop h O c x)).(\lambda (H4: (clear x
+e)).(getl_clear_conf (minus i h) x a (getl_drop_conf_ge i a c H x h O H3 H1)
+e H4)))) H2))))))))).
+
+theorem getl_drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
+(\forall (b: B).(\forall (c2: C).(\forall (v2: T).(\forall (i: nat).((getl i
+c2 (CHead e2 (Bind b) v2)) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2))
+(\lambda (c1: C).(drop (S i) j c1 e1)))))))))))
+\def
+ \lambda (j: nat).(\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop j O e1
+e2)).(\lambda (b: B).(\lambda (c2: C).(\lambda (v2: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c2 (CHead e2 (Bind b) v2))).(drop_conf_rev j e1 e2
+H c2 (S i) (getl_drop b c2 e2 v2 i H0)))))))))).
+
+theorem drop_getl_trans_lt:
+ \forall (i: nat).(\forall (d: nat).((lt i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (b: B).(\forall (e2:
+C).(\forall (v: T).((getl i c2 (CHead e2 (Bind b) v)) \to (ex2 C (\lambda
+(e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda
+(e1: C).(drop h (minus d (S i)) e1 e2)))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (lt i d)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
+c2)).(\lambda (b: B).(\lambda (e2: C).(\lambda (v: T).(\lambda (H1: (getl i
+c2 (CHead e2 (Bind b) v))).(let H2 \def (getl_gen_all c2 (CHead e2 (Bind b)
+v) i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e:
+C).(clear e (CHead e2 (Bind b) v))) (ex2 C (\lambda (e1: C).(getl i c1 (CHead
+e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h (minus d
+(S i)) e1 e2))) (\lambda (x: C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4:
+(clear x (CHead e2 (Bind b) v))).(ex2_ind C (\lambda (e1: C).(drop i O c1
+e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex2 C (\lambda (e1:
+C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1:
+C).(drop h (minus d (S i)) e1 e2))) (\lambda (x0: C).(\lambda (H5: (drop i O
+c1 x0)).(\lambda (H6: (drop h (minus d i) x0 x)).(let H7 \def (eq_ind nat
+(minus d i) (\lambda (n: nat).(drop h n x0 x)) H6 (S (minus d (S i)))
+(minus_x_Sy d i H)) in (let H8 \def (drop_clear_S x x0 h (minus d (S i)) H7 b
+e2 v H4) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind b) (lift h
+(minus d (S i)) v)))) (\lambda (c3: C).(drop h (minus d (S i)) c3 e2)) (ex2 C
+(\lambda (e1: C).(getl i c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v))))
+(\lambda (e1: C).(drop h (minus d (S i)) e1 e2))) (\lambda (x1: C).(\lambda
+(H9: (clear x0 (CHead x1 (Bind b) (lift h (minus d (S i)) v)))).(\lambda
+(H10: (drop h (minus d (S i)) x1 e2)).(ex_intro2 C (\lambda (e1: C).(getl i
+c1 (CHead e1 (Bind b) (lift h (minus d (S i)) v)))) (\lambda (e1: C).(drop h
+(minus d (S i)) e1 e2)) x1 (getl_intro i c1 (CHead x1 (Bind b) (lift h (minus
+d (S i)) v)) x0 H5 H9) H10)))) H8)))))) (drop_trans_le i d (le_S_n i d (le_S
+(S i) d H)) c1 c2 h H0 x H3))))) H2)))))))))))).
+
+theorem drop_getl_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2
+e2) \to (ex3_2 C C (\lambda (e0: C).(\lambda (_: C).(drop i O c1 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 e2))))))))))))
+\def
+ \lambda (i: nat).(\lambda (d: nat).(\lambda (H: (le i d)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1
+c2)).(\lambda (e2: C).(\lambda (H1: (getl i c2 e2)).(let H2 \def
+(getl_gen_all c2 e2 i H1) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
+(\lambda (e: C).(clear e e2)) (ex3_2 C C (\lambda (e0: C).(\lambda (_:
+C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i)
+e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 e2)))) (\lambda (x:
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(let H5 \def
+(drop_trans_le i d H c1 c2 h H0 x H3) in (ex2_ind C (\lambda (e1: C).(drop i
+O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 x)) (ex3_2 C C (\lambda
+(e0: C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1:
+C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1
+e2)))) (\lambda (x0: C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h
+(minus d i) x0 x)).(ex3_2_intro C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 e2))) x0 x H6 H7 H4)))) H5)))))
+H2)))))))))).
+
+theorem drop_getl_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((getl i c2 e2)
+\to ((le d i) \to (getl (plus i h) c1 e2)))))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2:
+C).(\lambda (H0: (getl i c2 e2)).(\lambda (H1: (le d i)).(let H2 \def
+(getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e))
+(\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x:
+C).(\lambda (H3: (drop i O c2 x)).(\lambda (H4: (clear x e2)).(getl_intro
+(plus i h) c1 e2 x (drop_trans_ge i c1 c2 d h H x H3 H1) H4)))) H2)))))))))).
+
+theorem getl_drop_trans:
+ \forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c2 e2) \to (drop (S (plus i
+h)) O c1 e2)))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
+nat).((getl h c c2) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c2
+e2) \to (drop (S (plus i h)) O c e2)))))))) (\lambda (n: nat).(\lambda (c2:
+C).(\lambda (h: nat).(\lambda (H: (getl h (CSort n) c2)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (_: (drop (S i) O c2 e2)).(getl_gen_sort n h c2
+H (drop (S (plus i h)) O (CSort n) e2))))))))) (\lambda (c2: C).(\lambda
+(IHc: ((\forall (c3: C).(\forall (h: nat).((getl h c2 c3) \to (\forall (e2:
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O c2
+e2))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall
+(c3: C).(\forall (h: nat).((getl h (CHead c2 k0 t) c3) \to (\forall (e2:
+C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i h)) O (CHead
+c2 k0 t) e2))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((getl n (CHead c2 (Bind b)
+t) c3) \to (\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop
+(S (plus i n)) O (CHead c2 (Bind b) t) e2)))))) (\lambda (H: (getl O (CHead
+c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H0: (drop (S
+i) O c3 e2)).(let H1 \def (eq_ind C c3 (\lambda (c: C).(drop (S i) O c e2))
+H0 (CHead c2 (Bind b) t) (clear_gen_bind b c2 c3 t (getl_gen_O (CHead c2
+(Bind b) t) c3 H))) in (eq_ind nat i (\lambda (n: nat).(drop (S n) O (CHead
+c2 (Bind b) t) e2)) (drop_drop (Bind b) i c2 e2 (drop_gen_drop (Bind b) c2 e2
+t i H1) t) (plus i O) (plus_n_O i))))))) (\lambda (n: nat).(\lambda (_:
+(((getl n (CHead c2 (Bind b) t) c3) \to (\forall (e2: C).(\forall (i:
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Bind b) t)
+e2))))))).(\lambda (H0: (getl (S n) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H1: (drop (S i) O c3 e2)).(eq_ind nat (plus (S
+i) n) (\lambda (n0: nat).(drop (S n0) O (CHead c2 (Bind b) t) e2)) (drop_drop
+(Bind b) (plus (S i) n) c2 e2 (IHc c3 n (getl_gen_S (Bind b) c2 c3 t n H0) e2
+i H1) t) (plus i (S n)) (plus_Snm_nSm i n)))))))) h))))) (\lambda (f:
+F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((getl n (CHead c2 (Flat f) t) c3) \to (\forall (e2: C).(\forall (i:
+nat).((drop (S i) O c3 e2) \to (drop (S (plus i n)) O (CHead c2 (Flat f) t)
+e2)))))) (\lambda (H: (getl O (CHead c2 (Flat f) t) c3)).(\lambda (e2:
+C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O c3 e2)).(drop_drop (Flat f)
+(plus i O) c2 e2 (IHc c3 O (getl_intro O c2 c3 c2 (drop_refl c2)
+(clear_gen_flat f c2 c3 t (getl_gen_O (CHead c2 (Flat f) t) c3 H))) e2 i H0)
+t))))) (\lambda (n: nat).(\lambda (_: (((getl n (CHead c2 (Flat f) t) c3) \to
+(\forall (e2: C).(\forall (i: nat).((drop (S i) O c3 e2) \to (drop (S (plus i
+n)) O (CHead c2 (Flat f) t) e2))))))).(\lambda (H0: (getl (S n) (CHead c2
+(Flat f) t) c3)).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop (S i)
+O c3 e2)).(drop_drop (Flat f) (plus i (S n)) c2 e2 (IHc c3 (S n) (getl_gen_S
+(Flat f) c2 c3 t n H0) e2 i H1) t))))))) h))))) k)))) c1).
+
+theorem getl_trans:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl
+h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1
+e2)))))))
+\def
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2
+e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e:
+C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2)
+(\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x
+e2)).((match i return (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h)
+c1 e2))) with [O \Rightarrow (\lambda (H4: (drop O O c2 x)).(let H5 \def
+(eq_ind_r C x (\lambda (c: C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in
+(getl_clear_trans (plus O h) c1 c2 H e2 H5))) | (S n) \Rightarrow (\lambda
+(H4: (drop (S n) O c2 x)).(let H_y \def (getl_drop_trans c1 c2 h H x n H4) in
+(getl_intro (plus (S n) h) c1 e2 x H_y H3)))]) H2)))) H1)))))))).
+
+theorem drop1_getl_trans:
+ \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
+i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2
+(CHead e2 (Bind b) (ctrans hds i v)))))))))))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
+C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
+C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
+(ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i
+v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2
+c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H
+return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_:
+(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex
+C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2:
+(eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C
+c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))
+(\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2:
+C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i
+c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
+H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3:
+(eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
+(eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
+PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False
+| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c2)
+\to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (ex C
+(\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1
+H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C
+c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0:
+PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 c1)
+\to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
+i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds0 i)
+c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda
+(b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl
+i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0 return (\lambda (p:
+PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
+PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex C
+(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
+(Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus
+d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i
+v)])))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil
+(PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
+c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return
+(\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
+\Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to
+((eq C c c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d)
+with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0
+i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true
+\Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false
+\Rightarrow (ctrans hds0 i v)]))))))) H5)) H3 H4)))) | (drop1_cons c0 c3 h0
+d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds0)
+(PCons h d hds0))).(\lambda (H5: (eq C c0 c2)).(\lambda (H6: (eq C c4
+c1)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return
+(\lambda (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p)
+\Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H8 \def
+(f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_:
+PList).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons
+h0 d0 hds0) (PCons h d hds0) H4) in ((let H9 \def (f_equal PList nat (\lambda
+(e: PList).(match e return (\lambda (_: PList).nat) with [PNil \Rightarrow h0
+| (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in
+(eq_ind nat h (\lambda (n: nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to
+((eq C c0 c2) \to ((eq C c4 c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3
+c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift
+h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans
+hds0 i v)])))))))))))) (\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda
+(n: nat).((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop
+h n c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match
+(blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
+\Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt
+(trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i)))
+(ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda
+(H11: (eq PList hds0 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0
+c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C
+(\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow
+(trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
+(Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus
+d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i
+v)])))))))))) (\lambda (H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq
+C c4 c1) \to ((drop h d c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2:
+C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
+| false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match
+(blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))
+(\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3)
+\to ((drop1 hds0 c3 c) \to (ex C (\lambda (e2: C).(getl (match (blt (trans
+hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus
+(trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with
+[true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) |
+false \Rightarrow (ctrans hds0 i v)])))))))) (\lambda (H14: (drop h d c2
+c3)).(\lambda (H15: (drop1 hds0 c3 c1)).(xinduction bool (blt (trans hds0 i)
+d) (\lambda (b0: bool).(ex C (\lambda (e2: C).(getl (match b0 with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S
+(trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i
+v)])))))) (\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt
+(trans hds0 i) d) b0) \to (ex C (\lambda (e2: C).(getl (match b0 with [true
+\Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2
+(CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S
+(trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i
+v)]))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def
+(H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2:
+C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C
+(\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d
+(S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x: C).(\lambda (H17:
+(getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H_x0 \def
+(drop_getl_trans_lt (trans hds0 i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S
+(trans hds0 i)) (S d) (blt_lt (S d) (S (trans hds0 i)) H0))) c2 c3 h H14 b x
+(ctrans hds0 i v) H17) in (let H \def H_x0 in (ex2_ind C (\lambda (e1:
+C).(getl (trans hds0 i) c2 (CHead e1 (Bind b) (lift h (minus d (S (trans hds0
+i))) (ctrans hds0 i v))))) (\lambda (e1: C).(drop h (minus d (S (trans hds0
+i))) e1 x)) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b)
+(lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x0:
+C).(\lambda (H1: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h (minus d
+(S (trans hds0 i))) (ctrans hds0 i v))))).(\lambda (_: (drop h (minus d (S
+(trans hds0 i))) x0 x)).(ex_intro C (\lambda (e2: C).(getl (trans hds0 i) c2
+(CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))
+x0 H1)))) H))))) H16)))) (\lambda (H0: (eq bool (blt (trans hds0 i) d)
+false)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in
+(ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans
+hds0 i v)))) (ex C (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead
+e2 (Bind b) (ctrans hds0 i v))))) (\lambda (x: C).(\lambda (H17: (getl (trans
+hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H \def
+(drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (ctrans
+hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus (trans hds0 i) h)
+c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d (trans hds0 i)
+H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2 H12)))
+hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0 (sym_eq nat
+h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons h d
+hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).
+
+theorem cimp_getl_conf:
+ \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
+(d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w))
+\to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind b) w)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
+C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
+(ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
+(b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def
+H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C
+(\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall
+(h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4:
+C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x
+(Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3:
+C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0))
+\to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0))))))))))
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0:
+B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h
+d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1
+(Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0
+(Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in
+(let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (plus (S h) i) c2
+(CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind
+b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (plus (S h) i) c2 (CHead x0
+(Bind b0) w0))).(let H_y0 \def (getl_conf_le (plus (S h) i) (CHead x0 (Bind
+b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (eq_ind nat (minus
+(plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind b) w) (CHead x0
+(Bind b0) w0))) (H_y0 (le_plus_r (S h) i)) (S h) (minus_plus_r (S h) i)) in
+(ex_intro C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0
+(getl_gen_S (Bind b) x (CHead x0 (Bind b0) w0) w h H6)))))) H4))))))))) H2)))
+H1)))))))))).
inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
-| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift (S i) O v)))
-| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v (THead k u1 t) (THead k u2 t))))))))
-| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i v (THead k u t1) (THead k u t2))))))))
-| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2 t2)))))))))).
-
-axiom subst0_gen_sort: \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 i v (TSort n) x) \to (\forall (P: Prop).P))))) .
-
-axiom subst0_gen_lref: \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) .
-
-axiom subst0_gen_head: \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))))) .
-
-axiom subst0_refl: \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to (\forall (P: Prop).P)))) .
-
-axiom subst0_gen_lift_lt: \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t1 t2))))))))) .
-
-axiom subst0_gen_lift_false: \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u (lift h d t) x) \to (\forall (P: Prop).P))))))))) .
-
-axiom subst0_gen_lift_ge: \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t1 t2)))))))))) .
-
-axiom subst0_lift_lt: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) .
-
-axiom subst0_lift_ge: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall (h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (plus i h) u (lift h d t1) (lift h d t2))))))))) .
-
-axiom subst0_lift_ge_S: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d t1) (lift (S O) d t2)))))))) .
-
-axiom subst0_lift_ge_s: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s (Bind b) i) u (lift (S O) d t1) (lift (S O) d t2))))))))) .
-
-axiom subst0_subst0: \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t t2))))))))))) .
-
-axiom subst0_subst0_back: \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0 j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u t2 t))))))))))) .
-
-axiom subst0_trans: \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0 i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1 t3))))))) .
-
-axiom subst0_confluence_neq: \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t)))))))))))) .
-
-axiom subst0_confluence_eq: \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2) (ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t1 t2) (subst0 i u t2 t1)))))))) .
-
-axiom subst0_confluence_lift: \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0 i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i t2)) \to (eq T t1 t2))))))) .
-
-axiom subst0_weight_le: \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t)))))))))) .
-
-axiom subst0_weight_lt: \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t)))))))))) .
-
-axiom subst0_tlt_head: \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt (THead (Bind Abbr) u z) (THead (Bind Abbr) u t))))) .
-
-axiom subst0_tlt: \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z (THead (Bind Abbr) u t))))) .
-
-axiom dnf_dec: \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or (subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v))))))) .
+| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift
+(S i) O v)))
+| subst0_fst: \forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i:
+nat).((subst0 i v u1 u2) \to (\forall (t: T).(\forall (k: K).(subst0 i v
+(THead k u1 t) (THead k u2 t))))))))
+| subst0_snd: \forall (k: K).(\forall (v: T).(\forall (t2: T).(\forall (t1:
+T).(\forall (i: nat).((subst0 (s k i) v t1 t2) \to (\forall (u: T).(subst0 i
+v (THead k u t1) (THead k u t2))))))))
+| subst0_both: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i:
+nat).((subst0 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2:
+T).((subst0 (s k i) v t1 t2) \to (subst0 i v (THead k u1 t1) (THead k u2
+t2)))))))))).
+
+theorem subst0_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
+i v (TSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return
+(\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda
+(_: (subst0 n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TSort
+n)) \to ((eq T t1 x) \to P))))))))) with [(subst0_lref v0 i0) \Rightarrow
+(\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T
+(TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat
+i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef n0) (TSort n)) \to ((eq T
+(lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T v0 v)).(eq_ind T v
+(\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T (lift (S i) O t) x)
+\to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let H6 \def (eq_ind T
+(TLRef i) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (TSort n) H5) in (False_ind ((eq T (lift (S i) O v) x) \to P)
+H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) |
+(subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0
+i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TSort
+n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0:
+nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2
+t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind
+T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2
+t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T (THead k u1 t)
+(TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in
+(False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to P)) H7))) v0
+(sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k
+v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq
+T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort n))).(\lambda (H4: (eq T
+(THead k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T
+(THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0)
+v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t:
+T).((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0
+(s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T (THead k u t1) (TSort
+n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in
+(False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to P))
+H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) |
+(subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0
+i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) (TSort
+n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda (n0:
+nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T (THead k
+u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to
+P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead
+k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to
+((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1 t1)
+(TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in
+(False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s
+k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2)
+H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal
+T (TSort n)) (refl_equal T x)))))))).
+
+theorem subst0_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
+i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst0 i v (TLRef n) x)).(let H0 \def (match H return (\lambda (n0:
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (subst0
+n0 t t0 t1)).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n)) \to ((eq
+T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))))))) with
+[(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1:
+(eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3: (eq T
+(lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq
+T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land (eq nat n
+i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v
+(\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O t) x)
+\to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda (H5: (eq T
+(TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e: T).(match e
+return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n)
+\Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H5) in
+(eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to (land (eq
+nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S n) O v)
+x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq T t
+(lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) O
+v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq nat
+i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) |
+(subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0
+i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef
+n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0:
+nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2
+t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O
+v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T
+(THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1
+u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T
+(THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
+H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to (land
+(eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0
+(sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u)
+\Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda
+(H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead k u t2)
+x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u t1)
+(TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to
+(land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0
+v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T
+(THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T
+x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let
+H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in (False_ind ((eq T (THead
+k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift
+(S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4
+H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2:
+(eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1)
+(TLRef n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda
+(n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TLRef n)) \to ((eq T
+(THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2)
+\to (land (eq nat n i) (eq T x (lift (S n) O v))))))))) (\lambda (H6: (eq T
+v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq
+T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2)
+\to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H7: (eq T
+(THead k u1 t1) (TLRef n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef n) H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2)
+\to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O
+v)))))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0
+H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n))
+(refl_equal T x))))))).
+
+theorem subst0_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
+(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) x)).(let H0
+\def (match H return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (_: (subst0 n t t0 t2)).((eq nat n i) \to ((eq T
+t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 x) \to (or3 (ex2 T (\lambda
+(u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2
+T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i)
+v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))))))) with [(subst0_lref
+v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0
+v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda (H3: (eq T
+(lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq
+T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) x) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))
+(\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i)
+(THead k u1 t1)) \to ((eq T (lift (S i) O t) x) \to (or3 (ex2 T (\lambda (u2:
+T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T
+(\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v
+t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) (\lambda (H5: (eq T
+(TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k u1
+t1) H5) in (False_ind ((eq T (lift (S i) O v) x) \to (or3 (ex2 T (\lambda
+(u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2
+T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i)
+v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) H6))) v0 (sym_eq T v0 v
+H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u0 i0 H0 t
+k0) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
+v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(\lambda (H4: (eq T
+(THead k0 u2 t) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T
+(THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 n
+v0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda
+(u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1
+t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T
+(THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 i
+t0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda
+(u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1
+t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2))))))))) (\lambda (H6: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H7
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t)
+\Rightarrow t])) (THead k0 u0 t) (THead k u1 t1) H6) in ((let H8 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k0 u0 t) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda
+(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
+(TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t)
+(THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T
+t t1) \to ((eq T (THead k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T
+(\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1
+u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1
+u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))))
+(\lambda (H10: (eq T u0 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to
+((eq T (THead k u2 t) x) \to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda
+(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2
+T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i)
+v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3
+t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t
+t1)).(eq_ind T t1 (\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v
+u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda
+(u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1
+t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2)))))))) (\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1)
+(\lambda (t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T
+t0 (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda
+(t2: T).(eq T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1
+t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3
+t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v
+u1 u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
+t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) (THead k
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3:
+T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1
+u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11)))
+u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0
+v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0
+H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
+v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T
+(THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T
+(THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0
+(s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq
+T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0
+(s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1)))
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(let
+H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H8 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t]))
+(THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda
+(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
+(TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u t0)
+(THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u u1) \to ((eq T
+t0 t1) \to ((eq T (THead k1 u t2) x) \to ((subst0 (s k1 i) v t0 t2) \to (or3
+(ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i
+v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))
+(\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to
+((eq T (THead k t t2) x) \to ((subst0 (s k i) v t0 t2) \to (or3 (ex2 T
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))
+(\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead k u1
+t2) x) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3:
+T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H12: (eq T
+(THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda (t: T).((subst0 (s k
+i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2 t1)))
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead k u2 t3)))) (\lambda (u2: T).(\lambda
+(_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3))))))) (\lambda (H13: (subst0 (s k i) v t1 t2)).(or3_intro1 (ex2 T
+(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3)))) (ex_intro2 T (\lambda (t3: T).(eq T (THead k
+u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)) t2
+(refl_equal T (THead k u1 t2)) H13))) x H12)) t0 (sym_eq T t0 t1 H11))) u
+(sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 v
+H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u0 u2 i0 H0
+k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0
+v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1 t1))).(\lambda (H5: (eq
+T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq
+T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to
+((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda
+(u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2
+T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i)
+v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))) (\lambda (H6: (eq T v0
+v)).(eq_ind T v (\lambda (t: T).((eq T (THead k0 u0 t0) (THead k u1 t1)) \to
+((eq T (THead k0 u2 t2) x) \to ((subst0 i t u0 u2) \to ((subst0 (s k0 i) t t0
+t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1
+t3)))))))))) (\lambda (H7: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H8
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H7) in ((let H9 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k0 u0 t0) (THead k u1 t1) H7) in ((let H10 \def (f_equal T K (\lambda
+(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
+(TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t0)
+(THead k u1 t1) H7) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T
+t0 t1) \to ((eq T (THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s
+k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead
+k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t:
+T).((eq T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to
+((subst0 (s k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k
+u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T
+x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T
+t1 (\lambda (t: T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to
+((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3
+t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H13: (eq T (THead k u2 t2)
+x)).(eq_ind T (THead k u2 t2) (\lambda (t: T).((subst0 i v u1 u2) \to
+((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t (THead k
+u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T
+t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H14: (subst0 i v u1
+u2)).(\lambda (H15: (subst0 (s k i) v t1 t2)).(or3_intro2 (ex2 T (\lambda
+(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v
+u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i) v t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i)
+v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14 H15)))) x H13)) t0
+(sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0 (sym_eq K k0 k H10)))
+H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0
+H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (THead k
+u1 t1)) (refl_equal T x))))))))).
+
+theorem subst0_refl:
+ \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (u: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d:
+nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort
+n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda
+(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef
+n))).(\lambda (P: Prop).(and_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O
+u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O
+u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1
+P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).((subst0 d u t0 t0) \to (\forall (P:
+Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).((subst0 d u
+t1 t1) \to (\forall (P: Prop).P))))).(\lambda (d: nat).(\lambda (H1: (subst0
+d u (THead k t0 t1) (THead k t0 t1))).(\lambda (P: Prop).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 d u t0 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead
+k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1 t2))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k d) u t1 t2)))) P (\lambda (H2: (ex2 T (\lambda (u2: T).(eq T
+(THead k t0 t1) (THead k u2 t1))) (\lambda (u2: T).(subst0 d u t0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T (THead k t0 t1) (THead k u2 t1)))
+(\lambda (u2: T).(subst0 d u t0 u2)) P (\lambda (x: T).(\lambda (H3: (eq T
+(THead k t0 t1) (THead k x t1))).(\lambda (H4: (subst0 d u t0 x)).(let H5
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t _)
+\Rightarrow t])) (THead k t0 t1) (THead k x t1) H3) in (let H6 \def (eq_ind_r
+T x (\lambda (t: T).(subst0 d u t0 t)) H4 t0 H5) in (H d H6 P)))))) H2))
+(\lambda (H2: (ex2 T (\lambda (t2: T).(eq T (THead k t0 t1) (THead k t0 t2)))
+(\lambda (t2: T).(subst0 (s k d) u t1 t2)))).(ex2_ind T (\lambda (t2: T).(eq
+T (THead k t0 t1) (THead k t0 t2))) (\lambda (t2: T).(subst0 (s k d) u t1
+t2)) P (\lambda (x: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k t0
+x))).(\lambda (H4: (subst0 (s k d) u t1 x)).(let H5 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t]))
+(THead k t0 t1) (THead k t0 x) H3) in (let H6 \def (eq_ind_r T x (\lambda (t:
+T).(subst0 (s k d) u t1 t)) H4 t1 H5) in (H0 (s k d) H6 P)))))) H2)) (\lambda
+(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 t1)
+(THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead k t0 t1) (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 d u t0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k d) u t1 t2))) P (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H3: (eq T (THead k t0 t1) (THead k x0 x1))).(\lambda (H4:
+(subst0 d u t0 x0)).(\lambda (H5: (subst0 (s k d) u t1 x1)).(let H6 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ t _) \Rightarrow t]))
+(THead k t0 t1) (THead k x0 x1) H3) in ((let H7 \def (f_equal T T (\lambda
+(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t1 |
+(TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k t0 t1)
+(THead k x0 x1) H3) in (\lambda (H8: (eq T t0 x0)).(let H9 \def (eq_ind_r T
+x1 (\lambda (t: T).(subst0 (s k d) u t1 t)) H5 t1 H7) in (let H10 \def
+(eq_ind_r T x0 (\lambda (t: T).(subst0 d u t0 t)) H4 t0 H8) in (H d H10
+P))))) H6))))))) H2)) (subst0_gen_head k u t0 t1 (THead k t0 t1) d
+H1)))))))))) t)).
+
+theorem subst0_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d
+u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n:
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n))
+x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t:
+T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d))))
+in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n)
+t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S
+(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2:
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
+n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
+(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
+(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (and_ind (eq nat n i) (eq
+T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
+(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
+(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))))
+(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0)
+O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T
+(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O
+(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3)))
+(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
+n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
+T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
+(plus i d)) H0)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
+h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
+(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
+u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n))
+H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2:
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
+n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4))))
+(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
+(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift
+h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0
+t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t
+t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0))
+(\lambda (t: T).(subst0 i (lift h d u) t x)) H1 (THead k (lift h (S (plus i
+d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d))))
+in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus
+i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d))
+t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t)
+t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
+d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S
+(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h
+d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
+t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k
+(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S
+(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h
+(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
+T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d)))
+t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t)
+x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T
+x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T
+(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
+(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda
+(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d)))
+t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
+t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h
+(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k
+(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1
+t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3:
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2)))
+(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d)))
+t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i
+d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
+(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0:
+T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda
+(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0)
+x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2
+T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3:
+T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i
+d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S
+(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus
+i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0))
+H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
+(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))
+(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d))
+x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s
+k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h
+(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i
+u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T
+(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1))
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0)
+t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda
+(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind
+T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
+(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i
+d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d))
+(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i
+d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d))))
+(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0
+H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S
+(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d
+u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
+(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda
+(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x
+(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d))
+t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i
+d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u
+(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda
+(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i
+d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d))
+(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7
+(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2))
+(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda
+(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k
+i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d))
+t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T
+(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u
+(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S
+(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S
+(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2
+x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0)
+t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S
+(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind
+nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k
+(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus
+i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S
+(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2:
+T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead
+k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus
+i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T
+(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k
+x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u
+t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S
+(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d)))
+(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0
+i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k
+(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i
+H2))))))))))))) t1)).
+
+theorem subst0_gen_lift_false:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u
+(lift h d t) x) \to (\forall (P: Prop).P)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x:
+T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i
+(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P:
+Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda
+(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n))
+x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda
+(t: T).(subst0 i u t x)) H1 (TSort n) (lift_sort n h d)) in (subst0_gen_sort
+u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x:
+T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H: (le d
+i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d
+(TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P (\lambda (H2: (lt n d)).(let
+H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H1
+(TLRef n) (lift_lref_lt n h d H2)) in (and_ind (eq nat n i) (eq T x (lift (S
+n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O
+u))).(let H6 \def (eq_ind nat n (\lambda (n: nat).(lt n d)) H2 i H4) in
+(le_false d i P H H6)))) (subst0_gen_lref u x i n H3)))) (\lambda (H2: (le d
+n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t
+x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d H2)) in (and_ind (eq nat (plus
+n h) i) (eq T x (lift (S (plus n h)) O u)) P (\lambda (H4: (eq nat (plus n h)
+i)).(\lambda (_: (eq T x (lift (S (plus n h)) O u))).(let H6 \def (eq_ind_r
+nat i (\lambda (n: nat).(lt n (plus d h))) H0 (plus n h) H4) in (le_false d n
+P H2 (lt_le_S n d (simpl_lt_plus_r h n d H6)))))) (subst0_gen_lref u x i
+(plus n h) H3))))))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H:
+((\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).(\forall
+(i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u (lift h d t0) x)
+\to (\forall (P: Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall
+(u: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i:
+nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to
+(\forall (P: Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda
+(H2: (lt i (plus d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1))
+x)).(\lambda (P: Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1))
+(\lambda (t: T).(subst0 i u t x)) H3 (THead k (lift h d t0) (lift h (s k d)
+t1)) (lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x
+(THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0)
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0
+(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2:
+T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u
+(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda
+(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7:
+(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda
+(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2:
+T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u
+(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k
+(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1)
+x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h))
+(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i
+(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7:
+(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d)
+t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d
+t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
+
+theorem subst0_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h)
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x:
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h
+d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n:
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus
+d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0
+i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2
+T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i
+h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d
+(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
+n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
+(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
+(and_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq T
+x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
+(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
+\def (eq_ind nat n (\lambda (n: nat).(lt n d)) H1 i H3) in (le_false (plus d
+h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
+(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
+T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
+(lift_lref_ge n h d H1)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S
+(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
+h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
+h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S
+(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2))))
+(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S
+(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n)
+t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h
+d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u)
+(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n
+h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
+u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
+nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
+(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n
+h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n)
+(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (le_O_n d))) (subst0_lref
+u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i H3))) (subst0_gen_lref
+u x i (plus n h) H2)))))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H:
+((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
+nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t
+t2))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (x: T).(\forall (i:
+nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h d t0) x) \to
+((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus i h) u t0 t2))))))))))).(\lambda (x:
+T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(subst0 i u (lift h d (THead k t t0)) x)).(\lambda (H2: (le (plus d h)
+i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) (\lambda (t: T).(subst0
+i u t x)) H1 (THead k (lift h d t) (lift h (s k d) t0)) (lift_head k t t0 h
+d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d)
+t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2))) (ex2 T (\lambda (t2:
+T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u
+(lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
+x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d
+t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d)
+t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (H4: (ex2 T (\lambda
+(u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i
+u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h
+(s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) u2)) (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
+k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x (THead k x0 (lift h (s k
+d) t0)))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(eq_ind_r T (THead k x0
+(lift h (s k d) t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift
+h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
+(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 (lift h (s k
+d) t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0)
+t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h d x1))).(\lambda (H8:
+(subst0 (minus i h) u t x1)).(eq_ind_r T (lift h d x1) (\lambda (t2: T).(ex2
+T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k d) t0)) (lift h d t3)))
+(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift
+h d (THead k x1 t0)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift
+h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3))))
+(ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k x1 t0)) (lift h d
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x1
+t0) (refl_equal T (lift h d (THead k x1 t0))) (subst0_fst u x1 t (minus i h)
+H8 t0 k)) (THead k (lift h d x1) (lift h (s k d) t0)) (lift_head k x1 t0 h
+d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) (\lambda (H4: (ex2 T (\lambda
+(t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i)
+u (lift h (s k d) t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k
+(lift h d t) t2))) (\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0)
+t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (H5: (eq T x
+(THead k (lift h d t) x0))).(\lambda (H6: (subst0 (s k i) u (lift h (s k d)
+t0) x0)).(eq_ind_r T (THead k (lift h d t) x0) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i
+h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (s k
+d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda
+(t2: T).(eq T (THead k (lift h d t) x0) (lift h d t2))) (\lambda (t2:
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7:
+(eq T x0 (lift h (s k d) x1))).(\lambda (H8: (subst0 (minus (s k i) h) u t0
+x1)).(eq_ind_r T (lift h (s k d) x1) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead k (lift h d t) t2) (lift h d t3))) (\lambda (t3: T).(subst0
+(minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d (THead k t x1))
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H9 \def (eq_ind_r
+nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x1)) H8 (s k (minus i
+h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2:
+T).(eq T (lift h d (THead k t x1)) (lift h d t2))) (\lambda (t2: T).(subst0
+(minus i h) u (THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h d
+(THead k t x1))) (subst0_snd k u x1 t0 (minus i h) H9 t))) (THead k (lift h d
+t) (lift h (s k d) x1)) (lift_head k t x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s
+k d) H6 (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le
+k (plus d h) i H2) (plus (s k d) h) (s_plus k d h)))) x H5)))) H4)) (\lambda
+(H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u (lift h d t) u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex2 T
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h)
+u (THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T
+x (THead k x0 x1))).(\lambda (H6: (subst0 i u (lift h d t) x0)).(\lambda (H7:
+(subst0 (s k i) u (lift h (s k d) t0) x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2:
+T).(eq T x1 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 (minus (s k i) h)
+u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x2:
+T).(\lambda (H8: (eq T x1 (lift h (s k d) x2))).(\lambda (H9: (subst0 (minus
+(s k i) h) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h d t2)))
+(\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda (t2: T).(eq T
+(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead
+k t t0) t2))) (\lambda (x3: T).(\lambda (H10: (eq T x0 (lift h d
+x3))).(\lambda (H11: (subst0 (minus i h) u t x3)).(eq_ind_r T (lift h d x3)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 x1) (lift h d
+t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind_r
+T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k
+(lift h d x3) t2) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u
+(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x3 x2)) (\lambda (t2:
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0
+(minus i h) u (THead k t t0) t3)))) (let H12 \def (eq_ind_r nat (minus (s k
+i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 (s k (minus i h)) (s_minus k i
+h (le_trans_plus_r d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h
+d (THead k x3 x2)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u
+(THead k t t0) t2)) (THead k x3 x2) (refl_equal T (lift h d (THead k x3 x2)))
+(subst0_both u t x3 (minus i h) H11 k t0 x2 H12))) (THead k (lift h d x3)
+(lift h (s k d) x2)) (lift_head k x3 x2 h d)) x1 H8) x0 H10)))) (H x0 i h d
+H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind nat (s k (plus d h)) (\lambda
+(n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) (s_plus k
+d h)))) x H5)))))) H4)) (subst0_gen_head k u (lift h d t) (lift h (s k d) t0)
+x i H3)))))))))))))) t1)).
+
+theorem subst0_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst0 i
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((lt n d) \to (\forall
+(h: nat).(subst0 n (lift h (minus d (S n)) t) (lift h d t0) (lift h d
+t3))))))))) (\lambda (v: T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda
+(H0: (lt i0 d)).(\lambda (h: nat).(eq_ind_r T (TLRef i0) (\lambda (t:
+T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (lift (S i0) O v))))
+(let w \def (minus d (S i0)) in (eq_ind nat (plus (S i0) (minus d (S i0)))
+(\lambda (n: nat).(subst0 i0 (lift h w v) (TLRef i0) (lift h n (lift (S i0) O
+v)))) (eq_ind_r T (lift (S i0) O (lift h (minus d (S i0)) v)) (\lambda (t:
+T).(subst0 i0 (lift h w v) (TLRef i0) t)) (subst0_lref (lift h (minus d (S
+i0)) v) i0) (lift h (plus (S i0) (minus d (S i0))) (lift (S i0) O v)) (lift_d
+v h (S i0) (minus d (S i0)) O (le_O_n (minus d (S i0))))) d (le_plus_minus_r
+(S i0) d H0))) (lift h d (TLRef i0)) (lift_lref_lt i0 h d H0))))))) (\lambda
+(v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_:
+(subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to (\forall
+(h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift h d
+u2))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (lt
+i0 d)).(\lambda (h: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d)
+t)) (\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) t0 (lift h d
+(THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t))
+(\lambda (t0: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d
+u1) (lift h (s k d) t)) t0)) (subst0_fst (lift h (minus d (S i0)) v) (lift h
+d u2) (lift h d u1) i0 (H1 d H2 h) (lift h (s k d) t) k) (lift h d (THead k
+u2 t)) (lift_head k u2 t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h
+d))))))))))))) (\lambda (k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1:
+((\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k i0)
+(lift h (minus d (S (s k i0))) v) (lift h d t3) (lift h d t0))))))).(\lambda
+(u0: T).(\lambda (d: nat).(\lambda (H2: (lt i0 d)).(\lambda (h: nat).(let H3
+\def (eq_ind_r nat (S (s k i0)) (\lambda (n: nat).(\forall (d: nat).((lt (s k
+i0) d) \to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d n) v) (lift h
+d t3) (lift h d t0)))))) H1 (s k (S i0)) (s_S k i0)) in (eq_ind_r T (THead k
+(lift h d u0) (lift h (s k d) t3)) (\lambda (t: T).(subst0 i0 (lift h (minus
+d (S i0)) v) t (lift h d (THead k u0 t0)))) (eq_ind_r T (THead k (lift h d
+u0) (lift h (s k d) t0)) (\lambda (t: T).(subst0 i0 (lift h (minus d (S i0))
+v) (THead k (lift h d u0) (lift h (s k d) t3)) t)) (eq_ind nat (minus (s k d)
+(s k (S i0))) (\lambda (n: nat).(subst0 i0 (lift h n v) (THead k (lift h d
+u0) (lift h (s k d) t3)) (THead k (lift h d u0) (lift h (s k d) t0))))
+(subst0_snd k (lift h (minus (s k d) (s k (S i0))) v) (lift h (s k d) t0)
+(lift h (s k d) t3) i0 (H3 (s k d) (s_lt k i0 d H2) h) (lift h d u0)) (minus
+d (S i0)) (minus_s_s k d (S i0))) (lift h d (THead k u0 t0)) (lift_head k u0
+t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d))))))))))))))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda
+(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((lt i0 d) \to
+(\forall (h: nat).(subst0 i0 (lift h (minus d (S i0)) v) (lift h d u1) (lift
+h d u2))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (d: nat).((lt (s k i0) d)
+\to (\forall (h: nat).(subst0 (s k i0) (lift h (minus d (S (s k i0))) v)
+(lift h d t0) (lift h d t3))))))).(\lambda (d: nat).(\lambda (H4: (lt i0
+d)).(\lambda (h: nat).(let H5 \def (eq_ind_r nat (S (s k i0)) (\lambda (n:
+nat).(\forall (d: nat).((lt (s k i0) d) \to (\forall (h: nat).(subst0 (s k
+i0) (lift h (minus d n) v) (lift h d t0) (lift h d t3)))))) H3 (s k (S i0))
+(s_S k i0)) in (eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t0))
+(\lambda (t: T).(subst0 i0 (lift h (minus d (S i0)) v) t (lift h d (THead k
+u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t3)) (\lambda
+(t: T).(subst0 i0 (lift h (minus d (S i0)) v) (THead k (lift h d u1) (lift h
+(s k d) t0)) t)) (subst0_both (lift h (minus d (S i0)) v) (lift h d u1) (lift
+h d u2) i0 (H1 d H4 h) k (lift h (s k d) t0) (lift h (s k d) t3) (eq_ind nat
+(minus (s k d) (s k (S i0))) (\lambda (n: nat).(subst0 (s k i0) (lift h n v)
+(lift h (s k d) t0) (lift h (s k d) t3))) (H5 (s k d) (lt_le_S (s k i0) (s k
+d) (s_lt k i0 d H4)) h) (minus d (S i0)) (minus_s_s k d (S i0)))) (lift h d
+(THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0))
+(lift_head k u1 t0 h d))))))))))))))))) i u t1 t2 H))))).
+
+theorem subst0_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
+(h: nat).((subst0 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (H: (subst0 i u t1 t2)).(subst0_ind (\lambda (n:
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t3: T).(\forall (d: nat).((le
+d n) \to (subst0 (plus n h) t (lift h d t0) (lift h d t3)))))))) (\lambda (v:
+T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T
+(TLRef (plus i0 h)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (lift
+(S i0) O v)))) (eq_ind_r T (lift (plus h (S i0)) O v) (\lambda (t: T).(subst0
+(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda
+(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r
+nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0))
+O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_comm i0 h)) (plus h (S
+i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d
+(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d
+H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le
+d i0) \to (subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (t:
+T).(\lambda (k: K).(\lambda (d: nat).(\lambda (H2: (le d i0)).(eq_ind_r T
+(THead k (lift h d u1) (lift h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0
+h) v t0 (lift h d (THead k u2 t)))) (eq_ind_r T (THead k (lift h d u2) (lift
+h (s k d) t)) (\lambda (t0: T).(subst0 (plus i0 h) v (THead k (lift h d u1)
+(lift h (s k d) t)) t0)) (subst0_fst v (lift h d u2) (lift h d u1) (plus i0
+h) (H1 d H2) (lift h (s k d) t) k) (lift h d (THead k u2 t)) (lift_head k u2
+t h d)) (lift h d (THead k u1 t)) (lift_head k u1 t h d)))))))))))) (\lambda
+(k: K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (d:
+nat).((le d (s k i0)) \to (subst0 (plus (s k i0) h) v (lift h d t3) (lift h d
+t0)))))).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H2: (le d i0)).(let H3
+\def (eq_ind_r nat (plus (s k i0) h) (\lambda (n: nat).(\forall (d: nat).((le
+d (s k i0)) \to (subst0 n v (lift h d t3) (lift h d t0))))) H1 (s k (plus i0
+h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift h d u0) (lift h (s k d)
+t3)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h d (THead k u0 t0))))
+(eq_ind_r T (THead k (lift h d u0) (lift h (s k d) t0)) (\lambda (t:
+T).(subst0 (plus i0 h) v (THead k (lift h d u0) (lift h (s k d) t3)) t))
+(subst0_snd k v (lift h (s k d) t0) (lift h (s k d) t3) (plus i0 h) (H3 (s k
+d) (s_le k d i0 H2)) (lift h d u0)) (lift h d (THead k u0 t0)) (lift_head k
+u0 t0 h d)) (lift h d (THead k u0 t3)) (lift_head k u0 t3 h d)))))))))))))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda
+(_: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (d: nat).((le d i0) \to
+(subst0 (plus i0 h) v (lift h d u1) (lift h d u2)))))).(\lambda (k:
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i0) v t0
+t3)).(\lambda (H3: ((\forall (d: nat).((le d (s k i0)) \to (subst0 (plus (s k
+i0) h) v (lift h d t0) (lift h d t3)))))).(\lambda (d: nat).(\lambda (H4: (le
+d i0)).(let H5 \def (eq_ind_r nat (plus (s k i0) h) (\lambda (n:
+nat).(\forall (d: nat).((le d (s k i0)) \to (subst0 n v (lift h d t0) (lift h
+d t3))))) H3 (s k (plus i0 h)) (s_plus k i0 h)) in (eq_ind_r T (THead k (lift
+h d u1) (lift h (s k d) t0)) (\lambda (t: T).(subst0 (plus i0 h) v t (lift h
+d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) (lift h (s k d) t3))
+(\lambda (t: T).(subst0 (plus i0 h) v (THead k (lift h d u1) (lift h (s k d)
+t0)) t)) (subst0_both v (lift h d u1) (lift h d u2) (plus i0 h) (H1 d H4) k
+(lift h (s k d) t0) (lift h (s k d) t3) (H5 (s k d) (s_le k d i0 H4))) (lift
+h d (THead k u2 t3)) (lift_head k u2 t3 h d)) (lift h d (THead k u1 t0))
+(lift_head k u1 t0 h d)))))))))))))))) i u t1 t2 H)))))).
+
+theorem subst0_lift_ge_S:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst0 (S i) u (lift (S O) d
+t1) (lift (S O) d t2))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(eq_ind nat
+(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d
+t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O))
+(plus_comm i (S O)))))))))).
+
+theorem subst0_lift_ge_s:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t1 t2) \to (\forall (d: nat).((le d i) \to (\forall (b: B).(subst0 (s
+(Bind b) i) u (lift (S O) d t1) (lift (S O) d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t1 t2)).(\lambda (d: nat).(\lambda (H0: (le d i)).(\lambda
+(_: B).(subst0_lift_ge_S t1 t2 u i H d H0)))))))).
+
+theorem subst0_subst0:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
+T).(subst0 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
+T).(\forall (i: nat).((subst0 i u u1 t) \to (ex2 T (\lambda (t4: T).(subst0 n
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t4 t3)))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 u u1 v)).(eq_ind nat (plus i0 (S i))
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
+(t: T).(subst0 n u t (lift (S i) O v))))) (ex_intro2 T (\lambda (t:
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t
+(lift (S i) O v))) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge u1 v
+u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i))
+(plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t:
+T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0:
+nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i
+u3 u1 t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u t0 u0)) (ex2 T (\lambda
+(t0: T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0
+i)) u t0 (THead k u0 t)))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1
+x)).(\lambda (H4: (subst0 (S (plus i0 i)) u x u0)).(ex_intro2 T (\lambda (t0:
+T).(subst0 i u3 (THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i))
+u t0 (THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst
+u u0 x (S (plus i0 i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k:
+K).(\lambda (v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i:
+nat).(\lambda (_: (subst0 (s k i) v t3 t0)).(\lambda (H1: ((\forall (u1:
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
+i))) u t t0))))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (u0:
+T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u0 u1 v)).(ex2_ind T (\lambda
+(t: T).(subst0 (s k i) u1 t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k
+i))) u0 t t0)) (ex2 T (\lambda (t: T).(subst0 i u1 (THead k u t3) t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u0 t (THead k u t0)))) (\lambda (x:
+T).(\lambda (H3: (subst0 (s k i) u1 t3 x)).(\lambda (H4: (subst0 (S (plus i0
+(s k i))) u0 x t0)).(let H5 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n:
+nat).(subst0 (S n) u0 x t0)) H4 (s k (plus i0 i)) (s_plus_sym k i0 i)) in
+(let H6 \def (eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n
+u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T
+(\lambda (t: T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S
+(plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3
+u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0
+H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u2:
+T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda
+(t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t
+u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
+(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u:
+T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda (t: T).(subst0
+(s k i) u1 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t
+t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4:
+(subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t:
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
+t (THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0
+x)).(\lambda (H6: (subst0 (S (plus i0 (s k i))) u x t3)).(ex2_ind T (\lambda
+(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))
+(ex2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t:
+T).(subst0 (S (plus i0 i)) u t (THead k u0 t3)))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i u3 u1 x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u x0
+u0)).(let H9 \def (eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0
+(S n) u x t3)) H6 (s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H10 \def
+(eq_ind_r nat (S (s k (plus i0 i))) (\lambda (n: nat).(subst0 n u x t3)) H9
+(s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t:
+T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u
+t (THead k u0 t3))) (THead k x0 x) (subst0_both u3 u1 x0 i H7 k t0 x H5)
+(subst0_both u x0 u0 (S (plus i0 i)) H8 k x t3 H10))))))) (H1 u3 u i0 H4)))))
+(H3 u3 u i0 H4))))))))))))))))) j u2 t1 t2 H))))).
+
+theorem subst0_subst0_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst0 i
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t:
+T).(subst0 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst0 j u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).(\forall (u:
+T).(\forall (i: nat).((subst0 i u t u1) \to (ex2 T (\lambda (t4: T).(subst0 n
+u1 t0 t4)) (\lambda (t4: T).(subst0 (S (plus i n)) u t3 t4)))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (u1: T).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 u v u1)).(eq_ind nat (plus i0 (S i))
+(\lambda (n: nat).(ex2 T (\lambda (t: T).(subst0 i u1 (TLRef i) t)) (\lambda
+(t: T).(subst0 n u (lift (S i) O v) t)))) (ex_intro2 T (\lambda (t:
+T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift
+(S i) O v) t)) (lift (S i) O u1) (subst0_lref u1 i) (subst0_lift_ge v u1 u i0
+(S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0
+(S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda
+(u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1:
+((\forall (u2: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u2) \to
+(ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus
+i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3:
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v
+u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0:
+T).(subst0 (S (plus i0 i)) u u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i u3
+(THead k u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t)
+t0))) (\lambda (x: T).(\lambda (H3: (subst0 i u3 u1 x)).(\lambda (H4: (subst0
+(S (plus i0 i)) u u0 x)).(ex_intro2 T (\lambda (t0: T).(subst0 i u3 (THead k
+u1 t) t0)) (\lambda (t0: T).(subst0 (S (plus i0 i)) u (THead k u0 t) t0))
+(THead k x t) (subst0_fst u3 x u1 i H3 t k) (subst0_fst u x u0 (S (plus i0
+i)) H4 t k))))) (H1 u3 u i0 H2)))))))))))))) (\lambda (k: K).(\lambda (v:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (_: (subst0
+(s k i) v t3 t0)).(\lambda (H1: ((\forall (u1: T).(\forall (u: T).(\forall
+(i0: nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1
+t3 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t0 t))))))))).(\lambda
+(u: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H2:
+(subst0 i0 u0 v u1)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u1 t3 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u0 t0 t)) (ex2 T (\lambda (t:
+T).(subst0 i u1 (THead k u t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0
+(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i) u1 t3
+x)).(\lambda (H4: (subst0 (S (plus i0 (s k i))) u0 t0 x)).(let H5 \def
+(eq_ind_r nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u0 t0 x)) H4
+(s k (plus i0 i)) (s_plus_sym k i0 i)) in (let H6 \def (eq_ind_r nat (S (s k
+(plus i0 i))) (\lambda (n: nat).(subst0 n u0 t0 x)) H5 (s k (S (plus i0 i)))
+(s_S k (plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u1 (THead k u
+t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead
+k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6
+u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1:
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u v u2) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t))
+(\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k:
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0
+t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u: T).(\forall (i0:
+nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 t0 t))
+(\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3:
+T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v
+u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t:
+T).(subst0 (S (plus i0 (s k i))) u t3 t)) (ex2 T (\lambda (t: T).(subst0 i u3
+(THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3)
+t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i) u3 t0 x)).(\lambda (H6:
+(subst0 (S (plus i0 (s k i))) u t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i
+u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t)) (ex2 T (\lambda
+(t: T).(subst0 i u3 (THead k u1 t0) t)) (\lambda (t: T).(subst0 (S (plus i0
+i)) u (THead k u0 t3) t))) (\lambda (x0: T).(\lambda (H7: (subst0 i u3 u1
+x0)).(\lambda (H8: (subst0 (S (plus i0 i)) u u0 x0)).(let H9 \def (eq_ind_r
+nat (plus i0 (s k i)) (\lambda (n: nat).(subst0 (S n) u t3 x)) H6 (s k (plus
+i0 i)) (s_plus_sym k i0 i)) in (let H10 \def (eq_ind_r nat (S (s k (plus i0
+i))) (\lambda (n: nat).(subst0 n u t3 x)) H9 (s k (S (plus i0 i))) (s_S k
+(plus i0 i))) in (ex_intro2 T (\lambda (t: T).(subst0 i u3 (THead k u1 t0)
+t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u (THead k u0 t3) t)) (THead k x0
+x) (subst0_both u3 u1 x0 i H7 k t0 x H5) (subst0_both u u0 x0 (S (plus i0 i))
+H8 k t3 x H10))))))) (H1 u3 u i0 H4))))) (H3 u3 u i0 H4))))))))))))))))) j u2
+t1 t2 H))))).
+
+theorem subst0_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst0
+i v t1 t2) \to (\forall (t3: T).((subst0 i v t2 t3) \to (subst0 i v t1
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i v t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t3 t4) \to
+(subst0 n t t0 t4))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (t3:
+T).(\lambda (H0: (subst0 i0 v0 (lift (S i0) O v0) t3)).(subst0_gen_lift_false
+v0 v0 t3 (S i0) O i0 (le_O_n i0) (le_n (plus O (S i0))) H0 (subst0 i0 v0
+(TLRef i0) t3)))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u1:
+T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1 u2)).(\lambda (H1:
+((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0 u1 t3))))).(\lambda
+(t: T).(\lambda (k: K).(\lambda (t3: T).(\lambda (H2: (subst0 i0 v0 (THead k
+u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t)))
+(\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda (t4: T).(eq T t3
+(THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda
+(H3: (ex2 T (\lambda (u2: T).(eq T t3 (THead k u2 t))) (\lambda (u3:
+T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3
+t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3)
+(\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0
+i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead
+k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda
+(H3: (ex2 T (\lambda (t2: T).(eq T t3 (THead k u2 t2))) (\lambda (t2:
+T).(subst0 (s k i0) v0 t t2)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k
+u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k
+u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda
+(H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0:
+T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3
+H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
+t3 (THead k u2 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t t2))))).(ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead k x0 x1))).(\lambda (H5:
+(subst0 i0 v0 u2 x0)).(\lambda (H6: (subst0 (s k i0) v0 t x1)).(eq_ind_r T
+(THead k x0 x1) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0))
+(subst0_both v0 u1 x0 i0 (H1 x0 H5) k t x1 H6) t3 H4)))))) H3))
+(subst0_gen_head k v0 u2 t t3 i0 H2)))))))))))) (\lambda (k: K).(\lambda (v0:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (H0: (subst0
+(s k i0) v0 t3 t0)).(\lambda (H1: ((\forall (t4: T).((subst0 (s k i0) v0 t0
+t4) \to (subst0 (s k i0) v0 t3 t4))))).(\lambda (u: T).(\lambda (t4:
+T).(\lambda (H2: (subst0 i0 v0 (THead k u t0) t4)).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))
+(ex2 T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s
+k i0) v0 t0 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2)))
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))) (subst0 i0 v0
+(THead k u t3) t4) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2
+t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq
+T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0
+(THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x
+t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda
+(t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0)
+t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u
+t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t0 t2)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))
+(subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4
+(THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k
+u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3
+i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0
+t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3)
+t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0
+x1))).(\lambda (H5: (subst0 i0 v0 u x0)).(\lambda (H6: (subst0 (s k i0) v0 t0
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(subst0 i0 v0 (THead k u t3)
+t)) (subst0_both v0 u x0 i0 H5 k t3 x1 (H1 x1 H6)) t4 H4)))))) H3))
+(subst0_gen_head k v0 u t0 t4 i0 H2)))))))))))) (\lambda (v0: T).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (H0: (subst0 i0 v0 u1
+u2)).(\lambda (H1: ((\forall (t3: T).((subst0 i0 v0 u2 t3) \to (subst0 i0 v0
+u1 t3))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2:
+(subst0 (s k i0) v0 t0 t3)).(\lambda (H3: ((\forall (t4: T).((subst0 (s k i0)
+v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4:
+(subst0 i0 v0 (THead k u2 t3) t4)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t4
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3))) (ex2 T (\lambda
+(t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1
+t0) t4) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3)))
+(\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead
+k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda
+(H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0
+i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4
+H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u2 t2)))
+(\lambda (t2: T).(subst0 (s k i0) v0 t3 t2)))).(ex2_ind T (\lambda (t5:
+T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4
+(THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead
+k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1
+u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i0) v0 t3 t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda
+(t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0
+i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))
+(subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H6: (eq T t4 (THead k x0 x1))).(\lambda (H7: (subst0 i0 v0 u2 x0)).(\lambda
+(H8: (subst0 (s k i0) v0 t3 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t:
+T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x0 i0 (H1 x0 H7) k t0
+x1 (H3 x1 H8)) t4 H6)))))) H5)) (subst0_gen_head k v0 u2 t3 t4 i0
+H4))))))))))))))) i v t1 t2 H))))).
+
+theorem subst0_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
+nat).((subst0 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
+(i2: nat).((subst0 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
+(t: T).(subst0 i2 u2 t1 t)) (\lambda (t: T).(subst0 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
+nat).(\lambda (H: (subst0 i1 u1 t0 t1)).(subst0_ind (\lambda (n:
+nat).(\lambda (t: T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4:
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq
+nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5:
+T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda
+(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2
+(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(and_ind (eq nat i i2) (eq
+T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v)
+t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda
+(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n:
+nat).(not (eq nat n i2))) H1 i2 H2) in (eq_ind_r T (lift (S i) O u2) (\lambda
+(t: T).(ex2 T (\lambda (t3: T).(subst0 i2 u2 (lift (S i) O v) t3)) (\lambda
+(t3: T).(subst0 i v t t3)))) (let H5 \def (match (H4 (refl_equal nat i2))
+return (\lambda (_: False).(ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O
+v) t)) (\lambda (t: T).(subst0 i v (lift (S i) O u2) t)))) with []) in H5) t2
+H3)))) (subst0_gen_lref u2 t2 i2 i H0))))))))) (\lambda (v: T).(\lambda (u2:
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0
+u2)).(\lambda (H1: ((\forall (t2: T).(\forall (u3: T).(\forall (i2:
+nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t:
+T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda
+(t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda (u3: T).(\lambda (i2:
+nat).(\lambda (H2: (subst0 i2 u3 (THead k u0 t) t2)).(\lambda (H3: (not (eq
+nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t)))
+(\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T (\lambda (t3: T).(eq T t2
+(THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex3_2 T T
+(\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4:
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead
+k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead k u2 t))) (\lambda (u2: T).(subst0 i2 u3 u0
+u2)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4:
+T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t)
+t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq
+T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k
+x t) (\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t)
+t4)) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex2_ind T (\lambda (t3:
+T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i v x t3)) (ex2 T (\lambda
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
+k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda
+(H8: (subst0 i v x x0)).(ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t)
+(subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2
+H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead
+k u0 t3))) (\lambda (t2: T).(subst0 (s k i2) u3 t t2)))).(ex2_ind T (\lambda
+(t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t
+t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0
+x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x)
+(\lambda (t3: T).(ex2 T (\lambda (t4: T).(subst0 i2 u3 (THead k u2 t) t4))
+(\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0
+i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3))
+(THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k))
+t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i2) u3 t
+t2))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4
+t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3:
+T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead k x0
+x1))).(\lambda (H6: (subst0 i2 u3 u0 x0)).(\lambda (H7: (subst0 (s k i2) u3 t
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t3: T).(ex2 T (\lambda (t4:
+T).(subst0 i2 u3 (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i v t3 t4))))
+(ex2_ind T (\lambda (t3: T).(subst0 i2 u3 u2 t3)) (\lambda (t3: T).(subst0 i
+v x0 t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i v (THead k x0 x1) t3))) (\lambda (x: T).(\lambda (H8:
+(subst0 i2 u3 u2 x)).(\lambda (H9: (subst0 i v x0 x)).(ex_intro2 T (\lambda
+(t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead
+k x0 x1) t3)) (THead k x x1) (subst0_both u3 u2 x i2 H8 k t x1 H7)
+(subst0_fst v x x0 i H9 x1 k))))) (H1 x0 u3 i2 H6 H3)) t2 H5)))))) H4))
+(subst0_gen_head k u3 u0 t t2 i2 H2))))))))))))))) (\lambda (k: K).(\lambda
+(v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H0:
+(subst0 (s k i) v t3 t2)).(\lambda (H1: ((\forall (t4: T).(\forall (u2:
+T).(\forall (i2: nat).((subst0 i2 u2 t3 t4) \to ((not (eq nat (s k i) i2))
+\to (ex2 T (\lambda (t: T).(subst0 i2 u2 t2 t)) (\lambda (t: T).(subst0 (s k
+i) v t4 t)))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H2: (subst0 i2 u2 (THead k u t3)
+t4)).(\lambda (H3: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u3: T).(eq
+T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3))) (ex2 T (\lambda
+(t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3
+t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t:
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (H4: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3))) (\lambda
+(u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k
+u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0
+i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H5: (eq T t4 (THead k x t3))).(\lambda (H6: (subst0 i2 u2 u
+x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t:
+T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2
+k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T
+(\lambda (t2: T).(eq T t4 (THead k u t2))) (\lambda (t2: T).(subst0 (s k i2)
+u2 t3 t2)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda
+(t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2
+t3 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t))
+(\lambda (t: T).(subst0 i v (THead k u x) t))) (\lambda (x0: T).(\lambda (H7:
+(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2
+T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i
+v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u)
+(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (\lambda (H7: (eq nat
+(s k i) (s k i2))).(H3 (s_inj k i i2 H7))))) t4 H5)))) H4)) (\lambda (H4:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i2) u2 t3 t2))))).(ex3_2_ind T T (\lambda
+(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k
+u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t4 (THead k x0 x1))).(\lambda (H6: (subst0 i2 u2 u
+x0)).(\lambda (H7: (subst0 (s k i2) u2 t3 x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u2 (THead k u t2) t5))
+(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 (s k
+i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t:
+T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0
+x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda
+(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2
+(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k
+x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0)))))
+(H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k i2))).(H3 (s_inj k i
+i2 H8))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 t4 i2
+H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
+(i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: ((\forall (t2:
+T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq
+nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t:
+T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: ((\forall (t4:
+T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq
+nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t3 t)) (\lambda (t:
+T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: T).(\lambda (u3:
+T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k u0 t2)
+t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq
+T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T
+(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
+u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4
+t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda
+(u2: T).(subst0 i2 u3 u0 u2)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k
+u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0
+i2 u3 u0 x)).(eq_ind_r T (THead k x t2) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v x
+t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i v (THead k x t2) t))) (\lambda (x0: T).(\lambda (H9: (subst0 i2
+u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x
+t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0
+i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T
+(\lambda (t2: T).(eq T t4 (THead k u0 t2))) (\lambda (t3: T).(subst0 (s k i2)
+u3 t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda
+(t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x:
+T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3
+t2 x)).(eq_ind_r T (THead k u0 x) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i v (THead k u0 x) t))) (\lambda (x0: T).(\lambda
+(H9: (subst0 (s k i2) u3 t3 x0)).(\lambda (H10: (subst0 (s k i) v x
+x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3
+i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8
+(\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H9))))) t4 H7))))
+H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4
+(THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t2 t3))))).(ex3_2_ind T
+T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4:
+T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t4 (THead k x0 x1))).(\lambda (H8: (subst0 i2 u3 u0
+x0)).(\lambda (H9: (subst0 (s k i2) u3 t2 x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u3 (THead k u2 t3) t5))
+(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 i2
+u3 u2 t)) (\lambda (t: T).(subst0 i v x0 t)) (ex2 T (\lambda (t: T).(subst0
+i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)))
+(\lambda (x: T).(\lambda (H10: (subst0 i2 u3 u2 x)).(\lambda (H11: (subst0 i
+v x0 x)).(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t:
+T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t))) (\lambda (x2:
+T).(\lambda (H12: (subst0 (s k i2) u3 t3 x2)).(\lambda (H13: (subst0 (s k i)
+v x1 x2)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k x x2) (subst0_both
+u3 u2 x i2 H10 k t3 x2 H12) (subst0_both v x0 x i H11 k x1 x2 H13))))) (H3 x1
+u3 (s k i2) H9 (\lambda (H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2
+H12)))))))) (H1 x0 u3 i2 H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2
+t4 i2 H4)))))))))))))))))) i1 u1 t0 t1 H))))).
+
+theorem subst0_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t0 t1) \to (\forall (t2: T).((subst0 i u t0 t2) \to (or4 (eq T t1 t2)
+(ex2 T (\lambda (t: T).(subst0 i u t1 t)) (\lambda (t: T).(subst0 i u t2 t)))
+(subst0 i u t1 t2) (subst0 i u t2 t1))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t0 t1)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to
+(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5:
+T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda
+(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef
+i0) t2)).(and_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T (lift
+(S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t))
+(\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2)
+(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda
+(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2)
+(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t:
+T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2) (subst0 i0 v t2
+(lift (S i0) O v)) (sym_eq T t2 (lift (S i0) O v) H2)))) (subst0_gen_lref v
+t2 i0 i0 H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda
+(i0: nat).(\lambda (H0: (subst0 i0 v u1 u2)).(\lambda (H1: ((\forall (t2:
+T).((subst0 i0 v u1 t2) \to (or4 (eq T u2 t2) (ex2 T (\lambda (t: T).(subst0
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v u2 t2) (subst0
+i0 v t2 u2)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (t2: T).(\lambda
+(H2: (subst0 i0 v (THead k u1 t) t2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T
+t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t
+t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2)
+(ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2
+t))) (\lambda (u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq
+T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead
+k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2
+(THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2 (THead k x
+t))).(\lambda (H5: (subst0 i0 v u1 x)).(eq_ind_r T (THead k x t) (\lambda
+(t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4: T).(subst0 i0 v
+(THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4))) (subst0 i0 v
+(THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind (eq T u2 x)
+(ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x
+t3))) (subst0 i0 v u2 x) (subst0 i0 v x u2) (or4 (eq T (THead k u2 t) (THead
+k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k
+x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (H6: (eq T u2
+x)).(eq_ind_r T x (\lambda (t3: T).(or4 (eq T (THead k t3 t) (THead k x t))
+(ex2 T (\lambda (t4: T).(subst0 i0 v (THead k t3 t) t4)) (\lambda (t4:
+T).(subst0 i0 v (THead k x t) t4))) (subst0 i0 v (THead k t3 t) (THead k x
+t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x
+t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))
+(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t)
+(THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead
+k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v
+u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) (THead
+k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda
+(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k
+x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x x0)).(or4_intro1 (eq T
+(THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v
+(THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))
+(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3:
+T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) (subst0_fst v x0 u2 i0 H7 t
+k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) (\lambda (H6: (subst0 i0 v u2
+x)).(or4_intro2 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x
+t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x
+t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t k))) (\lambda (H6: (subst0 i0 v
+x u2)).(or4_intro3 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3:
+T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x
+t) t3))) (subst0 i0 v (THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x
+t) (THead k u2 t)) (subst0_fst v u2 x i0 H6 t k))) (H1 x H5)) t2 H4)))) H3))
+(\lambda (H3: (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda
+(t2: T).(subst0 (s k i0) v t t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t t3)) (or4 (eq T
+(THead k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3))
+(\lambda (t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2)
+(subst0 i0 v t2 (THead k u2 t))) (\lambda (x: T).(\lambda (H4: (eq T t2
+(THead k u1 x))).(\lambda (H5: (subst0 (s k i0) v t x)).(eq_ind_r T (THead k
+u1 x) (\lambda (t3: T).(or4 (eq T (THead k u2 t) t3) (ex2 T (\lambda (t4:
+T).(subst0 i0 v (THead k u2 t) t4)) (\lambda (t4: T).(subst0 i0 v t3 t4)))
+(subst0 i0 v (THead k u2 t) t3) (subst0 i0 v t3 (THead k u2 t)))) (or4_ind
+(eq T u2 u2) (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3:
+T).(subst0 i0 v u2 t3))) (subst0 i0 v u2 u2) (subst0 i0 v u2 u2) (or4 (eq T
+(THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k
+u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v
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+x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1
+x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3
+x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t2
+t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3
+t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3)
+t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4
+(THead k u2 t3))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t4
+(THead k x0 x1))).(\lambda (H7: (subst0 i0 v u1 x0)).(\lambda (H8: (subst0 (s
+k i0) v t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(or4 (eq T (THead
+k u2 t3) t) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t3) t5))
+(\lambda (t5: T).(subst0 i0 v t t5))) (subst0 i0 v (THead k u2 t3) t) (subst0
+i0 v t (THead k u2 t3)))) (or4_ind (eq T t3 x1) (ex2 T (\lambda (t:
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))
+(subst0 (s k i0) v t3 x1) (subst0 (s k i0) v x1 t3) (or4 (eq T (THead k u2
+t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda
+(H9: (eq T t3 x1)).(eq_ind_r T x1 (\lambda (t: T).(or4 (eq T (THead k u2 t)
+(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k u2 t) t5))
+(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k u2
+t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t)))) (or4_ind
+(eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T
+(THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v
+(THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2
+x1))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T
+(THead k t x1) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k
+t x1) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v
+(THead k t x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t
+x1)))) (or4_intro0 (eq T (THead k x0 x1) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k x0 x1) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k x0 x1)) (refl_equal T (THead k x0 x1))) u2 H10)) (\lambda
+(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u2 x1))) (\lambda (x: T).(\lambda (H11: (subst0 i0
+v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 x1)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+x1) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 x1)) (ex_intro2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t)) (THead k x x1) (subst0_fst v x u2 i0 H11 x1 k)
+(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2
+x0)).(or4_intro2 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 x1)) (subst0_fst v x0 u2 i0 H10 x1 k))) (\lambda (H10:
+(subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 x1) (THead k x0 x1)) (ex2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 x1) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 x1) (THead k x0 x1)) (subst0
+i0 v (THead k x0 x1) (THead k u2 x1)) (subst0_fst v u2 x0 i0 H10 x1 k))) (H1
+x0 H7)) t3 H9)) (\lambda (H9: (ex2 T (\lambda (t: T).(subst0 (s k i0) v t3
+t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)))).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i0) v t3 t)) (\lambda (t: T).(subst0 (s k i0) v x1 t)) (or4
+(eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v
+(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0
+i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k
+u2 t3))) (\lambda (x: T).(\lambda (H10: (subst0 (s k i0) v t3 x)).(\lambda
+(H11: (subst0 (s k i0) v x1 x)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t:
+T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2
+x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u2 t3))) (\lambda (H12: (eq T u2 x0)).(eq_ind_r T
+x0 (\lambda (t: T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda
+(t5: T).(subst0 i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead
+k x0 x1) t5))) (subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v
+(THead k x0 x1) (THead k t t3)))) (or4_intro1 (eq T (THead k x0 t3) (THead k
+x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t3)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t)) (THead k x0 x) (subst0_snd k v x t3 i0 H10 x0) (subst0_snd k v x x1
+i0 H11 x0))) u2 H12)) (\lambda (H12: (ex2 T (\lambda (t: T).(subst0 i0 v u2
+t)) (\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0
+i0 v u2 t)) (\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda
+(x2: T).(\lambda (H13: (subst0 i0 v u2 x2)).(\lambda (H14: (subst0 i0 v x0
+x2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2
+t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x2 x)
+(subst0_both v u2 x2 i0 H13 k t3 x H10) (subst0_both v x0 x2 i0 H14 k x1 x
+H11)))))) H12)) (\lambda (H12: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead
+k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3)
+t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k
+u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))
+(ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t)) (THead k x0 x) (subst0_both v u2 x0 i0
+H12 k t3 x H10) (subst0_snd k v x x1 i0 H11 x0)))) (\lambda (H12: (subst0 i0
+v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda
+(t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k
+x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead
+k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k
+u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k u2 x)
+(subst0_snd k v x t3 i0 H10 u2) (subst0_both v x0 u2 i0 H12 k x1 x H11))))
+(H1 x0 H7))))) H9)) (\lambda (H9: (subst0 (s k i0) v t3 x1)).(or4_ind (eq T
+u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0
+v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2) (or4 (eq T (THead k u2 t3)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda
+(H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t: T).(or4 (eq T (THead k t t3)
+(THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0 i0 v (THead k t t3) t5))
+(\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5))) (subst0 i0 v (THead k t
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k t t3))))
+(or4_intro2 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k x0 t3)) (subst0_snd k v x1 t3 i0 H9 x0)) u2 H10)) (\lambda
+(H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t: T).(subst0 i0 v
+x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda (H11: (subst0 i0
+v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq T (THead k u2 t3)
+(THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t))
+(\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2
+t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t)) (THead k x x1) (subst0_both v u2 x i0 H11 k t3 x1 H9)
+(subst0_fst v x x0 i0 H12 x1 k)))))) H10)) (\lambda (H10: (subst0 i0 v u2
+x0)).(or4_intro2 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k
+x0 x1) (THead k u2 t3)) (subst0_both v u2 x0 i0 H10 k t3 x1 H9))) (\lambda
+(H10: (subst0 i0 v x0 u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1))
+(ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t:
+T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0
+x1) t)) (THead k u2 x1) (subst0_snd k v x1 t3 i0 H9 u2) (subst0_fst v u2 x0
+i0 H10 x1 k)))) (H1 x0 H7))) (\lambda (H9: (subst0 (s k i0) v x1
+t3)).(or4_ind (eq T u2 x0) (ex2 T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x0 t))) (subst0 i0 v u2 x0) (subst0 i0 v x0 u2)
+(or4 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)))
+(subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1)
+(THead k u2 t3))) (\lambda (H10: (eq T u2 x0)).(eq_ind_r T x0 (\lambda (t:
+T).(or4 (eq T (THead k t t3) (THead k x0 x1)) (ex2 T (\lambda (t5: T).(subst0
+i0 v (THead k t t3) t5)) (\lambda (t5: T).(subst0 i0 v (THead k x0 x1) t5)))
+(subst0 i0 v (THead k t t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1)
+(THead k t t3)))) (or4_intro3 (eq T (THead k x0 t3) (THead k x0 x1)) (ex2 T
+(\lambda (t: T).(subst0 i0 v (THead k x0 t3) t)) (\lambda (t: T).(subst0 i0 v
+(THead k x0 x1) t))) (subst0 i0 v (THead k x0 t3) (THead k x0 x1)) (subst0 i0
+v (THead k x0 x1) (THead k x0 t3)) (subst0_snd k v t3 x1 i0 H9 x0)) u2 H10))
+(\lambda (H10: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda (t:
+T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t: T).(subst0 i0 v u2 t))
+(\lambda (t: T).(subst0 i0 v x0 t)) (or4 (eq T (THead k u2 t3) (THead k x0
+x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0
+x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3))) (\lambda (x: T).(\lambda
+(H11: (subst0 i0 v u2 x)).(\lambda (H12: (subst0 i0 v x0 x)).(or4_intro1 (eq
+T (THead k u2 t3) (THead k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead
+k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v
+(THead k u2 t3) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2
+t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i0 v (THead k x0 x1) t)) (THead k x t3) (subst0_fst v x u2 i0
+H11 t3 k) (subst0_both v x0 x i0 H12 k x1 t3 H9)))))) H10)) (\lambda (H10:
+(subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t3) (THead k x0 x1)) (ex2
+T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0
+v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead k x0 x1)) (subst0
+i0 v (THead k x0 x1) (THead k u2 t3)) (ex_intro2 T (\lambda (t: T).(subst0 i0
+v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x0 x1) t)) (THead
+k x0 t3) (subst0_fst v x0 u2 i0 H10 t3 k) (subst0_snd k v t3 x1 i0 H9 x0))))
+(\lambda (H10: (subst0 i0 v x0 u2)).(or4_intro3 (eq T (THead k u2 t3) (THead
+k x0 x1)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda
+(t: T).(subst0 i0 v (THead k x0 x1) t))) (subst0 i0 v (THead k u2 t3) (THead
+k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t3)) (subst0_both v x0 u2
+i0 H10 k x1 t3 H9))) (H1 x0 H7))) (H3 x1 H8)) t4 H6)))))) H5))
+(subst0_gen_head k v u1 t2 t4 i0 H4))))))))))))))) i u t0 t1 H))))).
+
+theorem subst0_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst0 i u t0 (lift (S O) i
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H0: (subst0
+i u t0 (lift (S O) i t2))).(or4_ind (eq T (lift (S O) i t2) (lift (S O) i
+t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t:
+T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S
+O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2)
+(\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def
+(sym_equal T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O)
+i H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2)
+t)) (\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda
+(t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S
+O) i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S
+O) i t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1)
+x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S
+O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O))
+(plus_comm i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift
+(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1)
+(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n))
+(le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H1 (eq T t1 t2)))
+(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i
+t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i)
+(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O)
+i)) (plus i (S O)) (plus_comm i (S O))) H1 (eq T t1 t2)))
+(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
+
+theorem subst0_weight_le:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (le (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (le (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).(le_S_n (weight_map f (lift
+(S i) O v)) (weight_map g (TLRef i)) (le_S (S (weight_map f (lift (S i) O
+v))) (weight_map g (TLRef i)) H1)))))))) (\lambda (v: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1
+u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H2
+H3)) n))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+(wadd g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall
+(m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O
+v)) (g i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (plus_le_compat (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le
+f g H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t0)) (S (plus
+(weight_map g u1) (weight_map g t0))) (le_lt_n_Sm (plus (weight_map f0 u2)
+(weight_map f0 t0)) (plus (weight_map g u1) (weight_map g t0))
+(plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t0)
+(weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g H2))))))))) k)))))))))
+(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v: T).(\forall (t2:
+T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1 t2) \to
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s
+k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(le (weight_map f (THead k0 u0 t2)) (weight_map g (THead k0 u0
+t1))))))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (v:
+T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s (Bind b0)
+i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2)
+(weight_map g t1))))))) \to (\forall (u0: T).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+(Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0 t1)))))))))))))))
+(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
+(_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le (weight_map f
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
+u0 f g H2)) m)) (lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S
+(S i)) O v)) (wadd g (S (weight_map g u0)) (S i)) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f (S
+(weight_map f u0))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f
+u0)) v (S i) f))))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda
+(t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (u0:
+T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2:
+((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift
+(S i) O v)) (g i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O)
+t2)) (plus (weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_compat
+(weight_map f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map
+(wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m:
+nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O
+v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i))
+O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda
+(t2: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda
+(u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0
+(lift (S i) O v)) (g i))).(lt_le_S (plus (weight_map f0 u0) (weight_map f0
+t2)) (S (plus (weight_map g u0) (weight_map g t1))) (le_lt_n_Sm (plus
+(weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g
+t1)) (plus_le_compat (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
+(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3)))))))))))))))) k))
+(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
+(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt
+(weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f u2) (weight_map
+g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1:
+T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
+k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v
+t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2)
+(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
+\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
+(lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
+(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f
+m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
+u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
+t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
+f g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
+(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1))
+(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (lt_le_S
+(weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S
+(weight_map g u1)) (S i)) (eq_ind nat (weight_map f (lift (S i) O v))
+(\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f u2)))
+(lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u2)) v (S i)
+f)))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i)
+v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (le (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f
+(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (plus_le_compat (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O
+O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall
+(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f
+m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H5: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(lt_le_S (plus (weight_map f0 u2) (weight_map f0 t2)) (S (plus
+(weight_map g u1) (weight_map g t1))) (le_lt_n_Sm (plus (weight_map f0 u2)
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
+(plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
+(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5))))))))))))) k)))))))) d u t
+z H))))).
+
+theorem subst0_weight_lt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
+u t z) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+d) O u)) (g d)) \to (lt (weight_map f z) (weight_map g t))))))))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (d: nat).(\lambda
+(H: (subst0 d u t z)).(subst0_ind (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+n) O t0)) (g n)) \to (lt (weight_map f t2) (weight_map g t1))))))))))
+(\lambda (v: T).(\lambda (i: nat).(\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (_: ((\forall (m: nat).(le (f m) (g m))))).(\lambda
+(H1: (lt (weight_map f (lift (S i) O v)) (g i))).H1)))))) (\lambda (v:
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(t0: T).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map f (THead
+k0 u2 t0)) (weight_map g (THead k0 u1 t0)))))))) (\lambda (b: B).(B_ind
+(\lambda (b0: B).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map g
+(THead (Bind b0) u1 t0)))))))) (\lambda (f: ((nat \to nat))).(\lambda (g:
+((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g
+m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S
+(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
+(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
+(plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
+g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
+(weight_map g u1)) (le_S (S (weight_map f u2)) (weight_map g u1) (lt_le_S
+(weight_map f u2) (weight_map g u1) (H1 f g H2 H3))) n))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (plus_lt_le_compat (weight_map
+f u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n
+(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O
+O (le_n O) n))))))))))) (\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
+t0)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd
+f O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd
+f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) b)) (\lambda
+(_: F).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0
+(lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0
+t0)) (plus (weight_map g u1) (weight_map g t0)) (plus_lt_le_compat
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t0) (weight_map g t0) (H1
+f0 g H2 H3) (weight_le t0 f0 g H2)))))))) k))))))))) (\lambda (k: K).(K_ind
+(\lambda (k0: K).(\forall (v: T).(\forall (t2: T).(\forall (t1: T).(\forall
+(i: nat).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
+\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (lt
+(weight_map f t2) (weight_map g t1))))))) \to (\forall (u0: T).(\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map
+f (THead k0 u0 t2)) (weight_map g (THead k0 u0 t1))))))))))))))) (\lambda (b:
+B).(B_ind (\lambda (b0: B).(\forall (v: T).(\forall (t2: T).(\forall (t1:
+T).(\forall (i: nat).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind
+b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (u0:
+T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to
+(lt (weight_map f (THead (Bind b0) u0 t2)) (weight_map g (THead (Bind b0) u0
+t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
+u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
+t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd
+f (S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
+(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
+(S (weight_map g u0)) (lt_le_S (weight_map f u0) (S (weight_map g u0))
+(le_lt_n_Sm (weight_map f u0) (weight_map g u0) (weight_le u0 f g H2))) m))
+(lt_le_S (weight_map (wadd f (S (weight_map f u0))) (lift (S (S i)) O v))
+(wadd g (S (weight_map g u0)) (S i)) (eq_ind nat (weight_map f (lift (S i) O
+v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f (S (weight_map f
+u0))) (lift (S (S i)) O v)) (lift_weight_add_O (S (weight_map f u0)) v (S i)
+f))))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H1: ((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S i))) \to (lt
+(weight_map f t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
+(weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_lt_compat (weight_map
+f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
+g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
+(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
+(lift_weight_add_O O v (S i) f)))))))))))))))) (\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 (S i) v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (u0: T).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat
+\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
+(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0)
+(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
+t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd
+f O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O)
+(wadd g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
+(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
+f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda
+(u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda
+(H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H3: (lt (weight_map f0
+(lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f0 u0) (weight_map f0
+t2)) (plus (weight_map g u0) (weight_map g t1)) (plus_le_lt_compat
+(weight_map f0 u0) (weight_map g u0) (weight_map f0 t2) (weight_map g t1)
+(weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k)) (\lambda (v:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (_: (subst0 i
+v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f u2) (weight_map g u1)))))))).(\lambda
+(k: K).(K_ind (\lambda (k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s
+k0 i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(s k0 i)) O v)) (g (s k0 i))) \to (lt (weight_map f t2) (weight_map g
+t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f (THead k0 u2 t2)) (weight_map g (THead
+k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind (\lambda (b0: B).(\forall (t1:
+T).(\forall (t2: T).((subst0 (s (Bind b0) i) v t1 t2) \to (((\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S (s (Bind b0) i)) O v)) (g (s (Bind
+b0) i))) \to (lt (weight_map f t2) (weight_map g t1))))))) \to (\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (lt (weight_map
+f (THead (Bind b0) u2 t2)) (weight_map g (THead (Bind b0) u1 t1))))))))))))
+(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (subst0 (S i) v t1
+t2)).(\lambda (_: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+(S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f (S (weight_map f u2))) t2)) (plus (weight_map g u1)
+(weight_map (wadd g (S (weight_map g u1))) t1)) (plus_lt_le_compat
+(weight_map f u2) (weight_map g u1) (weight_map (wadd f (S (weight_map f
+u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f g H4 H5)
+(subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2))) (wadd g (S
+(weight_map g u1))) (\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2))
+(S (weight_map g u1)) (le_S (S (weight_map f u2)) (weight_map g u1) (lt_le_S
+(weight_map f u2) (weight_map g u1) (H1 f g H4 H5))) m)) (lt_le_S (weight_map
+(wadd f (S (weight_map f u2))) (lift (S (S i)) O v)) (wadd g (S (weight_map g
+u1)) (S i)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt
+n (g i))) H5 (weight_map (wadd f (S (weight_map f u2))) (lift (S (S i)) O v))
+(lift_weight_add_O (S (weight_map f u2)) v (S i) f)))))))))))))) (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3:
+((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (S i)) O v)) (g (S
+i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
+(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
+i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
+(weight_map g u1) (weight_map (wadd g O) t1)) (plus_lt_compat (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O
+m) (wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n
+O) m)))) (lt_le_S (weight_map (wadd f O) (lift (S (S i)) O v)) (wadd g O (S
+i)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v
+(S i) f)))))))))))))) (\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0
+(S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S (S i)) O v)) (g (S i))) \to (lt (weight_map f t2) (weight_map g
+t1)))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
+(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
+(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
+t1)) (plus_lt_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m)
+(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (lt_le_S (weight_map (wadd f
+O) (lift (S (S i)) O v)) (wadd g O (S i)) (eq_ind nat (weight_map f (lift (S
+i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S
+(S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))) b)) (\lambda (_:
+F).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 i v t1
+t2)).(\lambda (H3: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
+i) O v)) (g i)) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda
+(f0: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall
+(m: nat).(le (f0 m) (g m))))).(\lambda (H5: (lt (weight_map f0 (lift (S i) O
+v)) (g i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t2)) (plus
+(weight_map g u1) (weight_map g t1)) (plus_lt_compat (weight_map f0 u2)
+(weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1 f0 g H4 H5) (H3 f0
+g H4 H5)))))))))))) k)))))))) d u t z H))))).
+
+theorem subst0_tlt_head:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt
+(THead (Bind Abbr) u z) (THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus
+(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
+(weight_map (\lambda (_: nat).O) u))) t)) (plus_le_lt_compat (weight_map
+(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n
+(weight_map (\lambda (_: nat).O) u)) (subst0_weight_lt u t z O H (wadd
+(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (wadd (\lambda
+(_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) (\lambda (m: nat).(le_n
+(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u)) m)))
+(eq_ind nat (weight_map (\lambda (_: nat).O) (lift O O u)) (\lambda (n:
+nat).(lt n (S (weight_map (\lambda (_: nat).O) u)))) (eq_ind_r T u (\lambda
+(t0: T).(lt (weight_map (\lambda (_: nat).O) t0) (S (weight_map (\lambda (_:
+nat).O) u)))) (le_n (S (weight_map (\lambda (_: nat).O) u))) (lift O O u)
+(lift_r u O)) (weight_map (wadd (\lambda (_: nat).O) (S (weight_map (\lambda
+(_: nat).O) u))) (lift (S O) O u)) (lift_weight_add_O (S (weight_map (\lambda
+(_: nat).O) u)) u O (\lambda (_: nat).O))))))))).
+
+theorem subst0_tlt:
+ \forall (u: T).(\forall (t: T).(\forall (z: T).((subst0 O u t z) \to (tlt z
+(THead (Bind Abbr) u t)))))
+\def
+ \lambda (u: T).(\lambda (t: T).(\lambda (z: T).(\lambda (H: (subst0 O u t
+z)).(tlt_trans (THead (Bind Abbr) u z) z (THead (Bind Abbr) u t) (tlt_head_dx
+(Bind Abbr) u z) (subst0_tlt_head u t z H))))).
+
+theorem dnf_dec:
+ \forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or
+(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
+\def
+ \lambda (w: T).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(ex
+T (\lambda (v: T).(or (subst0 d w t0 (lift (S O) d v)) (eq T t0 (lift (S O) d
+v))))))) (\lambda (n: nat).(\lambda (d: nat).(ex_intro T (\lambda (v: T).(or
+(subst0 d w (TSort n) (lift (S O) d v)) (eq T (TSort n) (lift (S O) d v))))
+(TSort n) (eq_ind_r T (TSort n) (\lambda (t0: T).(or (subst0 d w (TSort n)
+t0) (eq T (TSort n) t0))) (or_intror (subst0 d w (TSort n) (TSort n)) (eq T
+(TSort n) (TSort n)) (refl_equal T (TSort n))) (lift (S O) d (TSort n))
+(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
+(ex T (\lambda (v: T).(or (subst0 d w (TLRef n) (lift (S O) d v)) (eq T
+(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(ex_intro T (\lambda
+(v: T).(or (subst0 d w (TLRef n) (lift (S O) d v)) (eq T (TLRef n) (lift (S
+O) d v)))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (subst0 d w
+(TLRef n) t0) (eq T (TLRef n) t0))) (or_intror (subst0 d w (TLRef n) (TLRef
+n)) (eq T (TLRef n) (TLRef n)) (refl_equal T (TLRef n))) (lift (S O) d (TLRef
+n)) (lift_lref_lt n (S O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n
+(\lambda (n0: nat).(ex T (\lambda (v: T).(or (subst0 n0 w (TLRef n) (lift (S
+O) n0 v)) (eq T (TLRef n) (lift (S O) n0 v)))))) (ex_intro T (\lambda (v:
+T).(or (subst0 n w (TLRef n) (lift (S O) n v)) (eq T (TLRef n) (lift (S O) n
+v)))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w) (\lambda (t0: T).(or
+(subst0 n w (TLRef n) t0) (eq T (TLRef n) t0))) (or_introl (subst0 n w (TLRef
+n) (lift (S n) O w)) (eq T (TLRef n) (lift (S n) O w)) (subst0_lref w n))
+(lift (S O) n (lift n O w)) (lift_free w n (S O) O n (le_n (plus O n))
+(le_O_n n)))) d H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (v: T).(or
+(subst0 d w (TLRef n) (lift (S O) d v)) (eq T (TLRef n) (lift (S O) d v))))
+(TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (subst0 d w
+(TLRef n) t0) (eq T (TLRef n) t0))) (or_intror (subst0 d w (TLRef n) (TLRef
+n)) (eq T (TLRef n) (TLRef n)) (refl_equal T (TLRef n))) (lift (S O) d (TLRef
+(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (v: T).(or (subst0 d w t0
+(lift (S O) d v)) (eq T t0 (lift (S O) d v)))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (d: nat).(ex T (\lambda (v: T).(or (subst0 d w t1 (lift (S O)
+d v)) (eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
+in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 d w t0 (lift (S
+O) d v)) (eq T t0 (lift (S O) d v)))) (ex T (\lambda (v: T).(or (subst0 d w
+(THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x: T).(\lambda (H2: (or (subst0 d w t0 (lift (S O) d x)) (eq T t0
+(lift (S O) d x)))).(or_ind (subst0 d w t0 (lift (S O) d x)) (eq T t0 (lift
+(S O) d x)) (ex T (\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O)
+d v)) (eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (H3: (subst0 d w t0
+(lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H4 \def H_x0 in
+(ex_ind T (\lambda (v: T).(or (subst0 (s k d) w t1 (lift (S O) (s k d) v))
+(eq T t1 (lift (S O) (s k d) v)))) (ex T (\lambda (v: T).(or (subst0 d w
+(THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x0: T).(\lambda (H5: (or (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0)) (eq T t1 (lift (S O) (s k d) x0)))).(or_ind (subst0 (s k d) w t1 (lift
+(S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)) (ex T (\lambda (v:
+T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O)
+d v)) (eq T (THead k t0 t1) (lift (S O) d v)))) (THead k x x0) (eq_ind_r T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(or
+(subst0 d w (THead k t0 t1) t2) (eq T (THead k t0 t1) t2))) (or_introl
+(subst0 d w (THead k t0 t1) (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0))) (eq T (THead k t0 t1) (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0))) (subst0_both w t0 (lift (S O) d x) d H3 k t1 (lift (S O) (s k d) x0)
+H6)) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d)))) (\lambda
+(H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T (lift (S O) (s k d) x0)
+(\lambda (t2: T).(ex T (\lambda (v: T).(or (subst0 d w (THead k t0 t2) (lift
+(S O) d v)) (eq T (THead k t0 t2) (lift (S O) d v)))))) (ex_intro T (\lambda
+(v: T).(or (subst0 d w (THead k t0 (lift (S O) (s k d) x0)) (lift (S O) d v))
+(eq T (THead k t0 (lift (S O) (s k d) x0)) (lift (S O) d v)))) (THead k x x0)
+(eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (\lambda (t2:
+T).(or (subst0 d w (THead k t0 (lift (S O) (s k d) x0)) t2) (eq T (THead k t0
+(lift (S O) (s k d) x0)) t2))) (or_introl (subst0 d w (THead k t0 (lift (S O)
+(s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) x0))) (eq T (THead
+k t0 (lift (S O) (s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0))) (subst0_fst w (lift (S O) d x) t0 d H3 (lift (S O) (s k d) x0) k))
+(lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d))) t1 H6)) H5)))
+H4)))) (\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d))
+in (let H4 \def H_x0 in (ex_ind T (\lambda (v: T).(or (subst0 (s k d) w t1
+(lift (S O) (s k d) v)) (eq T t1 (lift (S O) (s k d) v)))) (ex T (\lambda (v:
+T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (x0: T).(\lambda (H5: (or (subst0 (s k d) w t1
+(lift (S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)))).(or_ind (subst0
+(s k d) w t1 (lift (S O) (s k d) x0)) (eq T t1 (lift (S O) (s k d) x0)) (ex T
+(\lambda (v: T).(or (subst0 d w (THead k t0 t1) (lift (S O) d v)) (eq T
+(THead k t0 t1) (lift (S O) d v))))) (\lambda (H6: (subst0 (s k d) w t1 (lift
+(S O) (s k d) x0))).(eq_ind_r T (lift (S O) d x) (\lambda (t2: T).(ex T
+(\lambda (v: T).(or (subst0 d w (THead k t2 t1) (lift (S O) d v)) (eq T
+(THead k t2 t1) (lift (S O) d v)))))) (ex_intro T (\lambda (v: T).(or (subst0
+d w (THead k (lift (S O) d x) t1) (lift (S O) d v)) (eq T (THead k (lift (S
+O) d x) t1) (lift (S O) d v)))) (THead k x x0) (eq_ind_r T (THead k (lift (S
+O) d x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(or (subst0 d w (THead k
+(lift (S O) d x) t1) t2) (eq T (THead k (lift (S O) d x) t1) t2))) (or_introl
+(subst0 d w (THead k (lift (S O) d x) t1) (THead k (lift (S O) d x) (lift (S
+O) (s k d) x0))) (eq T (THead k (lift (S O) d x) t1) (THead k (lift (S O) d
+x) (lift (S O) (s k d) x0))) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H6
+(lift (S O) d x))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d)))
+t0 H3)) (\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T (lift (S
+O) (s k d) x0) (\lambda (t2: T).(ex T (\lambda (v: T).(or (subst0 d w (THead
+k t0 t2) (lift (S O) d v)) (eq T (THead k t0 t2) (lift (S O) d v))))))
+(eq_ind_r T (lift (S O) d x) (\lambda (t2: T).(ex T (\lambda (v: T).(or
+(subst0 d w (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)) (eq T
+(THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (ex_intro T
+(\lambda (v: T).(or (subst0 d w (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0)) (lift (S O) d v)) (eq T (THead k (lift (S O) d x) (lift (S O) (s k d)
+x0)) (lift (S O) d v)))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) (\lambda (t2: T).(or (subst0 d w (THead k (lift (S
+O) d x) (lift (S O) (s k d) x0)) t2) (eq T (THead k (lift (S O) d x) (lift (S
+O) (s k d) x0)) t2))) (or_intror (subst0 d w (THead k (lift (S O) d x) (lift
+(S O) (s k d) x0)) (THead k (lift (S O) d x) (lift (S O) (s k d) x0))) (eq T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0))) (refl_equal T (THead k (lift (S O) d x) (lift (S O)
+(s k d) x0)))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O) d))) t0
+H3) t1 H6)) H5))) H4)))) H2))) H1))))))))) t)).
inductive subst1 (i:nat) (v:T) (t1:T): T \to Prop \def
| subst1_refl: subst1 i v t1 t1
| subst1_single: \forall (t2: T).((subst0 i v t1 t2) \to (subst1 i v t1 t2)).
-axiom subst1_head: \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1 i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2)))))))))) .
-
-axiom subst1_gen_sort: \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 i v (TSort n) x) \to (eq T x (TSort n)))))) .
-
-axiom subst1_gen_lref: \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1 i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift (S n) O v)))))))) .
-
-axiom subst1_gen_head: \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2)))))))))) .
-
-axiom subst1_gen_lift_lt: \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst1 i u t1 t2))))))))) .
-
-axiom subst1_gen_lift_eq: \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u (lift h d t) x) \to (eq T x (lift h d t)))))))))) .
-
-axiom subst1_gen_lift_ge: \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)))))))))) .
+theorem subst1_head:
+ \forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
+i v u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((subst1 (s
+k i) v t1 t2) \to (subst1 i v (THead k u1 t1) (THead k u2 t2))))))))))
+\def
+ \lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t: T).(\forall (k:
+K).(\forall (t1: T).(\forall (t2: T).((subst1 (s k i) v t1 t2) \to (subst1 i
+v (THead k u1 t1) (THead k t t2))))))) (\lambda (k: K).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (subst1 (s k i) v t1 t2)).(subst1_ind (s k
+i) v t1 (\lambda (t: T).(subst1 i v (THead k u1 t1) (THead k u1 t)))
+(subst1_refl i v (THead k u1 t1)) (\lambda (t3: T).(\lambda (H1: (subst0 (s k
+i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k u1 t3) (subst0_snd k
+v t3 t1 i H1 u1)))) t2 H0))))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
+t2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H1: (subst1
+(s k i) v t1 t0)).(subst1_ind (s k i) v t1 (\lambda (t: T).(subst1 i v (THead
+k u1 t1) (THead k t2 t))) (subst1_single i v (THead k u1 t1) (THead k t2 t1)
+(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k
+i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both
+v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
+
+theorem subst1_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
+i v (TSort n) x) \to (eq T x (TSort n))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst1 i v (TSort n) x)).(subst1_ind i v (TSort n) (\lambda (t: T).(eq T
+t (TSort n))) (refl_equal T (TSort n)) (\lambda (t2: T).(\lambda (H0: (subst0
+i v (TSort n) t2)).(subst0_gen_sort v t2 i n H0 (eq T t2 (TSort n))))) x
+H))))).
+
+theorem subst1_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
+i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
+(S n) O v))))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (subst1 i v (TLRef n) x)).(subst1_ind i v (TLRef n) (\lambda (t: T).(or
+(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
+(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
+v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
+(TLRef n) t2)).(and_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
+nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
+(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
+(eq T t2 (lift (S n) O v)) H1 H2)))) (subst0_gen_lref v t2 i n H0)))) x
+H))))).
+
+theorem subst1_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
+(x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst1 (s k i) v t1 t2))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1)
+x)).(subst1_ind i v (THead k u1 t1) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k u1
+t1) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst1 (s k i) v t1 t2))) u1 t1 (refl_equal
+T (THead k u1 t1)) (subst1_refl i v u1) (subst1_refl (s k i) v t1)) (\lambda
+(t2: T).(\lambda (H0: (subst0 i v (THead k u1 t1) t2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1
+u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst1 (s k i) v t1 t3)))) (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2
+(THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda
+(H2: (eq T t2 (THead k x0 t1))).(\lambda (H3: (subst0 i v u1
+x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0 t1 H2 (subst1_single i v u1
+x0 H3) (subst1_refl (s k i) v t1))))) H1)) (\lambda (H1: (ex2 T (\lambda (t3:
+T).(eq T t2 (THead k u1 t3))) (\lambda (t2: T).(subst0 (s k i) v t1
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3:
+T).(subst0 (s k i) v t1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3)))) (\lambda (x0:
+T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
+t1 x0)).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) u1 x0 H2 (subst1_refl i v u1)
+(subst1_single (s k i) v t1 x0 H3))))) H1)) (\lambda (H1: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i) v t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst1 (s k i) v t1 t3)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H2: (eq T t2 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda
+(H4: (subst0 (s k i) v t1 x1)).(ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1
+i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s k i) v t1 t3))) x0
+x1 H2 (subst1_single i v u1 x0 H3) (subst1_single (s k i) v t1 x1 H4)))))))
+H1)) (subst0_gen_head k v u1 t1 t2 i H0)))) x H))))))).
+
+theorem subst1_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
+(t2: T).(subst1 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
+(plus i d)) t1) x)).(subst1_ind i (lift h d u) (lift h (S (plus i d)) t1)
+(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2)))
+(\lambda (t2: T).(subst1 i u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T
+(lift h (S (plus i d)) t1) (lift h (S (plus i d)) t2))) (\lambda (t2:
+T).(subst1 i u t1 t2)) t1 (refl_equal T (lift h (S (plus i d)) t1))
+(subst1_refl i u t1)) (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u)
+(lift h (S (plus i d)) t1) t2)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h
+(S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u t1 t3)) (ex2 T (\lambda
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1
+t3))) (\lambda (x0: T).(\lambda (H1: (eq T t2 (lift h (S (plus i d))
+x0))).(\lambda (H2: (subst0 i u t1 x0)).(ex_intro2 T (\lambda (t3: T).(eq T
+t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst1 i u t1 t3)) x0 H1
+(subst1_single i u t1 x0 H2))))) (subst0_gen_lift_lt u t1 t2 i h d H0)))) x
+H))))))).
+
+theorem subst1_gen_lift_eq:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
+(lift h d t) x) \to (eq T x (lift h d t))))))))))
+\def
+ \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
+h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(subst1_ind i u (lift h d t)
+(\lambda (t0: T).(eq T t0 (lift h d t))) (refl_equal T (lift h d t)) (\lambda
+(t2: T).(\lambda (H2: (subst0 i u (lift h d t) t2)).(subst0_gen_lift_false t
+u t2 h d i H H0 H2 (eq T t2 (lift h d t))))) x H1))))))))).
+
+theorem subst1_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
+(h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
+T).(subst1 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
+x)).(\lambda (H0: (le (plus d h) i)).(subst1_ind i u (lift h d t1) (\lambda
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d t2))) (\lambda (t2:
+T).(subst1 (minus i h) u t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift
+h d t1) (lift h d t2))) (\lambda (t2: T).(subst1 (minus i h) u t1 t2)) t1
+(refl_equal T (lift h d t1)) (subst1_refl (minus i h) u t1)) (\lambda (t2:
+T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(ex2_ind T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u t1 t3))
+(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1
+(minus i h) u t1 t3))) (\lambda (x0: T).(\lambda (H2: (eq T t2 (lift h d
+x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(ex_intro2 T (\lambda (t3:
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst1 (minus i h) u t1 t3)) x0
+H2 (subst1_single (minus i h) u t1 x0 H3))))) (subst0_gen_lift_ge u t1 t2 i h
+d H1 H0)))) x H)))))))).
+
+theorem subst1_lift_lt:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1
+i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i
+(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t: T).(\forall (d:
+nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u)
+(lift h d t1) (lift h d t)))))) (\lambda (d: nat).(\lambda (_: (lt i
+d)).(\lambda (h: nat).(subst1_refl i (lift h (minus d (S i)) u) (lift h d
+t1))))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda (d:
+nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
+(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d
+H1 h))))))) t2 H))))).
+
+theorem subst1_lift_ge:
+ \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
+(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1
+(plus i h) u (lift h d t1) (lift h d t2)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(h: nat).(\lambda (H: (subst1 i u t1 t2)).(subst1_ind i u t1 (\lambda (t:
+T).(\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h
+d t))))) (\lambda (d: nat).(\lambda (_: (le d i)).(subst1_refl (plus i h) u
+(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda
+(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1)
+(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
+
+theorem subst1_ex:
+ \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2:
+T).(subst1 d u t1 (lift (S O) d t2))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (d: nat).(ex
+T (\lambda (t2: T).(subst1 d u t (lift (S O) d t2)))))) (\lambda (n:
+nat).(\lambda (d: nat).(ex_intro T (\lambda (t2: T).(subst1 d u (TSort n)
+(lift (S O) d t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(subst1 d
+u (TSort n) t)) (subst1_refl d u (TSort n)) (lift (S O) d (TSort n))
+(lift_sort n (S O) d))))) (\lambda (n: nat).(\lambda (d: nat).(lt_eq_gt_e n d
+(ex T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O) d t2)))) (\lambda
+(H: (lt n d)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n) (lift (S O)
+d t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t: T).(subst1 d u (TLRef n)
+t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef n)) (lift_lref_lt n (S
+O) d H)))) (\lambda (H: (eq nat n d)).(eq_ind nat n (\lambda (n0: nat).(ex T
+(\lambda (t2: T).(subst1 n0 u (TLRef n) (lift (S O) n0 t2))))) (ex_intro T
+(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u)
+(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n)
+t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S
+O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d
+H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n)
+(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef
+(pred n))) (lift_lref_gt d n H))))))) (\lambda (k: K).(\lambda (t:
+T).(\lambda (H: ((\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t (lift
+(S O) d t2))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (d: nat).(ex T
+(\lambda (t2: T).(subst1 d u t0 (lift (S O) d t2))))))).(\lambda (d:
+nat).(let H_x \def (H d) in (let H1 \def H_x in (ex_ind T (\lambda (t2:
+T).(subst1 d u t (lift (S O) d t2))) (ex T (\lambda (t2: T).(subst1 d u
+(THead k t t0) (lift (S O) d t2)))) (\lambda (x: T).(\lambda (H2: (subst1 d u
+t (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
+(ex_ind T (\lambda (t2: T).(subst1 (s k d) u t0 (lift (S O) (s k d) t2))) (ex
+T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d t2)))) (\lambda
+(x0: T).(\lambda (H4: (subst1 (s k d) u t0 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (t2: T).(subst1 d u (THead k t t0) (lift (S O) d
+t2))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift (S O) (s k
+d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
+(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k
+x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
+
+theorem subst1_subst1:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
+u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0
+t)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
+(subst1 i u u1 u2)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(t: T).(subst1 (S (plus i j)) u t t1)) t1 (subst1_refl j u1 t1) (subst1_refl
+(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
+t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
+i u u1 u2)).(insert_eq T u2 (\lambda (t: T).(subst1 i u u1 t)) (ex2 T
+(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
+t t3))) (\lambda (y: T).(\lambda (H2: (subst1 i u u1 y)).(subst1_ind i u u1
+(\lambda (t: T).((eq T t u2) \to (ex2 T (\lambda (t0: T).(subst1 j u1 t1 t0))
+(\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3))))) (\lambda (H3: (eq T u1
+u2)).(eq_ind_r T u2 (\lambda (t: T).(ex2 T (\lambda (t0: T).(subst1 j t t1
+t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t0 t3)))) (ex_intro2 T
+(\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
+t t3)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i j)) u t3)) u1
+H3)) (\lambda (t0: T).(\lambda (H3: (subst0 i u u1 t0)).(\lambda (H4: (eq T
+t0 u2)).(let H5 \def (eq_ind T t0 (\lambda (t: T).(subst0 i u u1 t)) H3 u2
+H4) in (ex2_ind T (\lambda (t: T).(subst0 j u1 t1 t)) (\lambda (t: T).(subst0
+(S (plus i j)) u t t3)) (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(t: T).(subst1 (S (plus i j)) u t t3))) (\lambda (x: T).(\lambda (H6: (subst0
+j u1 t1 x)).(\lambda (H7: (subst0 (S (plus i j)) u x t3)).(ex_intro2 T
+(\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u
+t t3)) x (subst1_single j u1 t1 x H6) (subst1_single (S (plus i j)) u x t3
+H7))))) (subst0_subst0 t1 t3 u2 j H0 u1 u i H5)))))) y H2))) H1))))))) t2
+H))))).
+
+theorem subst1_subst1_back:
+ \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
+j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
+u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
+T).(subst1 (S (plus i j)) u t2 t)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
+(H: (subst1 j u2 t1 t2)).(subst1_ind j u2 t1 (\lambda (t: T).(\forall (u1:
+T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda
+(t0: T).(subst1 j u1 t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t
+t0)))))))) (\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_:
+(subst1 i u u2 u1)).(ex_intro2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda
+(t: T).(subst1 (S (plus i j)) u t1 t)) t1 (subst1_refl j u1 t1) (subst1_refl
+(S (plus i j)) u t1)))))) (\lambda (t3: T).(\lambda (H0: (subst0 j u2 t1
+t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (subst1
+i u u2 u1)).(subst1_ind i u u2 (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(subst1 j t t1 t0)) (\lambda (t0: T).(subst1 (S (plus i j)) u t3 t0))))
+(ex_intro2 T (\lambda (t: T).(subst1 j u2 t1 t)) (\lambda (t: T).(subst1 (S
+(plus i j)) u t3 t)) t3 (subst1_single j u2 t1 t3 H0) (subst1_refl (S (plus i
+j)) u t3)) (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(ex2_ind T
+(\lambda (t: T).(subst0 j t0 t1 t)) (\lambda (t: T).(subst0 (S (plus i j)) u
+t3 t)) (ex2 T (\lambda (t: T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S
+(plus i j)) u t3 t))) (\lambda (x: T).(\lambda (H3: (subst0 j t0 t1
+x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(ex_intro2 T (\lambda (t:
+T).(subst1 j t0 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t3 t)) x
+(subst1_single j t0 t1 x H3) (subst1_single (S (plus i j)) u t3 x H4)))))
+(subst0_subst0_back t1 t3 u2 j H0 t0 u i H2)))) u1 H1))))))) t2 H))))).
+
+theorem subst1_trans:
+ \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1
+i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1
+t3)))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i v t1 t2)).(subst1_ind i v t1 (\lambda (t: T).(\forall (t3:
+T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) (\lambda (t3: T).(\lambda
+(H0: (subst1 i v t1 t3)).H0)) (\lambda (t3: T).(\lambda (H0: (subst0 i v t1
+t3)).(\lambda (t4: T).(\lambda (H1: (subst1 i v t3 t4)).(subst1_ind i v t3
+(\lambda (t: T).(subst1 i v t1 t)) (subst1_single i v t1 t3 H0) (\lambda (t0:
+T).(\lambda (H2: (subst0 i v t3 t0)).(subst1_single i v t1 t0 (subst0_trans
+t3 t1 v i H0 t0 H2)))) t4 H1))))) t2 H))))).
+
+theorem subst1_confluence_neq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
+nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
+(i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
+(t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
+nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(subst1_ind i1 u1 t0 (\lambda (t:
+T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2)
+\to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t3: T).(subst1 i2 u2 t t3))
+(\lambda (t3: T).(subst1 i1 u1 t2 t3))))))))) (\lambda (t2: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not
+(eq nat i1 i2))).(ex_intro2 T (\lambda (t: T).(subst1 i2 u2 t0 t)) (\lambda
+(t: T).(subst1 i1 u1 t2 t)) t2 H0 (subst1_refl i1 u1 t2))))))) (\lambda (t2:
+T).(\lambda (H0: (subst0 i1 u1 t0 t2)).(\lambda (t3: T).(\lambda (u2:
+T).(\lambda (i2: nat).(\lambda (H1: (subst1 i2 u2 t0 t3)).(\lambda (H2: (not
+(eq nat i1 i2))).(subst1_ind i2 u2 t0 (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(subst1 i2 u2 t2 t4)) (\lambda (t4: T).(subst1 i1 u1 t t4)))) (ex_intro2 T
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t0 t)) t2
+(subst1_refl i2 u2 t2) (subst1_single i1 u1 t0 t2 H0)) (\lambda (t4:
+T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(ex2_ind T (\lambda (t: T).(subst0 i1
+u1 t4 t)) (\lambda (t: T).(subst0 i2 u2 t2 t)) (ex2 T (\lambda (t: T).(subst1
+i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t))) (\lambda (x: T).(\lambda
+(H4: (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(ex_intro2 T
+(\lambda (t: T).(subst1 i2 u2 t2 t)) (\lambda (t: T).(subst1 i1 u1 t4 t)) x
+(subst1_single i2 u2 t2 x H5) (subst1_single i1 u1 t4 x H4)))))
+(subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 (sym_not_eq nat i1 i2
+H2))))) t3 H1)))))))) t1 H))))).
+
+theorem subst1_confluence_eq:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
+i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t:
+T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u t0 t1)).(subst1_ind i u t0 (\lambda (t: T).(\forall (t2:
+T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t3: T).(subst1 i u t t3))
+(\lambda (t3: T).(subst1 i u t2 t3)))))) (\lambda (t2: T).(\lambda (H0:
+(subst1 i u t0 t2)).(ex_intro2 T (\lambda (t: T).(subst1 i u t0 t)) (\lambda
+(t: T).(subst1 i u t2 t)) t2 H0 (subst1_refl i u t2)))) (\lambda (t2:
+T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i
+u t0 t3)).(subst1_ind i u t0 (\lambda (t: T).(ex2 T (\lambda (t4: T).(subst1
+i u t2 t4)) (\lambda (t4: T).(subst1 i u t t4)))) (ex_intro2 T (\lambda (t:
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t0 t)) t2 (subst1_refl i u
+t2) (subst1_single i u t0 t2 H0)) (\lambda (t4: T).(\lambda (H2: (subst0 i u
+t0 t4)).(or4_ind (eq T t4 t2) (ex2 T (\lambda (t: T).(subst0 i u t4 t))
+(\lambda (t: T).(subst0 i u t2 t))) (subst0 i u t4 t2) (subst0 i u t2 t4)
+(ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)))
+(\lambda (H3: (eq T t4 t2)).(eq_ind_r T t2 (\lambda (t: T).(ex2 T (\lambda
+(t5: T).(subst1 i u t2 t5)) (\lambda (t5: T).(subst1 i u t t5)))) (ex_intro2
+T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t2 t)) t2
+(subst1_refl i u t2) (subst1_refl i u t2)) t4 H3)) (\lambda (H3: (ex2 T
+(\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i
+u t2 t)) (ex2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i
+u t4 t))) (\lambda (x: T).(\lambda (H4: (subst0 i u t4 x)).(\lambda (H5:
+(subst0 i u t2 x)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda
+(t: T).(subst1 i u t4 t)) x (subst1_single i u t2 x H5) (subst1_single i u t4
+x H4))))) H3)) (\lambda (H3: (subst0 i u t4 t2)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u t2 t)) (\lambda (t: T).(subst1 i u t4 t)) t2 (subst1_refl i u
+t2) (subst1_single i u t4 t2 H3))) (\lambda (H3: (subst0 i u t2
+t4)).(ex_intro2 T (\lambda (t: T).(subst1 i u t2 t)) (\lambda (t: T).(subst1
+i u t4 t)) t4 (subst1_single i u t2 t4 H3) (subst1_refl i u t4)))
+(subst0_confluence_eq t0 t4 u i H2 t2 H0)))) t3 H1))))) t1 H))))).
+
+theorem subst1_confluence_lift:
+ \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
+i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i
+t2)) \to (eq T t1 t2)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u t0 (lift (S O) i t1))).(insert_eq T (lift (S O) i t1)
+(\lambda (t: T).(subst1 i u t0 t)) (\forall (t2: T).((subst1 i u t0 (lift (S
+O) i t2)) \to (eq T t1 t2))) (\lambda (y: T).(\lambda (H0: (subst1 i u t0
+y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i t1)) \to
+(\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 t2)))))
+(\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H2:
+(subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda (t:
+T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 \def
+(sym_equal T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u
+(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O)))
+H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1:
+(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3:
+T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2
+(\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T
+(lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (eq T t1 t3) (\lambda
+(y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 (\lambda (t:
+T).((eq T t (lift (S O) i t3)) \to (eq T t1 t3))) (\lambda (H6: (eq T t0
+(lift (S O) i t3))).(let H7 \def (eq_ind T t0 (\lambda (t: T).(subst0 i u t
+(lift (S O) i t1))) H4 (lift (S O) i t3) H6) in (subst0_gen_lift_false t3 u
+(lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O)))
+H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6: (subst0 i u t0
+t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def (eq_ind T t4
+(\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in (sym_eq T t3
+t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5))) H3))))))) y
+H0))) H))))).
-axiom subst1_lift_lt: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i (lift h (minus d (S i)) u) (lift h d t1) (lift h d t2))))))))) .
-
-axiom subst1_lift_ge: \forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall (h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1 (plus i h) u (lift h d t1) (lift h d t2))))))))) .
-
-axiom subst1_ex: \forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2: T).(subst1 d u t1 (lift (S O) d t2)))))) .
-
-axiom subst1_subst1: \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t t2))))))))))) .
+inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
+| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1:
+T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i)
+v (CHead c k u1) (CHead c k u2))))))))
+| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s
+k i) v (CHead c1 k u) (CHead c2 k u))))))))
+| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall
+(u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall
+(c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2
+k u2)))))))))).
+
+theorem csubst0_snd_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c
+(Bind b) u1) (CHead c (Bind b) u2))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c: C).(eq_ind nat (s (Bind
+b) i) (\lambda (n: nat).(csubst0 n v (CHead c (Bind b) u1) (CHead c (Bind b)
+u2))) (csubst0_snd (Bind b) i v u1 u2 H c) (S i) (refl_equal nat (S
+i))))))))).
+
+theorem csubst0_fst_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall
+(v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1
+(Bind b) u) (CHead c2 (Bind b) u))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (H: (csubst0 i v c1 c2)).(\lambda (u: T).(eq_ind nat (s (Bind
+b) i) (\lambda (n: nat).(csubst0 n v (CHead c1 (Bind b) u) (CHead c2 (Bind b)
+u))) (csubst0_fst (Bind b) i c1 c2 v H u) (S i) (refl_equal nat (S i))))))))).
+
+theorem csubst0_both_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i
+v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b)
+u2))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst0 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst0 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n:
+nat).(csubst0 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))
+(csubst0_both (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S
+i))))))))))).
+
+theorem csubst0_gen_sort:
+ \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0
+i v (CSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
+(H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(let H0 \def (match H
+return (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0:
+C).(\lambda (_: (csubst0 n0 t c c0)).((eq nat n0 i) \to ((eq T t v) \to ((eq
+C c (CSort n)) \to ((eq C c0 x) \to P))))))))) with [(csubst0_snd k i0 v0 u1
+u2 H0 c) \Rightarrow (\lambda (H1: (eq nat (s k i0) i)).(\lambda (H2: (eq T
+v0 v)).(\lambda (H3: (eq C (CHead c k u1) (CSort n))).(\lambda (H4: (eq C
+(CHead c k u2) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to
+((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 i0
+v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t:
+T).((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0
+i0 t u1 u2) \to P)))) (\lambda (H6: (eq C (CHead c k u1) (CSort n))).(let H7
+\def (eq_ind C (CHead c k u1) (\lambda (e: C).(match e return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort n) H6) in (False_ind ((eq C (CHead c k u2) x) \to ((subst0
+i0 v u1 u2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) |
+(csubst0_fst k i0 c1 c2 v0 H0 u) \Rightarrow (\lambda (H1: (eq nat (s k i0)
+i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c1 k u) (CSort
+n))).(\lambda (H4: (eq C (CHead c2 k u) x)).(eq_ind nat (s k i0) (\lambda (_:
+nat).((eq T v0 v) \to ((eq C (CHead c1 k u) (CSort n)) \to ((eq C (CHead c2 k
+u) x) \to ((csubst0 i0 v0 c1 c2) \to P))))) (\lambda (H5: (eq T v0
+v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c1 k u) (CSort n)) \to ((eq C
+(CHead c2 k u) x) \to ((csubst0 i0 t c1 c2) \to P)))) (\lambda (H6: (eq C
+(CHead c1 k u) (CSort n))).(let H7 \def (eq_ind C (CHead c1 k u) (\lambda (e:
+C).(match e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ _ _) \Rightarrow True])) I (CSort n) H6) in (False_ind ((eq C (CHead
+c2 k u) x) \to ((csubst0 i0 v c1 c2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i
+H1 H2 H3 H4 H0))))) | (csubst0_both k i0 v0 u1 u2 H0 c1 c2 H1) \Rightarrow
+(\lambda (H2: (eq nat (s k i0) i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4:
+(eq C (CHead c1 k u1) (CSort n))).(\lambda (H5: (eq C (CHead c2 k u2)
+x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c1
+k u1) (CSort n)) \to ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v0 u1 u2) \to
+((csubst0 i0 v0 c1 c2) \to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v
+(\lambda (t: T).((eq C (CHead c1 k u1) (CSort n)) \to ((eq C (CHead c2 k u2)
+x) \to ((subst0 i0 t u1 u2) \to ((csubst0 i0 t c1 c2) \to P))))) (\lambda
+(H7: (eq C (CHead c1 k u1) (CSort n))).(let H8 \def (eq_ind C (CHead c1 k u1)
+(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H7) in
+(False_ind ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v u1 u2) \to ((csubst0
+i0 v c1 c2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))])
+in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal C (CSort n))
+(refl_equal C x)))))))).
+
+theorem csubst0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
+(v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
+u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))))
+(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2))))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) x)).(let
+H0 \def (match H return (\lambda (n: nat).(\lambda (t: T).(\lambda (c:
+C).(\lambda (c0: C).(\lambda (_: (csubst0 n t c c0)).((eq nat n i) \to ((eq T
+t v) \to ((eq C c (CHead c1 k u1)) \to ((eq C c0 x) \to (or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
+u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))))
+(\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2))))))))))))))) with [(csubst0_snd k0 i0 v0 u0 u2 H0 c) \Rightarrow
+(\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3:
+(eq C (CHead c k0 u0) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c k0 u2)
+x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c
+k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c k0 u2) x) \to ((subst0 i0 v0 u0
+u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k
+j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda
+(u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c2: C).(\lambda (_:
+nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j
+v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat n (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_:
+nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v c1 c2))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v
+(\lambda (t: T).((eq C (CHead c k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c
+k0 u2) x) \to ((subst0 i0 t u0 u2) \to (or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda
+(_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j:
+nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2
+k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k
+j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2
+k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2)))))))))) (\lambda (H6: (eq C (CHead c k0 u0) (CHead c1 k u1))).(let H7
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c k0 u0)
+(CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c k0 u0) (CHead c1 k u1) H6) in ((let H9 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow c | (CHead c _ _) \Rightarrow c])) (CHead c k0 u0) (CHead c1 k
+u1) H6) in (eq_ind C c1 (\lambda (c0: C).((eq K k0 k) \to ((eq T u0 u1) \to
+((eq C (CHead c0 k0 u2) x) \to ((subst0 i0 v u0 u2) \to (or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j:
+nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2
+k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k
+j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2
+k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T
+u0 u1) \to ((eq C (CHead c1 k1 u2) x) \to ((subst0 i0 v u0 u2) \to (or3
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))
+(\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3:
+T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda
+(_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda
+(c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))) (\lambda
+(H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c1 k u2) x)
+\to ((subst0 i0 v t u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x
+(CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3))))
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
+(u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3)))))
+(\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3))))
+(\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2))))))))) (\lambda (H12: (eq C (CHead c1 k u2) x)).(eq_ind C (CHead c1 k
+u2) (\lambda (c0: C).((subst0 i0 v u1 u2) \to (or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda
+(_: nat).(eq C c0 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j:
+nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2
+k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k
+j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2
+k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1
+c2)))))))) (\lambda (H13: (subst0 i0 v u1 u2)).(let H \def (eq_ind K k0
+(\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (or3_intro0 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3:
+T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3:
+T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda
+(_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda
+(j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda
+(c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u3))))) (\lambda
+(u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))) (ex3_2_intro T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda
+(u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda
+(u3: T).(\lambda (j: nat).(subst0 j v u1 u3))) u2 i0 (refl_equal nat (s k
+i0)) (refl_equal C (CHead c1 k u2)) H13)))) x H12)) u0 (sym_eq T u0 u1 H11)))
+k0 (sym_eq K k0 k H10))) c (sym_eq C c c1 H9))) H8)) H7))) v0 (sym_eq T v0 v
+H5))) i H1 H2 H3 H4 H0))))) | (csubst0_fst k0 i0 c0 c2 v0 H0 u) \Rightarrow
+(\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3:
+(eq C (CHead c0 k0 u) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c2 k0 u)
+x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0
+k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v0 c0
+c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat n (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c1 c3))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v
+(\lambda (t: T).((eq C (CHead c0 k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2
+k0 u) x) \to ((csubst0 i0 t c0 c2) \to (or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda
+(_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))))))))) (\lambda (H6: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H7
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u)
+(CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c0 k0 u) (CHead c1 k u1) H6) in ((let H9 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) (CHead c1
+k u1) H6) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u u1) \to
+((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v c c2) \to (or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T
+u u1) \to ((eq C (CHead c2 k1 u) x) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda
+(H11: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k t) x) \to
+((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x
+(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
+(u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2)))))
+(\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2))))
+(\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3))))))))) (\lambda (H12: (eq C (CHead c2 k u1) x)).(eq_ind C (CHead c2 k
+u1) (\lambda (c: C).((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda
+(_: nat).(eq C c (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))))))) (\lambda (H13: (csubst0 i0 v c1 c2)).(let H \def (eq_ind K k0
+(\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (or3_intro1 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda
+(j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2_intro C
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda
+(c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))) c2 i0 (refl_equal nat (s k
+i0)) (refl_equal C (CHead c2 k u1)) H13)))) x H12)) u (sym_eq T u u1 H11)))
+k0 (sym_eq K k0 k H10))) c0 (sym_eq C c0 c1 H9))) H8)) H7))) v0 (sym_eq T v0
+v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_both k0 i0 v0 u0 u2 H0 c0 c2 H1)
+\Rightarrow (\lambda (H2: (eq nat (s k0 i0) i)).(\lambda (H3: (eq T v0
+v)).(\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda (H5: (eq
+C (CHead c2 k0 u2) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v)
+\to ((eq C (CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x)
+\to ((subst0 i0 v0 u0 u2) \to ((csubst0 i0 v0 c0 c2) \to (or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u3:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j:
+nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat n (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k
+u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))))
+(\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C
+(CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0
+i0 t u0 u2) \to ((csubst0 i0 t c0 c2) \to (or3 (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda
+(_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j:
+nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k
+j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3))))))))))) (\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H8
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0)
+(CHead c1 k u1) H7) in ((let H9 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in ((let H10 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1
+k u1) H7) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to
+((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c c2)
+\to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s
+k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3))))
+(\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))))) (\lambda
+(H11: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C
+(CHead c2 k1 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c1 c2) \to
+(or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k
+j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda
+(u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda
+(H12: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x)
+\to ((subst0 i0 v t u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j:
+nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k
+j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))))))))) (\lambda (H13: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k
+u2) (\lambda (c: C).((subst0 i0 v u1 u2) \to ((csubst0 i0 v c1 c2) \to (or3
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (u3: T).(\lambda (_: nat).(eq C c (CHead c1 k u3)))) (\lambda (u3:
+T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda
+(_: nat).(eq C c (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda
+(c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u3))))) (\lambda (u3:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))) (\lambda
+(H14: (subst0 i0 v u1 u2)).(\lambda (H15: (csubst0 i0 v c1 c2)).(let H \def
+(eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H2 k H11) in (or3_intro2
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j))))
+(\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c1 k u3))))
+(\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) (\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda
+(u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k
+u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda
+(_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c1 c3)))) u2 c2 i0 (refl_equal nat (s k
+i0)) (refl_equal C (CHead c2 k u2)) H14 H15))))) x H13)) u0 (sym_eq T u0 u1
+H12))) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c1 H10))) H9)) H8))) v0
+(sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i)
+(refl_equal T v) (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))))).
+
+theorem csubst0_drop_gt:
+ \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
+c1 e) \to (drop n O c2 e)))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda
+(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
+O c1 e)).(let H2 \def (match H return (\lambda (n: nat).(\lambda (_: (le ?
+n)).((eq nat n O) \to (drop O O c2 e)))) with [le_n \Rightarrow (\lambda (H2:
+(eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind (drop O O c2 e) H3))) | (le_S m H2) \Rightarrow
+(\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c2
+e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\lambda (n0:
+nat).(\lambda (H: ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O
+c1 e) \to (drop n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S
+n0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S
+n0) O c2 e))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (_: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (H2:
+(drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O)
+(eq nat O O) (drop (S n0) O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda
+(H4: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1)
+(\lambda (c: C).(drop (S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0)
+(\lambda (ee: nat).(match ee return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (S
+n0) O c2 (CSort n1)) H6)) e H3)))) (drop_gen_sort n1 (S n0) O e H2))))))))
+(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v
+c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2
+e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
+(H3: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O
+c2 e) (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i
+(s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2))))
+(\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2))) (drop (S n0) O c2 e) (\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k
+x0))).(\lambda (_: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda
+(c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind nat i (\lambda (n:
+nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x1) H5) in
+(let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5)
+in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall
+(c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x1) (S
+n0)) \to (drop (S n0) O (CHead c k0 x0) e))))) with [(Bind b) \Rightarrow
+(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2:
+C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (_: (lt (s
+(Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c e H10 x0)))) | (Flat f)
+\Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (_:
+((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to
+(\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda
+(H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O
+(CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c
+e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) x0) e)
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
+n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) (lt_gen_xS x1 n0 H12)))))])
+(drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) (\lambda (H4: (ex3_2
+C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S
+n0) O c2 e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k
+x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c
+x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let
+H8 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v:
+T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S
+n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n:
+nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (k0:
+K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0
+(s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O
+c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t)
+e))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c
+e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1)
+v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2
+e)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0
+x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t)))) | (Flat f) \Rightarrow
+(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c2:
+C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s
+(Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq
+nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat
+f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7
+e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m)))
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e)
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x
+n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS
+x1 n0 H12)))))]) (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4))
+(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0:
+T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k
+x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t
+x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda
+(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n:
+nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x2) H5) in
+(let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5)
+in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall
+(c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x2) (S
+n0)) \to (drop (S n0) O (CHead x1 k0 x0) e))))) with [(Bind b) \Rightarrow
+(\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2:
+C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H13: (lt (s
+(Bind b) x2) (S n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c
+x1 v H8 e H11) x0)))) | (Flat f) \Rightarrow (\lambda (H11: (drop (r (Flat f)
+n0) O c e)).(\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s
+(Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S
+n0) O c2 e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq
+nat x2 O) (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt
+m n0))) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2
+O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2
+nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m
+n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m:
+nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x:
+nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat
+f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))])
+(drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head
+k c c2 t v i H2))))))))))) c1)))))) n).
+
+theorem csubst0_drop_gt_back:
+ \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
+c2 e) \to (drop n O c1 e)))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda
+(i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O
+O c2 e)).(let H2 \def (match H return (\lambda (n: nat).(\lambda (_: (le ?
+n)).((eq nat n O) \to (drop O O c1 e)))) with [le_n \Rightarrow (\lambda (H2:
+(eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind (drop O O c1 e) H3))) | (le_S m H2) \Rightarrow
+(\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c1
+e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\lambda (n0:
+nat).(\lambda (H: ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O
+c2 e) \to (drop n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S
+n0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S
+n0) O c e))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H1: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O
+c2 e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e))))))))
+(\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v
+c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
+e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
+(H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3:
+C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e)
+(\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k
+x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda (c:
+C).(drop (S n0) O c e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind nat i
+(\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to
+(\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k
+x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s
+k x1) H5) in ((match k return (\lambda (k0: K).(((\forall (c2: C).(\forall
+(v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e)
+\to (drop (S n0) O c e))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0)
+O c e) \to (drop (S n0) O (CHead c k0 t) e))))) with [(Bind b) \Rightarrow
+(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c
+c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
+e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r
+(Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))) | (Flat f)
+\Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat
+f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c
+e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda (H13: (drop (r
+(Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat
+x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t)
+e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e H13 t)) (\lambda
+(H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m
+n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m:
+nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x:
+nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat
+f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k
+c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j
+v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O (CHead c k t)
+e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k
+x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c
+x0)).(let H8 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead
+x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2:
+C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2
+e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat
+i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return
+(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c
+c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))
+\to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to (drop (S n0) O
+(CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2:
+C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e:
+C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s
+(Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0
+e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13)
+t)))) | (Flat f) \Rightarrow (\lambda (H11: ((\forall (c2: C).(\forall (v:
+T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2
+e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S
+n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O)
+(ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))
+(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop
+(Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m:
+nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
+(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13)
+t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k x0 e t n0
+H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O
+(CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
+nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k
+x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c
+x1)).(let H9 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead
+x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2:
+C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2
+e) \to (drop (S n0) O c e))))))) H1 (s k x2) H5) in (let H11 \def (eq_ind nat
+i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k return
+(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c
+c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))
+\to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to (drop (S n0) O
+(CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2:
+C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e:
+C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s
+(Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1
+e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14)
+t)))) | (Flat f) \Rightarrow (\lambda (H12: ((\forall (c2: C).(\forall (v:
+T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2
+e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S
+n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O)
+(ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))
+(drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop
+(Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m:
+nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
+(m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O
+(CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S
+x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14)
+t)))) H15)) (lt_gen_xS x2 n0 H13)))))]) H10 H11 (drop_gen_drop k x1 e x0 n0
+H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
+
+theorem csubst0_drop_lt:
+ \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O
+c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k
+w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
+n)) v e1 e2))))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i)
+\to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2)
+\to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T
+T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
+O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i:
+nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O
+c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0:
+nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r
+nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k i0) (s k O))
+(s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1))
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1)
+(CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C
+(CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
+T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda
+(k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0
+(s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k
+O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(\lambda
+(u: T).(let H4 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n v0 c3 c4))
+H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H5 \def (eq_ind_r nat i0
+(\lambda (n: nat).(or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2))))))
+(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead
+e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus n (s k O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k
+i0)) in (or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead c3 k u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k
+u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u)
+(CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead c3 k u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0
+u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))) k c3 c4 u (refl_equal C
+(CHead c3 k u)) (drop_refl (CHead c4 k u)) H4)))))))))))) (\lambda (k:
+K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4:
+C).(\lambda (H3: (csubst0 i0 v0 c3 c4)).(\lambda (_: (or4 (drop O O c4 c3)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda
+(k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0
+(s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k
+O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(let H5
+\def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k
+i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r nat i0 (\lambda (n:
+nat).(csubst0 n v0 c3 c4)) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in
+(or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) (ex3_4 K C T T
+(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0
+w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k
+u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s
+k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1)
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0
+w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda
+(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k
+u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4
+u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5
+H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda
+(n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1:
+C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e:
+C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))
+(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead
+e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H:
+(lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c
+e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1:
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1)
+c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind
+(eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S
+n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq
+nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c:
+C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind
+nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: nat).Prop) with
+[O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4
+(drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort
+n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1
+k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c:
+C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to
+(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C
+T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda
+(H2: (drop (S n0) O (CHead c k t) e)).(let H3 \def (match H1 return (\lambda
+(n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_:
+(csubst0 n t0 c0 c1)).((eq nat n i) \to ((eq T t0 v) \to ((eq C c0 (CHead c k
+t)) \to ((eq C c1 c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
+(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k
+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1
+e2))))))))))))))))) with [(csubst0_snd k0 i0 v0 u1 u2 H3 c0) \Rightarrow
+(\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6:
+(eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2)
+c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead
+c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0
+u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda
+(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq
+T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t))
+\to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to (or4 (drop (S
+n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u
+w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1
+e2)))))))))))) (\lambda (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1)
+(CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1)
+(CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1
+t) \to ((eq C (CHead c k0 u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop (S
+n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u
+w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1
+e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq
+T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop
+(S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s
+k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t:
+T).((eq C (CHead c k u2) c2) \to ((subst0 i0 v t u2) \to (or4 (drop (S n0) O
+c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u
+w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))
+(\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda
+(c: C).((subst0 i0 v t u2) \to (or4 (drop (S n0) O c e) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O c (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c (CHead e2 k u))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c (CHead e2 k w)))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (_: (subst0 i0 v t
+u2)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k H13)
+in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall
+(v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4
+(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2
+(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
+w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def
+(eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind
+(\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v:
+T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to
+(or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1
+e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c k
+u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k u2) (CHead
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c k u2) (CHead e2 k1 u)))))) (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0)
+(s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c k u2) (CHead e2 k1 w)))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda
+(H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall
+(v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O
+c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda
+(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s
+(Bind b) i0))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) u2) e) (ex3_4 K C
+T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S
+n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c e H2 u2)))))) (\lambda (f:
+F).(\lambda (H2: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2:
+C).(\forall (v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e:
+C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
+(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt
+(S n0) (s (Flat f) i0))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) u2) e)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c
+e H2 u2)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u1
+(sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11))
+H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_fst k0 i0 c1
+c0 v0 H3 u) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq
+T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H7:
+(eq C (CHead c0 k0 u) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0
+v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2)
+\to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S
+n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k
+u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S
+n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda
+(t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2)
+\to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0)
+(s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead
+e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
+w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda
+(H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T
+(\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t)
+H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k]))
+(CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda
+(e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 |
+(CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) H9) in (eq_ind C
+c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C (CHead c0 k0 u) c2)
+\to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0)
+(s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead
+e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
+w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda
+(H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead
+c0 k u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C
+T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0)
+(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2
+(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))
+(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v
+u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1
+e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C
+(CHead c0 k t) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0)
+(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead
+e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda
+(H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2:
+C).((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda
+(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k
+u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v
+u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0))))))
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (H16: (csubst0 i0 v
+c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k
+H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2:
+C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c
+e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1)
+in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0)
+H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2:
+C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S
+n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
+(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k
+i0)) \to (or4 (drop (S n0) O (CHead c0 k t) e) (ex3_4 K C T T (\lambda (k1:
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1
+u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 k t) (CHead e0 k1 w)))))) (\lambda (k1:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0)
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+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S
+n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2))))))) (ex3_4_intro K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0
+(Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0
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+nat).(csubst0 (minus (s (Bind b) i0) n) v x1 x2)) H23 (s x0 (S n0)) (s_S x0
+n0)))) e H21)))))))) H20)) (\lambda (H20: (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k
+n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
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+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0
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+e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda
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+u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind
+b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S
+n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda
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+(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w)))))))
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+T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k:
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+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k
+u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k:
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+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
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+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3)
+(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
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+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3)
+(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0))))))
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+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_:
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+n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
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+e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
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+n0)) (s_S x0 n0)))) e H21)))))))))) H20)) H19)))))) (\lambda (f: F).(\lambda
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+(v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: C).((drop (S n0) O
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+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
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+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k
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+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k
+u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
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+u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
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+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
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+f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S
+n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2)
+(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x2)
+(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C (CHead x1 x0 x2) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2))))))) (ex3_4_intro K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))
+x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead
+x1 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1
+e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k u0))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i0 (s k (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f)
+t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t)
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e
+(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t)
+(CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0
+w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
+T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0
+(CHead x2 x0 x3))).(\lambda (H23: (csubst0 (minus i0 (s x0 (S n0))) v x1
+x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead
+c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t)
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c
+(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t)
+(CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0
+w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0
+x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t)
+(CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t)
+(CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))
+x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead
+x2 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k
+(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0
+(minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k
+(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K
+C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k
+u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0:
+K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0
+(CHead x2 x0 x4))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x3
+x4)).(\lambda (H24: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C
+(CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k
+u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S
+n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3)
+(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3)
+(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal
+C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H22 t) H23
+H24)) e H21)))))))))) H20)) H19)))))) k (drop_gen_drop k c e t n0 H2) H17
+H18))))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq
+C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) |
+(csubst0_both k0 i0 v0 u1 u2 H3 c1 c0 H4) \Rightarrow (\lambda (H5: (eq nat
+(s k0 i0) i)).(\lambda (H6: (eq T v0 v)).(\lambda (H7: (eq C (CHead c1 k0 u1)
+(CHead c k t))).(\lambda (H8: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0
+i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t))
+\to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0
+c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda
+(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H9:
+(eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k
+t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to ((csubst0
+i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H10: (eq C
+(CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0:
+C).(match e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1 |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let
+H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0
+u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match
+e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _)
+\Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda
+(c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to
+((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e)
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0)
+(s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead
+e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H14: (eq K k0
+k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2)
+\to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2
+e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s
+k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t:
+T).((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v t u2) \to ((csubst0 i0 v c
+c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda
+(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
+(CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u
+w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1
+e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0
+k u2) (\lambda (c2: C).((subst0 i0 v t u2) \to ((csubst0 i0 v c c0) \to (or4
+(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u
+w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))
+(\lambda (_: (subst0 i0 v t u2)).(\lambda (H18: (csubst0 i0 v c c0)).(let H1
+\def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H5 k H14) in (let H19
+\def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v:
+T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4
+(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2
+(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C
+T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
+w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H20 \def
+(eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind
+(\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v:
+T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to
+(or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u
+w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1
+e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c0
+k u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k u2) (CHead
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c0 k u2) (CHead e2 k1 u)))))) (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0)
+(s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 k u2) (CHead e2 k1 w)))))))
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda
+(H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall
+(v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O
+c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda
+(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s
+(Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (H: (lt (S n0) (s
+(Bind b) i0))).(let H21 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H18 e H2) in
+(or4_ind (drop n0 O c0 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k
+w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k n0)) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n0 O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
+O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))))) (or4 (drop (S n0) O (CHead
+c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2)
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop n0
+O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H22 u2)))
+(\lambda (H22: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k
+n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k:
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k
+w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k n0)) v u w))))) (or4 (drop (S n0) O (CHead c0 (Bind
+b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2)
+(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0:
+K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e
+(CHead x1 x0 x2))).(\lambda (H24: (drop n0 O c0 (CHead x1 x0 x3))).(\lambda
+(H25: (subst0 (minus i0 (s x0 n0)) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2)
+(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Bind b) u2) c) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0
+(Bind b) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
+x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Bind b) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C
+(CHead x1 x0 x2)) (drop_drop (Bind b) n0 c0 (CHead x1 x0 x3) H24 u2)
+(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0)
+n) v x2 x3)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda
+(H22: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i0 (s k n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n0 O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))
+(or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0
+x3))).(\lambda (H24: (drop n0 O c0 (CHead x2 x0 x3))).(\lambda (H25: (csubst0
+(minus i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c:
+C).(or4 (drop (S n0) O (CHead c0 (Bind b) u2) c) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0
+(Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C
+(CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0 (CHead x2 x0 x3) H24 u2)
+(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind b) i0)
+n) v x1 x2)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda
+(H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
+O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))))).(ex4_5_ind K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k
+n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T
+T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e
+(CHead x1 x0 x3))).(\lambda (H24: (drop n0 O c0 (CHead x2 x0 x4))).(\lambda
+(H25: (subst0 (minus i0 (s x0 n0)) v x3 x4)).(\lambda (H26: (csubst0 (minus
+i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4
+(drop (S n0) O (CHead c0 (Bind b) u2) c) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind
+b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0
+(Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S
+n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1
+e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b)
+n0 c0 (CHead x2 x0 x4) H24 u2) (eq_ind_r nat (S (s x0 n0)) (\lambda (n:
+nat).(subst0 (minus (s (Bind b) i0) n) v x3 x4)) H25 (s x0 (S n0)) (s_S x0
+n0)) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind
+b) i0) n) v x1 x2)) H26 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))))) H22))
+H21)))))) (\lambda (f: F).(\lambda (H2: (drop (r (Flat f) n0) O c
+e)).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) i0)
+v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2
+e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead
+e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T
+T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k
+w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) i0))).(let H21 \def (H0
+c0 v H18 e H2) in (or4_ind (drop (S n0) O c0 e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i0 (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k
+(S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4
+K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop (S
+n0) O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C
+T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H22 u2)))
+(\lambda (H22: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i0 (s k (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))
+(or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k:
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0
+x2))).(\lambda (H24: (drop (S n0) O c0 (CHead x1 x0 x3))).(\lambda (H25:
+(subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2)
+(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0
+(Flat f) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead
+x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus (s (Flat f) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C
+(CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead x1 x0 x3) H24 u2) H25)) e
+H23)))))))) H22)) (\lambda (H22: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead
+e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))))).(ex3_4_ind K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1
+e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda
+(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k
+u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0
+x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 x3))).(\lambda (H25:
+(csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3)
+(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T
+(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0
+(Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C
+(CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H24 u2) H25)) e
+H23)))))))) H22)) (\lambda (H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1
+e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u
+w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop
+(S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda
+(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead
+c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u
+w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2)
+(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e
+(CHead x1 x0 x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0
+x4))).(\lambda (H25: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda
+(H26: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0
+x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C
+T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f)
+i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s
+(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat
+f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0
+(Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u))))))
+(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0)
+O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w)))))))
+(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k:
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0
+(Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S
+n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1
+e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f)
+n0 c0 (CHead x2 x0 x4) H24 u2) H25 H26)) e H23)))))))))) H22)) H21)))))) k
+(drop_gen_drop k c e t n0 H2) H19 H20)))))) c2 H16)) u1 (sym_eq T u1 t H15)))
+k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T
+v0 v H9))) i H5 H6 H7 H8 H3 H4)))))]) in (H3 (refl_equal nat i) (refl_equal T
+v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) c1)))))) n).
+
+theorem csubst0_drop_eq:
+ \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
+n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1
+e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
+c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1
+e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
+(or4 (drop O O c2 c1) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c1 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 y v c1 c2)).(csubst0_ind
+(\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat
+n0 O) \to (or4 (drop O O c0 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O t u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop O O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O t e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O t u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O t e1
+e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i:
+nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2)
+\to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c k0 u2)
+(CHead c k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k0
+u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1)
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O (CHead c k0 u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O (CHead c k0 u2) (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))
+(\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq
+nat (S i) O)).(let H4 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H3) in (False_ind (or4 (drop O O (CHead c (Bind b) u2) (CHead c
+(Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+c (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Bind b) u2) (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind
+b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Bind b) u2)
+(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda
+(v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1
+u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i
+(\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O O
+(CHead c (Flat f) u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f)
+u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Flat f) u1) (CHead e1
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e1
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))) f c u1 u2
+(refl_equal C (CHead c (Flat f) u1)) (drop_refl (CHead c (Flat f) u2))
+H4))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i:
+nat).(\forall (c3: C).(\forall (c4: C).(\forall (v0: T).((csubst0 i v0 c3 c4)
+\to ((((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O)
+\to (or4 (drop O O (CHead c4 k0 u) (CHead c3 k0 u)) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0
+u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O (CHead c4 k0 u) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0
+u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O
+(CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O
+(CHead c4 k0 u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0
+i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H4) in (False_ind (or4 (drop O O (CHead c4 (Bind b) u) (CHead c3 (Bind b)
+u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind
+b) u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Bind b)
+u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop O O (CHead c4 (Bind b) u) (CHead e2 (Flat f)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b)
+u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3:
+C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3
+c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i
+O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop
+O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in
+(let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4)
+in (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4
+F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat
+f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) (drop_refl (CHead c4
+(Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0:
+K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2:
+T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0
+c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop
+O O (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1)
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O (CHead c4 k0 u2) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0
+u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4
+k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c3 k0 u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in
+(False_ind (or4 (drop O O (CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind
+b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Bind b)
+u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop O O (CHead c4 (Bind b) u2) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b)
+u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u2) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
+c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6
+\def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c4 c3)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0
+u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in (let H7 \def
+(eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8
+\def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in
+(or4_intro3 (drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f)
+u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f)
+u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2)
+(CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1))
+(drop_refl (CHead c4 (Flat f) u2)) H8 H7)))))))))))))))) k)) y v c1 c2 H1)))
+H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn:
+((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to
+(\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda
+(c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall
+(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1:
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort
+n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1)
+e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop
+(S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0)
+O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4
+(drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 F C T
+T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f)
+u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2)
+\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda
+(t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v
+(CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t)
+e)).(let H2 \def (match H0 return (\lambda (n: nat).(\lambda (t0: T).(\lambda
+(c0: C).(\lambda (c1: C).(\lambda (_: (csubst0 n t0 c0 c1)).((eq nat n (S
+n0)) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) \to (or4
+(drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0)
+O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))))))) with [(csubst0_snd k0 i v0 u1 u2 H2 c0) \Rightarrow
+(\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda
+(H5: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0
+u2) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S
+n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C
+(CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0
+i v0 u1 u2) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq
+C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to
+((subst0 i t0 u1 u2) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda
+(H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T
+(\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k
+t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow
+k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C
+(\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k
+t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq
+C (CHead c k0 u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k0 i) O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0
+i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq
+T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop
+(s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k
+i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C
+(CHead c k u2) c2) \to ((subst0 i v t u2) \to (or4 (drop (s k i) O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k
+i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k
+u2) (\lambda (c: C).((subst0 i v t u2) \to (or4 (drop (s k i) O c e) (ex3_4 F
+C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k i) O c (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k i) O c (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (subst0 i v t
+u2)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k
+H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S
+n0)) \to (or4 (drop (s k i) O (CHead c k u2) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s k i) O (CHead c k u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O (CHead c k u2) (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s k i) O (CHead c k u2) (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c
+e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O
+\Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def
+(eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 n0 H18) in (eq_ind_r
+nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c (Bind b) u2) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c (Bind b)
+u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0
+(drop (s (Bind b) n0) O (CHead c (Bind b) u2) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O
+(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H1
+u2)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c
+e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat
+nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i
+(\lambda (n: nat).(subst0 n v t u2)) H16 (S n0) H18) in (eq_ind_r nat (S n0)
+(\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c (Flat f) u2) e) (ex3_4
+F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e0
+(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s
+(Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Flat f)
+(S n0)) O (CHead c (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c
+(Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0))
+O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H1 u2)) i
+H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u1 (sym_eq T u1 t
+H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0
+(sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_fst k0 i c1 c0
+v0 H2 u) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4:
+(eq T v0 v)).(\lambda (H5: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda
+(H6: (eq C (CHead c0 k0 u) c2)).((let H7 \def (f_equal nat nat (\lambda (e0:
+nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T
+v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u)
+c2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop n O c2 (CHead e2 (Flat f) u0))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v
+(\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0
+k0 u) c2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F
+C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2
+(CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u)
+(CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal
+C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t)
+H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C
+(CHead c0 k0 u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0
+i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat
+f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq
+T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i v c c0) \to (or4 (drop
+(s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k
+i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C
+(CHead c0 k t) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k
+i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k
+t) (\lambda (c2: C).((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
+f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda
+(H16: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat
+(s k i) (S n0))) H7 k H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e)
+\to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 k t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k
+i) O (CHead c0 k t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0
+k t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c
+e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O
+\Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def
+(eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 n0 H18) in (eq_ind_r
+nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b)
+t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat
+f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def
+(IHn c c0 v H19 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0
+(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
+O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H20: (drop n0 O c0
+e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T
+T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H20
+t))) (\lambda (H20: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O
+c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Bind b) n0) O (CHead
+c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2
+(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H22: (drop n0 O
+c0 (CHead x1 (Flat x0) x3))).(\lambda (H23: (subst0 O v x2 x3)).(eq_ind_r C
+(CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead
+c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2
+(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro1 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f)
+u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v
+u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f)
+u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0
+x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0
+(CHead x1 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20:
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2
+(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind
+b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2
+(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
+T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H22: (drop n0 O
+c0 (CHead x2 (Flat x0) x3))).(\lambda (H23: (csubst0 O v x1 x2)).(eq_ind_r C
+(CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead
+c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2
+(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f)
+u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v
+u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f)
+u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0
+(CHead x2 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20:
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0
+O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
+f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O
+(CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0
+(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1
+(Flat x0) x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 (Flat x0)
+x4))).(\lambda (H23: (subst0 O v x3 x4)).(\lambda (H24: (csubst0 O v x1
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind
+b) n0) O (CHead c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(eq C c (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C c (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0
+(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Bind b) n0) O (CHead
+c0 (Bind b) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat
+x0) x3) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead
+e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1
+(Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0
+(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0)
+O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C
+(CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4)
+H22 t) H23 H24)) e H21)))))))))) H20)) H)) i H18)))))) (\lambda (f:
+F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s
+(Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i
+(S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c
+c0)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s
+(Flat f) n) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v
+u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) n) O
+(CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H20 \def (H c0 v H19 e
+H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat
+f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S
+n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0))
+O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead
+e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H21: (drop (S n0) O c0
+e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F
+C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat
+f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
+f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H21 t))) (\lambda (H21: (ex3_4 F
+C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Flat f) (S n0)) O
+(CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S
+n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H22: (eq C e
+(CHead x1 (Flat x0) x2))).(\lambda (H23: (drop (S n0) O c0 (CHead x1 (Flat
+x0) x3))).(\lambda (H24: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0)
+x2) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead
+e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c
+(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O
+(CHead c0 (Flat f) t) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat
+x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S
+n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0)
+u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2))
+(drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H23 t) H24)) e H22))))))))
+H21)) (\lambda (H21: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead
+e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat
+f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0:
+T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop
+(s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
+T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S
+n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H24: (csubst0 O v x1
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat
+f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t)
+(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0)
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat
+x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3
+(refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2
+(Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O
+(CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S
+n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w:
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S
+n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H24: (subst0 O v x3
+x4)).(\lambda (H25: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3)
+(\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead
+e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c
+(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O
+(CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat
+x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S
+n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0)
+u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat
+x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H23 t) H24 H25)) e
+H22)))))))))) H21)) H20)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0)))
+c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c
+H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) |
+(csubst0_both k0 i v0 u1 u2 H2 c1 c0 H3) \Rightarrow (\lambda (H4: (eq nat (s
+k0 i) (S n0))).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0
+u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).((let H8 \def
+(f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H4) in (eq_ind nat (s
+k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k
+t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to ((csubst0 i
+v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq
+C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to
+((subst0 i t0 u1 u2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0
+i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let
+H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0
+u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match
+e0 return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1)
+(CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1
+t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v
+c c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0
+i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K
+k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i
+v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T
+T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda
+(H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to
+((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k
+i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0
+k u2) (\lambda (c2: C).((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4
+(drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k
+i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H17: (subst0 i v t
+u2)).(\lambda (H18: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda
+(k: K).(eq nat (s k i) (S n0))) H8 k H14) in (K_ind (\lambda (k: K).((drop (r
+k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0
+k u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0
+k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s k i) O (CHead c0 k u2) (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+k i) O (CHead c0 k u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r
+(Bind b) n0) O c e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20
+\def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_:
+nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H19) in
+(let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 n0 H20)
+in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 n0
+H20) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead
+c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O
+(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (let H \def (IHn c c0 v H21 e H1) in (or4_ind (drop n0 O c0 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O
+(CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (H23: (drop n0 O c0 e)).(or4_intro0 (drop (s (Bind
+b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H23 u2))) (\lambda (H23: (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2)
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b)
+u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u)))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H24: (eq C e
+(CHead x1 (Flat x0) x2))).(\lambda (H25: (drop n0 O c0 (CHead x1 (Flat x0)
+x3))).(\lambda (H26: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2)
+(\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1
+(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x2))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x1 (Flat x0) x2) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0)
+O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x2) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2))
+(drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H25 u2) H26)) e H24))))))))
+H23)) (\lambda (H23: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O
+c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O
+(CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop
+n0 O c0 (CHead x2 (Flat x0) x3))).(\lambda (H26: (csubst0 O v x1
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind
+b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2)
+(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b)
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3))
+(drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x3) H25 u2) H26)) e H24))))))))
+H23)) (\lambda (H23: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O
+c0 (CHead x2 (Flat x0) x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda
+(H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c:
+C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3
+(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O
+(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead
+x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0)
+O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0)
+O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat
+x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H25 u2) H26 H27))
+e H24)))))))))) H23)) H)) i H20))))))) (\lambda (f: F).(\lambda (H1: (drop (r
+(Flat f) n0) O c e)).(\lambda (H19: (eq nat (s (Flat f) i) (S n0))).(let H20
+\def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H19) in (let H21 \def
+(eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 (S n0) H20) in (let
+H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 (S n0) H20)
+in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead
+c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O
+(CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e2
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (let H23 \def (H c0 v H21 e H1) in (or4_ind (drop (S n0) O
+c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S
+n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead
+e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H24: (drop (S n0) O c0
+e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4
+F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat
+f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H24 u2))) (\lambda (H24: (ex3_4
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq
+C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead
+e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat
+f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0))
+O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead
+e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0)
+x2))).(\lambda (H26: (drop (S n0) O c0 (CHead x1 (Flat x0) x3))).(\lambda
+(H27: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c:
+C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead
+e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x2) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1
+(Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H26 u2)
+H27)) e H25)))))))) H24)) (\lambda (H24: (ex3_4 F C C T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop (S n0) O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
+C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead
+e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3:
+T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S
+n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H27: (csubst0 O v x1
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat
+f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0)
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead
+x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H26 u2)
+H27)) e H25)))))))) H24)) (\lambda (H24: (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O
+(CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0))
+O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S
+n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H27: (subst0 O v x3
+x4)).(\lambda (H28: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3)
+(\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead
+e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O
+(CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0)
+x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2)
+(CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
+(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u)))))))
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0)
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0
+c0 (CHead x2 (Flat x0) x4) H26 u2) H27 H28)) e H25)))))))))) H24)) H23)) i
+H20))))))) k (drop_gen_drop k c e t n0 H1) H0)))) c2 H16)) u1 (sym_eq T u1 t
+H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0
+(sym_eq T v0 v H9))) (S n0) H8)) H5 H6 H7 H2 H3)))))]) in (H2 (refl_equal nat
+(S n0)) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C
+c2)))))))))))) c1)))) n).
+
+theorem csubst0_drop_eq_back:
+ \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0
+n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2
+e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O
+c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2
+e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O
+O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2))
+(or4 (drop O O c1 c2) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
+c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 y v c1 c2)).(csubst0_ind
+(\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat
+n0 O) \to (or4 (drop O O c c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop O O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O t e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O t e1
+e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i:
+nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2)
+\to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c k0 u1)
+(CHead c k0 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u4: T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
+(CHead c k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead
+c k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(drop O O (CHead c k0 u1) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 u2) (CHead e2 (Flat f)
+u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3:
+T).(\lambda (_: T).(drop O O (CHead c k0 u1) (CHead e1 (Flat f) u3)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda
+(u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))
+(\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq
+nat (S i) O)).(let H4 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H3) in (False_ind (or4 (drop O O (CHead c (Bind b) u1) (CHead c
+(Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
+(CHead c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C
+C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Bind b) u1)
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
+(CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c
+(Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
+i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind
+nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O
+O (CHead c (Flat f) u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f)
+u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0)
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
+T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e1
+(Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
+(CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))) f c u1 u2
+(refl_equal C (CHead c (Flat f) u2)) (drop_refl (CHead c (Flat f) u1))
+H4))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i:
+nat).(\forall (c3: C).(\forall (c4: C).(\forall (v0: T).((csubst0 i v0 c3 c4)
+\to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O)
+\to (or4 (drop O O (CHead c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0
+u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop O O (CHead c3 k0 u) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O
+v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O
+(CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O
+O (CHead c3 k0 u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i:
+nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0
+i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat
+(S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H4) in (False_ind (or4 (drop O O (CHead c3 (Bind b) u) (CHead c4 (Bind b)
+u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
+(Bind b) u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C
+(CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind b) u)
+(CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+(CHead c4 (Bind b) u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3
+(Bind b) u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2:
+(csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c3 c4)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O
+v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda
+(u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n:
+nat).((eq nat n O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i
+(\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O
+(CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f)
+u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) (CHead e1
+(Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O
+(CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u
+(refl_equal C (CHead c4 (Flat f) u)) (drop_refl (CHead c3 (Flat f) u))
+H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i:
+nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2)
+\to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i
+O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O
+c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c3 k0 u1)
+(CHead c4 k0 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (u4: T).(eq C (CHead c4 k0 u2) (CHead e0 (Flat f) u4))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O
+(CHead c3 k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead
+c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 k0 u1) (CHead e1 (Flat
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 u2)
+(CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 k0 u1) (CHead e1
+(Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1
+e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3
+c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6
+\def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in
+(False_ind (or4 (drop O O (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2))
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4:
+T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3
+(Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead
+c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Bind b) u1) (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4
+(Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind
+b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_:
+T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i:
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3
+c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6
+\def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c3 c4)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O
+v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in
+(let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5)
+in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5)
+in (or4_intro3 (drop O O (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2))
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3
+(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Flat f) u1)
+(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C
+(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O
+O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0
+O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex4_5_intro F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_:
+T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4:
+T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) f c3 c4 u1 u2
+(refl_equal C (CHead c4 (Flat f) u2)) (drop_refl (CHead c3 (Flat f) u1)) H8
+H7)))))))))))))))) k)) y v c1 c2 H1))) H) e (drop_gen_refl c2 e H0))))))))
+(\lambda (n0: nat).(\lambda (IHn: ((\forall (c1: C).(\forall (c2: C).(\forall
+(v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (or4
+(drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O
+c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0)
+O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v:
+T).(\lambda (H: (csubst0 (S n0) v (CSort n1) c2)).(\lambda (e: C).(\lambda
+(_: (drop (S n0) O c2 e)).(csubst0_gen_sort c2 v (S n0) n1 H (or4 (drop (S
+n0) O (CSort n1) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CSort
+n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c: C).(\lambda (H:
+((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e:
+C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2:
+C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t)
+c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3))))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j))))
+(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v t u2))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda
+(H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def
+(eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead c k x0) H4) in
+((match k return (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0
+n0) O c e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) with
+[(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda
+(H8: (drop (r (Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda
+(e0: nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S
+n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1
+(\lambda (n: nat).(subst0 n v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) O
+(CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead
+e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t)))))) | (Flat f) \Rightarrow
+(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r (Flat
+f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) x1
+H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5
+(S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T
+T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e
+H8 t))))))]) H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) (\lambda (H2:
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))
+(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda
+(H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6
+\def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead x0 k t) H4)
+in ((match k return (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r
+k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
+C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0)
+(s (Bind b) x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def
+(f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat)
+with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10
+\def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 n0 H9) in (let
+H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
+(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
+x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15:
+(subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4
+(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c
+(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq
+C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2)
+x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
+(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5
+(refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3
+(Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O
+c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16:
+(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0
+c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))) |
+(Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda
+(H8: (drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda
+(e0: nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n:
+nat).(csubst0 n v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in
+(or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead
+e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0)
+O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e
+H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda
+(f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2)
+x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda
+(H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat
+f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5))
+(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13))))))))
+H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead
+e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0)
+O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
+(x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop
+(S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3
+x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4
+(Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0)
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0)
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c
+(CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12:
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0)
+O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2)
+x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda
+(H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C
+(CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat
+f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat
+f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat
+x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0)
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0)
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6))
+(drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e
+H13)))))))))) H12)) H11))))))]) H3 (drop_gen_drop k x0 e t n0 H6)))))))) H2))
+(\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
+(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
+C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0)
+O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda
+(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2
+(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2
+v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1
+(CHead x1 k x0) H4) in ((match k return (\lambda (k0: K).((eq nat (S n0) (s
+k0 x2)) \to ((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t)
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H8: (eq nat (S n0)
+(s (Bind b) x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def
+(f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat)
+with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x2) H8) in (let H11
+\def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 n0 H10) in
+(let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 n0
+H10) in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c
+(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))) (\lambda (H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead
+c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(drop_drop (Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+(or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3)
+x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17:
+(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4
+(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2))))))
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))
+x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c
+(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14:
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq
+C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3)
+x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O
+(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat
+x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6
+(refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4
+(Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T
+T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1
+(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1
+e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t)
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7:
+T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O
+c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18:
+(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind
+b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0)
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f)
+u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0
+c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))) |
+(Flat f) \Rightarrow (\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda
+(H9: (drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda
+(e0: nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2
+(\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 v
+H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda
+(H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e)
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop
+(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C
+e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda
+(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4
+(Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3)
+x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6)
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O
+(CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3)
+x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat
+f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat
+x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead
+x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t)
+H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C
+C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e
+(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
+(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3)
+x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda
+(H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0:
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat
+f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O
+v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop
+(Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14))
+(\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda
+(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq
+C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda
+(x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x5
+(Flat x3) x7))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3)
+x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4
+x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O
+(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2))))))
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5
+(Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0)
+u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u))))))
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S
+n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0)
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0)
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))
+(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2
+(Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7))
+(drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e
+H15)))))))))) H14)) H13)))))))]) H3 (drop_gen_drop k x1 e x0 n0 H7))))))))))
+H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).
+
+theorem csubst0_clear_O:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
+(\forall (c: C).((clear c1 c) \to (clear c2 c))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2
+c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
+(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n)
+c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda
+(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0:
+C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t)
+c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq
+nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0:
+T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C
+c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k
+x0) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (k0: K).((clear
+(CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0)
+c0)))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t)
+c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O
+(\lambda (ee: nat).(match ee return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind
+(clear (CHead c (Bind b) x0) c0) H8)))) | (Flat f) \Rightarrow (\lambda (H6:
+(clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s (Flat f)
+x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 O
+H7) in (clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))]) H1 H3) c2
+H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead
+c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c
+c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear c2 c0) (\lambda (x0:
+C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C
+c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0
+k t) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (k0:
+K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead x0
+k0 t) c0)))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b)
+t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O
+(\lambda (ee: nat).(match ee return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind
+(clear (CHead x0 (Bind b) t) c0) H8)))) | (Flat f) \Rightarrow (\lambda (H6:
+(clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s (Flat f)
+x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5
+O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t H6)) f
+t))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
+nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k
+x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c
+x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) ((match k
+return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2))
+\to (clear (CHead x1 k0 x0) c0)))) with [(Bind b) \Rightarrow (\lambda (_:
+(clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b)
+x2))).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda
+(_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2)
+H8) in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9)))) | (Flat f)
+\Rightarrow (\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq
+nat O (s (Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda
+(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0
+(clear_gen_flat f c c0 t H7)) f x0)))))]) H1 H3) c2 H4)))))))) H2))
+(csubst0_gen_head k c c2 t v O H0))))))))))) c1).
+
+theorem csubst0_clear_O_back:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
+(\forall (c: C).((clear c2 c) \to (clear c1 c))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c
+c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
+(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2
+c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c:
+C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to
+(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O
+v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind
+(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T
+nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead
+c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear
+(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O
+(s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v
+t x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead c k
+x0) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x1)) \to
+((clear (CHead c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b)
+\Rightarrow (\lambda (H7: (eq nat O (s (Bind b) x1))).(\lambda (_: (clear
+(CHead c (Bind b) x0) c0)).(let H9 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S
+_) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b)
+t) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat O (s (Flat f)
+x1))).(\lambda (H8: (clear (CHead c (Flat f) x0) c0)).(let H9 \def (eq_ind_r
+nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0
+(clear_gen_flat f c c0 x0 H8) f t))))]) H3 H6))))))) H2)) (\lambda (H2:
+(ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda
+(c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3))) (clear (CHead c k t) c0) (\lambda (x0: C).(\lambda (x1:
+nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k
+t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (eq_ind C c2 (\lambda
+(c: C).(clear c c0)) H1 (CHead x0 k t) H4) in ((match k return (\lambda (k0:
+K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 k0 t) c0) \to (clear (CHead c
+k0 t) c0)))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat O (s (Bind b)
+x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let H9 \def (eq_ind nat
+O (\lambda (ee: nat).(match ee return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind
+(clear (CHead c (Bind b) t) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7:
+(eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 (Flat f) t)
+c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 O
+H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t H8)) f
+t))))]) H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c
+c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda
+(x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1
+k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c
+x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead x1 k
+x0) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x2)) \to
+((clear (CHead x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b)
+\Rightarrow (\lambda (H8: (eq nat O (s (Bind b) x2))).(\lambda (_: (clear
+(CHead x1 (Bind b) x0) c0)).(let H10 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S
+_) \Rightarrow False])) I (S x2) H8) in (False_ind (clear (CHead c (Bind b)
+t) c0) H10)))) | (Flat f) \Rightarrow (\lambda (H8: (eq nat O (s (Flat f)
+x2))).(\lambda (H9: (clear (CHead x1 (Flat f) x0) c0)).(let H10 \def
+(eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 O H8) in (let H11
+\def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 O H8) in
+(clear_flat c c0 (H x1 v H10 c0 (clear_gen_flat f x1 c0 x0 H9)) f t)))))]) H3
+H7))))))))) H2)) (csubst0_gen_head k c c2 t v O H0))))))))))) c1).
+
+theorem csubst0_clear_S:
+ \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
+(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4
+B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2))))))))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
+T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c
+c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c:
+C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4
+(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i:
+nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4
+(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t)
+c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
+T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda
+(u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2:
+T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_:
+C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
+v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
+nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_:
+nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda
+(j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2
+(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t
+u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k
+j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda
+(u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k
+x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda
+(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to
+((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0)
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H6: (clear (CHead c (Bind
+b) t) c0)).(\lambda (H7: (eq nat (S i) (s (Bind b) x1))).(let H8 \def
+(f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: nat).nat)
+with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H7) in (let H9
+\def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in
+(eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind
+b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b)
+x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro1 (clear (CHead c
+(Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e
+(Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b)
+x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda
+(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C T T (\lambda
+(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind
+b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0)
+u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2))))) b c t x0 (refl_equal C (CHead c (Bind b) t))
+(clear_bind b c x0) H9)) c0 (clear_gen_bind b c c0 t H6)))))) | (Flat f)
+\Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq
+nat (S i) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
+nat).e) (S i) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda
+(n: nat).(subst0 n v t x0)) H5 (S i) H8) in (or4_intro0 (clear (CHead c (Flat
+f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Flat f)
+x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead c (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat c c0 (clear_gen_flat
+f c c0 t H6) f x0))))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j:
+nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
+nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2
+(CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))
+(or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda
+(x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda
+(H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C
+(CHead x0 k t) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+i v e1 e2))))))))) ((match k return (\lambda (k0: K).((clear (CHead c k0 t)
+c0) \to ((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4
+B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t)
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H6: (clear (CHead c (Bind
+b) t) c0)).(\lambda (H7: (eq nat (S i) (s (Bind b) x1))).(let H8 \def
+(f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: nat).nat)
+with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H7) in (let H9
+\def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in
+(eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind
+b) t) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b)
+t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro2 (clear (CHead x0
+(Bind b) t) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e
+(Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b)
+t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0
+(Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
+t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2))))) b c x0 t (refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t)
+H9)) c0 (clear_gen_bind b c c0 t H6)))))) | (Flat f) \Rightarrow (\lambda
+(H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f)
+x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f)
+x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c
+x0)) H5 (S i) H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t
+H6)) in (or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat
+f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
+t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t)))
+(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
+(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t)
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3:
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind
+x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14:
+(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
+t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
+t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0
+(CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
+t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0
+(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2
+(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t)
+H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2)
+x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14:
+(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear
+(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0
+(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))]) H1 H3)
+c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3:
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2:
+C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_:
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2:
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_:
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2
+c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda
+(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda
+(H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda
+(H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4
+(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind
+b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k
+return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0
+x2)) \to (or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) with
+[(Bind b) \Rightarrow (\lambda (H7: (clear (CHead c (Bind b) t) c0)).(\lambda
+(H8: (eq nat (S i) (s (Bind b) x2))).(let H9 \def (f_equal nat nat (\lambda
+(e: nat).(match e return (\lambda (_: nat).nat) with [O \Rightarrow i | (S n)
+\Rightarrow n])) (S i) (S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda
+(n: nat).(csubst0 n v c x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2
+(\lambda (n: nat).(subst0 n v t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind
+b) t) (\lambda (c3: C).(or4 (clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T
+(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
+(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Bind b)
+x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda
+(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead
+e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+i v e1 e2))))))))) (or4_intro3 (clear (CHead x1 (Bind b) x0) (CHead c (Bind
+b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1))))))
+(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
+(CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead
+e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1
+(Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1
+(Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b)
+t)) (clear_bind b x1 x0) H11 H10)) c0 (clear_gen_bind b c c0 t H7))))))) |
+(Flat f) \Rightarrow (\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda
+(H8: (eq nat (S i) (s (Flat f) x2))).(let H9 \def (f_equal nat nat (\lambda
+(e: nat).e) (S i) (s (Flat f) x2) H8) in (let H10 \def (eq_ind_r nat x2
+(\lambda (n: nat).(csubst0 n v c x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r
+nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) H9) in (let H12 \def (H
+x1 v i H10 c0 (clear_gen_flat f c c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4
+B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq
+C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 i v e1 e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H13: (clear x1
+c0)).(or4_intro0 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda
+(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
+(clear_flat x1 c0 H13 f x0))) (\lambda (H13: (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))).(ex3_4_ind B C
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))
+(or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6:
+T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x5))).(\lambda (H15: (clear x1
+(CHead x4 (Bind x3) x6))).(\lambda (H16: (subst0 i v x5 x6)).(or4_intro1
+(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
+T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
+(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
+u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x4 (Bind x3) x6) H15 f x0)
+H16))))))))) H13)) (\lambda (H13: (ex3_4 B C C T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 i v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) (or4 (clear
+(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3:
+B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C c0
+(CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3)
+x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2 (clear (CHead x1 (Flat
+f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
+x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1
+(Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))
+x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0) H16)))))))))
+H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
+b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
+T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
+(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
+u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
+e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
+T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3)
+x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16:
+(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear
+(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
+T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
+(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
+T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
+(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
+u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1
+(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))]) H1
+H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1).
+
+theorem csubst0_getl_ge:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1
+e) \to (getl n c2 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all
+c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0:
+C).(clear e0 e)) (getl n c2 e) (\lambda (x: C).(\lambda (H3: (drop n O c1
+x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c2 e) (\lambda (H5:
+(lt i n)).(getl_intro n c2 e x (csubst0_drop_gt n i H5 c1 c2 v H0 x H3) H4))
+(\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n:
+nat).(drop n O c1 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n:
+nat).(le i n)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c2 e))
+(let H8 \def (csubst0_drop_eq i c1 c2 v H0 x H6) in (or4_ind (drop i O c2 x)
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c2 e) (\lambda (H9:
+(drop i O c2 x)).(getl_intro i c2 e x H9 H4)) (\lambda (H9: (ex3_4 F C T T
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop i O c2 (CHead e0 (Flat f) w)))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u
+w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C x (CHead e0 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop i O c2 (CHead e0
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 O v u w))))) (getl i c2 e) (\lambda (x0: F).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat
+x0) x2))).(\lambda (H11: (drop i O c2 (CHead x1 (Flat x0) x3))).(\lambda (_:
+(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
+(CHead x1 (Flat x0) x2) H10) in (getl_intro i c2 e (CHead x1 (Flat x0) x3)
+H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x2 H13) x0 x3)))))))))) H9))
+(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2 (CHead e2
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 (Flat f) u))))))
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop i O c2
+(CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c2 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x
+(CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O c2 (CHead x2 (Flat x0)
+x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e
+(CHead x2 (Flat x0) x3) H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H12 e
+(clear_gen_flat x0 x1 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9: (ex4_5 F
+C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C x (CHead e1 (Flat f) u))))))) (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop i O
+c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 (Flat f)
+u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop i O c2 (CHead e2 (Flat f) w))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c2 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H10: (eq C x (CHead x1 (Flat x0) x3))).(\lambda (H11: (drop i O
+c2 (CHead x2 (Flat x0) x4))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13:
+(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e))
+H4 (CHead x1 (Flat x0) x3) H10) in (getl_intro i c2 e (CHead x2 (Flat x0) x4)
+H11 (clear_flat x2 e (csubst0_clear_O x1 x2 v H13 e (clear_gen_flat x0 x1 e
+x3 H14)) x0 x4)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n
+i)).(le_lt_false i n H H5 (getl n c2 e))))))) H2)))))))))).
+
+theorem csubst0_getl_lt:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1
+e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all
+c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0:
+C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x:
+C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def
+(csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K
+C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))
+(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1
+e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k:
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O
+c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))
+(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B
+C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C
+e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6:
+(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u
+w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k
+n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0:
+K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x
+(CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda
+(H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in ((match x0 return (\lambda (k:
+K).((drop n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to
+((clear (CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) with [(Bind
+b) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda
+(H12: (subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead
+x1 (Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4
+(getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda
+(w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind
+b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2
+(CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0
+(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C
+(CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2
+(CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12))
+e (clear_gen_bind b x1 e x2 H13))))) | (Flat f) \Rightarrow (\lambda (H11:
+(drop n O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat
+f) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0
+(getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1
+(Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))])
+H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: K).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead
+e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k:
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k
+u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))) (or4 (getl n
+c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda
+(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x (CHead x1 x0
+x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda (H9: (csubst0
+(minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda (c: C).(clear
+c e)) H4 (CHead x1 x0 x3) H7) in ((match x0 return (\lambda (k: K).((drop n O
+c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear
+(CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) with [(Bind
+b) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda
+(H12: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear
+(CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c:
+C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u))))))
+(\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u)))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2
+(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda
+(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C
+T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
+(CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0)
+u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1
+(Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3)
+H11 (clear_bind b x2 x3)) H12)) e (clear_gen_bind b x1 e x3 H13))))) | (Flat
+f) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x2 (Flat f) x3))).(\lambda
+(H12: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda (H13: (clear
+(CHead x1 (Flat f) x3) e)).(let H14 \def (eq_ind nat (minus i n) (\lambda (n:
+nat).(csubst0 n v x1 x2)) H12 (S (minus i (S n))) (minus_x_Sy i n H)) in (let
+H15 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H14 e (clear_gen_flat f x1
+e x3 H13)) in (or4_ind (clear x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2
+(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
+v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0
+(getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x2
+(Flat f) x3) H11 (clear_flat x2 e H16 f x3)))) (\lambda (H16: (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
+v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0
+(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T
+T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: T).(\lambda
+(x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x6))).(\lambda (H18:
+(clear x2 (CHead x5 (Bind x4) x7))).(\lambda (H19: (subst0 (minus i (S n)) v
+x6 x7)).(eq_ind_r C (CHead x5 (Bind x4) x6) (\lambda (c: C).(or4 (getl n c2
+c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 (CHead x5 (Bind x4)
+x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4)
+x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e1
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))
+(ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 x6 x7 (refl_equal
+C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind x4) x7) (CHead x2
+(Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 f x3)) H19)) e
+H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))).(ex3_4_ind B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4:
+B).(\lambda (x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H17: (eq C e
+(CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4)
+x7))).(\lambda (H19: (csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5
+(Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2
+(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b)
+u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4
+x5 x6 x7 (refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6
+(Bind x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4)
+x7) H18 f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2
+(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda
+(x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq
+C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4)
+x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20:
+(csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
+(getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7))
+(getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11
+(clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17))))))))))
+H16)) H15))))))]) H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T
+(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda
+(k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k
+n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k
+u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k
+n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n
+c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda
+(x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq
+C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda
+(H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s
+x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
+(CHead x1 x0 x3) H7) in ((match x0 return (\lambda (k: K).((drop n O c2
+(CHead x2 k x4)) \to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus
+i (s k n)) v x1 x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))))))))) with [(Bind b) \Rightarrow (\lambda
+(H12: (drop n O c2 (CHead x2 (Bind b) x4))).(\lambda (H13: (subst0 (minus i
+(s (Bind b) n)) v x3 x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v
+x1 x2)).(\lambda (H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead
+x1 (Bind b) x3) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda
+(b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0
+(Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u)))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
+(getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead
+e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda
+(b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C
+T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))))
+(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4
+(refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4)
+(CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind
+b x1 e x3 H15)))))) | (Flat f) \Rightarrow (\lambda (H12: (drop n O c2 (CHead
+x2 (Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3
+x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda
+(H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n)
+(\lambda (n: nat).(csubst0 n v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy i
+n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e
+(clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n))
+v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18:
+(clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e
+(CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18:
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e (Bind b) u2))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+(minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2
+(CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e)
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda
+(x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5)
+x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21:
+(subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1
+(getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
+x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b)
+u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6
+x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind
+x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20
+f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind
+b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda
+(x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20:
+(clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v
+x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2
+c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_:
+T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5)
+x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda
+(_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5)
+x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))
+(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2
+(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8
+(refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5)
+x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f
+x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e
+(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind
+b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+(minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
+T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda
+(x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq
+C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5)
+x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22:
+(csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8)
+(\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c
+(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v
+u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3
+(getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead
+e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq
+C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
+x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5)
+x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8))
+(getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12
+(clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19))))))))))
+H18)) H17)))))))]) H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
+
+theorem csubst0_getl_ge_back:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2
+e) \to (getl n c1 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1
+c2)).(\lambda (e: C).(\lambda (H1: (getl n c2 e)).(let H2 \def (getl_gen_all
+c2 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c2 e0)) (\lambda (e0:
+C).(clear e0 e)) (getl n c1 e) (\lambda (x: C).(\lambda (H3: (drop n O c2
+x)).(\lambda (H4: (clear x e)).(lt_eq_gt_e i n (getl n c1 e) (\lambda (H5:
+(lt i n)).(getl_intro n c1 e x (csubst0_drop_gt_back n i H5 c1 c2 v H0 x H3)
+H4)) (\lambda (H5: (eq nat i n)).(let H6 \def (eq_ind_r nat n (\lambda (n:
+nat).(drop n O c2 x)) H3 i H5) in (let H7 \def (eq_ind_r nat n (\lambda (n:
+nat).(le i n)) H i H5) in (eq_ind nat i (\lambda (n0: nat).(getl n0 c1 e))
+(let H8 \def (csubst0_drop_eq_back i c1 c2 v H0 x H6) in (or4_ind (drop i O
+c1 x) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f:
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1)))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (getl i c1
+e) (\lambda (H9: (drop i O c1 x)).(getl_intro i c1 e x H9 H4)) (\lambda (H9:
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2:
+T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0:
+C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0 (Flat f) u1))))))
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v
+u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (u2: T).(eq C x (CHead e0 (Flat f) u2)))))) (\lambda (f:
+F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop i O c1 (CHead e0
+(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(u2: T).(subst0 O v u1 u2))))) (getl i c1 e) (\lambda (x0: F).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C x (CHead x1 (Flat
+x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0) x2))).(\lambda (_:
+(subst0 O v x2 x3)).(let H13 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4
+(CHead x1 (Flat x0) x3) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x2)
+H11 (clear_flat x1 e (clear_gen_flat x0 x1 e x3 H13) x0 x2)))))))))) H9))
+(\lambda (H9: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u)))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1 (CHead e1
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(eq C x (CHead e2 (Flat f) u))))))
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop i O c1
+(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (getl i c1 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq C x
+(CHead x2 (Flat x0) x3))).(\lambda (H11: (drop i O c1 (CHead x1 (Flat x0)
+x3))).(\lambda (H12: (csubst0 O v x1 x2)).(let H13 \def (eq_ind C x (\lambda
+(c: C).(clear c e)) H4 (CHead x2 (Flat x0) x3) H10) in (getl_intro i c1 e
+(CHead x1 (Flat x0) x3) H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v
+H12 e (clear_gen_flat x0 x2 e x3 H13)) x0 x3)))))))))) H9)) (\lambda (H9:
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat f) u2))))))) (\lambda (f:
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop i
+O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C x (CHead e2 (Flat
+f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(drop i O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_:
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (getl i c1 e) (\lambda (x0:
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H10: (eq C x (CHead x2 (Flat x0) x4))).(\lambda (H11: (drop i O
+c1 (CHead x1 (Flat x0) x3))).(\lambda (_: (subst0 O v x3 x4)).(\lambda (H13:
+(csubst0 O v x1 x2)).(let H14 \def (eq_ind C x (\lambda (c: C).(clear c e))
+H4 (CHead x2 (Flat x0) x4) H10) in (getl_intro i c1 e (CHead x1 (Flat x0) x3)
+H11 (clear_flat x1 e (csubst0_clear_O_back x1 x2 v H13 e (clear_gen_flat x0
+x2 e x4 H14)) x0 x3)))))))))))) H9)) H8)) n H5)))) (\lambda (H5: (lt n
+i)).(le_lt_false i n H H5 (getl n c1 e))))))) H2)))))))))).
-axiom subst1_subst1_back: \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1 j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t: T).(subst1 (S (plus i j)) u t2 t))))))))))) .
+inductive csubst1 (i:nat) (v:T) (c1:C): C \to Prop \def
+| csubst1_refl: csubst1 i v c1 c1
+| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
-axiom subst1_trans: \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1 i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1 t3))))))) .
+theorem csubst1_head:
+ \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
+v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(subst1_ind i v u1 (\lambda (t:
+T).(\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k
+i) v (CHead c1 k u1) (CHead c2 k t)))))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst1 i v c1 c2)).(csubst1_ind i v c1 (\lambda (c:
+C).(csubst1 (s k i) v (CHead c1 k u1) (CHead c k u1))) (csubst1_refl (s k i)
+v (CHead c1 k u1)) (\lambda (c3: C).(\lambda (H1: (csubst0 i v c1
+c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k u1) (csubst0_fst k i
+c1 c3 v H1 u1)))) c2 H0)))) (\lambda (t2: T).(\lambda (H0: (subst0 i v u1
+t2)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(csubst1 (s k i) v (CHead c1 k u1)
+(CHead c k t2))) (csubst1_sing (s k i) v (CHead c1 k u1) (CHead c1 k t2)
+(csubst0_snd k i v u1 t2 H0 c1)) (\lambda (c3: C).(\lambda (H2: (csubst0 i v
+c1 c3)).(csubst1_sing (s k i) v (CHead c1 k u1) (CHead c3 k t2) (csubst0_both
+k i v u1 t2 H0 c1 c3 H2)))) c2 H1)))))) u2 H)))))).
+
+theorem csubst1_bind:
+ \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
+v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b)
+u2))))))))))
+\def
+ \lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Bind b) i) (\lambda (n:
+nat).(csubst1 n v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))
+(csubst1_head (Bind b) i v u1 u2 H c1 c2 H0) (S i) (refl_equal nat (S
+i))))))))))).
+
+theorem csubst1_flat:
+ \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
+(u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i
+v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f)
+u2))))))))))
+\def
+ \lambda (f: F).(\lambda (i: nat).(\lambda (v: T).(\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H: (subst1 i v u1 u2)).(\lambda (c1: C).(\lambda (c2:
+C).(\lambda (H0: (csubst1 i v c1 c2)).(eq_ind nat (s (Flat f) i) (\lambda (n:
+nat).(csubst1 n v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2)))
+(csubst1_head (Flat f) i v u1 u2 H c1 c2 H0) i (refl_equal nat i)))))))))).
+
+theorem csubst1_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
+(v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
+C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 i v c1 c2))))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
+x)).(let H0 \def (match H return (\lambda (c: C).(\lambda (_: (csubst1 ? ? ?
+c)).((eq C c x) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C x
+(CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))))) with [csubst1_refl
+\Rightarrow (\lambda (H0: (eq C (CHead c1 k u1) x)).(eq_ind C (CHead c1 k u1)
+(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead
+c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
+(_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda
+(u2: T).(\lambda (c2: C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda
+(u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 i v c1 c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i
+v u1) (csubst1_refl i v c1)) x H0)) | (csubst1_sing c2 H0) \Rightarrow
+(\lambda (H1: (eq C c2 x)).(eq_ind C x (\lambda (c: C).((csubst0 (s k i) v
+(CHead c1 k u1) c) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))))) (\lambda (H2:
+(csubst0 (s k i) v (CHead c1 k u1) x)).(or3_ind (ex3_2 T nat (\lambda (_:
+T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_:
+nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j
+v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s
+k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1))))
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat
+(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))))
+(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k
+u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3))))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k
+u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (H3: (ex3_2 T nat
+(\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2:
+T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j:
+nat).(subst0 j v u1 u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j:
+nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x
+(CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))
+(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2))))
+(\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C x (CHead c1
+k x0))).(\lambda (H5: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0)
+(\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead
+c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
+(_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x1
+(\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x1 H)) in (ex3_2_intro
+T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k
+u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k
+x0)) (subst1_single i v u1 x0 H6) (csubst1_refl i v c1))) x H4)))))) H3))
+(\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i)
+(s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1))))
+(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))).(ex3_2_ind C nat
+(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3:
+C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
+nat).(csubst0 j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C
+x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0:
+C).(\lambda (x1: nat).(\lambda (H: (eq nat (s k i) (s k x1))).(\lambda (H4:
+(eq C x (CHead x0 k u1))).(\lambda (H5: (csubst0 x1 v c1 x0)).(eq_ind_r C
+(CHead x0 k u1) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3:
+C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1
+u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H6 \def
+(eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0)) H5 i (s_inj k i x1
+H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k
+u1) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C
+(CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H6))) x
+H4)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
+C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
+(c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2:
+T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
+T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))).(ex4_3_ind T C
+nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
+j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
+k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
+u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
+c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k
+u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
+T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (x2: nat).(\lambda (H: (eq nat (s k i) (s k x2))).(\lambda (H4:
+(eq C x (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6:
+(csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C
+(\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2:
+T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
+C).(csubst1 i v c1 c3))))) (let H7 \def (eq_ind_r nat x2 (\lambda (n:
+nat).(csubst0 n v c1 x1)) H6 i (s_inj k i x2 H)) in (let H8 \def (eq_ind_r
+nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x2 H)) in
+(ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0)
+(CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
+(\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C
+(CHead x1 k x0)) (subst1_single i v u1 x0 H8) (csubst1_sing i v c1 x1 H7))))
+x H4)))))))) H3)) (csubst0_gen_head k c1 x u1 v (s k i) H2))) c2 (sym_eq C c2
+x H1) H0))]) in (H0 (refl_equal C x))))))))).
+
+theorem csubst1_getl_ge:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1
+e) \to (getl n c2 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c1 e) \to
+(getl n c e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
+(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
+(getl n c1 e)).(csubst0_getl_ge i n H c1 c3 v H1 e H2))))) c2 H0))))))).
+
+theorem csubst1_getl_lt:
+ \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1
+e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2:
+C).(getl n c2 e2)))))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e1: C).((getl n c1 e1) \to
+(ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl
+n c e2)))))) (\lambda (e1: C).(\lambda (H1: (getl n c1 e1)).(eq_ind_r nat (S
+(minus i (S n))) (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1
+e2)) (\lambda (e2: C).(getl n c1 e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c1 e2)) e1
+(csubst1_refl (S (minus i (S n))) v e1) H1) (minus i n) (minus_x_Sy i n H))))
+(\lambda (c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e1: C).(\lambda
+(H2: (getl n c1 e1)).(eq_ind_r nat (S (minus i (S n))) (\lambda (n0:
+nat).(ex2 C (\lambda (e2: C).(csubst1 n0 v e1 e2)) (\lambda (e2: C).(getl n
+c3 e2)))) (let H3 \def (csubst0_getl_lt i n H c1 c3 v H1 e1 H2) in (or4_ind
+(getl n c3 e1) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u:
+T).(getl n c3 (CHead e3 (Bind b) u)))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (e3: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e3: C).(\lambda (_: T).(\lambda (w: T).(getl n
+c3 (CHead e3 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e2 e3))))))) (ex2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)))
+(\lambda (H4: (getl n c3 e1)).(ex_intro2 C (\lambda (e2: C).(csubst1 (S
+(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2)) e1 (csubst1_refl
+(S (minus i (S n))) v e1) H4)) (\lambda (H4: (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind
+b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u:
+T).(\lambda (_: T).(eq C e1 (CHead e0 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i (S n)) v u w))))) (ex2 C (\lambda (e2: C).(csubst1 (S
+(minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq C e1
+(CHead x1 (Bind x0) x2))).(\lambda (H6: (getl n c3 (CHead x1 (Bind x0)
+x3))).(\lambda (H7: (subst0 (minus i (S n)) v x2 x3)).(eq_ind_r C (CHead x1
+(Bind x0) x2) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
+n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x2) e2)) (\lambda (e2:
+C).(getl n c3 e2)) (CHead x1 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
+v (CHead x1 (Bind x0) x2) (CHead x1 (Bind x0) x3) (csubst0_snd_bind x0 (minus
+i (S n)) v x2 x3 H7 x1)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(eq C e1
+(CHead e2 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl n c3 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e2: C).(\lambda
+(_: C).(\lambda (u: T).(eq C e1 (CHead e2 (Bind b) u)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e3: C).(\lambda (u: T).(getl n c3 (CHead e3
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) v e2 e3))))) (ex2 C (\lambda (e2: C).(csubst1
+(S (minus i (S n))) v e1 e2)) (\lambda (e2: C).(getl n c3 e2))) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H5: (eq C e1
+(CHead x1 (Bind x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0)
+x3))).(\lambda (H7: (csubst0 (minus i (S n)) v x1 x2)).(eq_ind_r C (CHead x1
+(Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S
+n))) v c e2)) (\lambda (e2: C).(getl n c3 e2)))) (ex_intro2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind x0) x3) e2)) (\lambda (e2:
+C).(getl n c3 e2)) (CHead x2 (Bind x0) x3) (csubst1_sing (S (minus i (S n)))
+v (CHead x1 (Bind x0) x3) (CHead x2 (Bind x0) x3) (csubst0_fst_bind x0 (minus
+i (S n)) x1 x2 v H7 x3)) H6) e1 H5)))))))) H4)) (\lambda (H4: (ex4_5 B C C T
+T (\lambda (b: B).(\lambda (e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(eq C e1 (CHead e2 (Bind b) u))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda
+(e2: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e1 (CHead e2
+(Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e3: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c3 (CHead e3 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (e3:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e2 e3))))))
+(ex2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v e1 e2)) (\lambda (e2:
+C).(getl n c3 e2))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H5: (eq C e1 (CHead x1 (Bind
+x0) x3))).(\lambda (H6: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H7:
+(subst0 (minus i (S n)) v x3 x4)).(\lambda (H8: (csubst0 (minus i (S n)) v x1
+x2)).(eq_ind_r C (CHead x1 (Bind x0) x3) (\lambda (c: C).(ex2 C (\lambda (e2:
+C).(csubst1 (S (minus i (S n))) v c e2)) (\lambda (e2: C).(getl n c3 e2))))
+(ex_intro2 C (\lambda (e2: C).(csubst1 (S (minus i (S n))) v (CHead x1 (Bind
+x0) x3) e2)) (\lambda (e2: C).(getl n c3 e2)) (CHead x2 (Bind x0) x4)
+(csubst1_sing (S (minus i (S n))) v (CHead x1 (Bind x0) x3) (CHead x2 (Bind
+x0) x4) (csubst0_both_bind x0 (minus i (S n)) v x3 x4 H7 x1 x2 H8)) H6) e1
+H5)))))))))) H4)) H3)) (minus i n) (minus_x_Sy i n H)))))) c2 H0))))))).
+
+theorem csubst1_getl_ge_back:
+ \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2
+e) \to (getl n c1 e)))))))))
+\def
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (le i n)).(\lambda (c1:
+C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst1 i v c1
+c2)).(csubst1_ind i v c1 (\lambda (c: C).(\forall (e: C).((getl n c e) \to
+(getl n c1 e)))) (\lambda (e: C).(\lambda (H1: (getl n c1 e)).H1)) (\lambda
+(c3: C).(\lambda (H1: (csubst0 i v c1 c3)).(\lambda (e: C).(\lambda (H2:
+(getl n c3 e)).(csubst0_getl_ge_back i n H c1 c3 v H1 e H2))))) c2 H0))))))).
+
+theorem getl_csubst1:
+ \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c
+(CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_:
+C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0
+a))))))))
+\def
+ \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e:
+C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind
+(\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind
+Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda
+(n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n)
+(CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2
+C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda
+(H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to
+(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda
+(a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(match k
+return (\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u:
+T).((getl O (CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda
+(a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) O a0 a))))))))) with [(Bind b) \Rightarrow
+(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead
+c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda
+(e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
+(CHead c _ _) \Rightarrow c])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t)
+(clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b)
+t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda (e0:
+C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
+(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead e (Bind Abbr) u) (CHead
+c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O
+(CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H3 \def (f_equal
+C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind Abbr) u) (CHead
+c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O
+(CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in (\lambda (H4: (eq B
+Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 (CHead c0 (Bind b) t) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) (eq_ind B Abbr
+(\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t
+(CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O
+a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t
+(CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O
+a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead c0 (Bind Abbr)
+t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u H3)))) H2))
+H1)))))) | (Flat f) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let
+H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T (\lambda (t2:
+T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_:
+C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a:
+C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: (subst1 O u t (lift
+(S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead e (Bind Abbr) u) c0
+(drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) u) t (getl_gen_O
+(CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in (ex2_2_ind C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda
+(_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a:
+C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H4:
+(csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 x1)).(ex2_2_intro C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead x0 (Flat f)
+(lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O x) H2 c0 x0 H4)
+(drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) H1)))))))]))))
+c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall (e: C).(\forall
+(u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0:
+C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a:
+C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind (\lambda (c0:
+C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u))
+\to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) (\lambda (n0:
+nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) (CSort n0)
+(CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind Abbr) u) H0
+(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CSort n0) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) (\lambda
+(c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead
+e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S
+n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0
+a))))))))).(\lambda (k: K).(match k return (\lambda (k0: K).(\forall (t:
+T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 t) (CHead e
+(Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S
+n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n)
+a0 a))))))))) with [(Bind b) \Rightarrow (\lambda (t: T).(\lambda (e:
+C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead e
+(Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in
+(ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
+T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u
+(getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda
+(_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda
+(a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda
+(H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C
+C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t)
+a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0
+(Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift
+(S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4))))
+H2))))))) | (Flat f) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u:
+T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr)
+u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T
+(\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0)))
+(\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x:
+T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e
+u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C
+(\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0:
+C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0:
+C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0
+x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u
+(CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S
+n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x)
+(csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat
+(S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))])))) c)))) d).
-axiom subst1_confluence_neq: \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1: nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda (t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t)))))))))))) .
+inductive fsubst0 (i:nat) (v:T) (c1:C) (t1:T): C \to (T \to Prop) \def
+| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1
+t2))
+| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2
+t1))
+| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2:
+C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))).
+
+theorem fsubst0_gen_base:
+ \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall
+(v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1
+c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0
+i v t1 t2) (csubst0 i v c1 c2)))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(let H0 \def
+(match H return (\lambda (c: C).(\lambda (t: T).(\lambda (_: (fsubst0 ? ? ? ?
+c t)).((eq C c c2) \to ((eq T t t2) \to (or3 (land (eq C c1 c2) (subst0 i v
+t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2)
+(csubst0 i v c1 c2)))))))) with [(fsubst0_snd t0 H0) \Rightarrow (\lambda
+(H1: (eq C c1 c2)).(\lambda (H2: (eq T t0 t2)).(eq_ind C c2 (\lambda (c:
+C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to (or3 (land (eq C c c2) (subst0 i
+v t1 t2)) (land (eq T t1 t2) (csubst0 i v c c2)) (land (subst0 i v t1 t2)
+(csubst0 i v c c2)))))) (\lambda (H3: (eq T t0 t2)).(eq_ind T t2 (\lambda (t:
+T).((subst0 i v t1 t) \to (or3 (land (eq C c2 c2) (subst0 i v t1 t2)) (land
+(eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v t1 t2) (csubst0 i v c2
+c2))))) (\lambda (H4: (subst0 i v t1 t2)).(or3_intro0 (land (eq C c2 c2)
+(subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v
+t1 t2) (csubst0 i v c2 c2)) (conj (eq C c2 c2) (subst0 i v t1 t2) (refl_equal
+C c2) H4))) t0 (sym_eq T t0 t2 H3))) c1 (sym_eq C c1 c2 H1) H2 H0))) |
+(fsubst0_fst c0 H0) \Rightarrow (\lambda (H1: (eq C c0 c2)).(\lambda (H2: (eq
+T t1 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t1 t2) \to ((csubst0 i v c1 c)
+\to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i
+v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))) (\lambda (H3: (eq
+T t1 t2)).(eq_ind T t2 (\lambda (t: T).((csubst0 i v c1 c2) \to (or3 (land
+(eq C c1 c2) (subst0 i v t t2)) (land (eq T t t2) (csubst0 i v c1 c2)) (land
+(subst0 i v t t2) (csubst0 i v c1 c2))))) (\lambda (H4: (csubst0 i v c1
+c2)).(or3_intro1 (land (eq C c1 c2) (subst0 i v t2 t2)) (land (eq T t2 t2)
+(csubst0 i v c1 c2)) (land (subst0 i v t2 t2) (csubst0 i v c1 c2)) (conj (eq
+T t2 t2) (csubst0 i v c1 c2) (refl_equal T t2) H4))) t1 (sym_eq T t1 t2 H3)))
+c0 (sym_eq C c0 c2 H1) H2 H0))) | (fsubst0_both t0 H0 c0 H1) \Rightarrow
+(\lambda (H2: (eq C c0 c2)).(\lambda (H3: (eq T t0 t2)).(eq_ind C c2 (\lambda
+(c: C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to ((csubst0 i v c1 c) \to (or3
+(land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1
+c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2))))))) (\lambda (H4: (eq T
+t0 t2)).(eq_ind T t2 (\lambda (t: T).((subst0 i v t1 t) \to ((csubst0 i v c1
+c2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2)
+(csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2))))))
+(\lambda (H5: (subst0 i v t1 t2)).(\lambda (H6: (csubst0 i v c1
+c2)).(or3_intro2 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2)
+(csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)) (conj
+(subst0 i v t1 t2) (csubst0 i v c1 c2) H5 H6)))) t0 (sym_eq T t0 t2 H4))) c0
+(sym_eq C c0 c2 H2) H3 H0 H1)))]) in (H0 (refl_equal C c2) (refl_equal T
+t2))))))))).
-axiom subst1_confluence_eq: \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t: T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t))))))))) .
+inductive tau0 (g:G): C \to (T \to (T \to Prop)) \def
+| tau0_sort: \forall (c: C).(\forall (n: nat).(tau0 g c (TSort n) (TSort
+(next g n))))
+| tau0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau0 g d v w)
+\to (tau0 g c (TLRef i) (lift (S i) O w))))))))
+| tau0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((tau0 g d v w)
+\to (tau0 g c (TLRef i) (lift (S i) O v))))))))
+| tau0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
+T).(\forall (t2: T).((tau0 g (CHead c (Bind b) v) t1 t2) \to (tau0 g c (THead
+(Bind b) v t1) (THead (Bind b) v t2)))))))
+| tau0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2:
+T).((tau0 g c t1 t2) \to (tau0 g c (THead (Flat Appl) v t1) (THead (Flat
+Appl) v t2))))))
+| tau0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((tau0 g c v1
+v2) \to (\forall (t1: T).(\forall (t2: T).((tau0 g c t1 t2) \to (tau0 g c
+(THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
+
+theorem tau0_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
+e) \to (tau0 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (tau0 g e t1 t2)).(tau0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
+\to (tau0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
+(n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
+(drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(tau0 g c0 t (lift h d
+(TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(tau0 g
+c0 (TSort n) t)) (tau0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
+(next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (tau0 g d v
+w)).(\lambda (H2: ((\forall (c: C).(\forall (h: nat).(\forall (d0:
+nat).((drop h d0 c d) \to (tau0 g c (lift h d0 v) (lift h d0
+w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
+(drop h d0 c0 c)).(lt_le_e i d0 (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0
+(lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
+i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (tau0 g c0 (lift h
+d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
+H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
+Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
+i)) c1 d)) (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
+d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
+(TLRef i) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
+nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 g c0 (TLRef i)
+(lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
+i)) w)) (\lambda (t: T).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
+nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
+(tau0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
+(Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
+w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
+i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
+h)) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
+(S i) (\lambda (_: nat).(tau0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
+O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(tau0 g c0
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
+c0 (TLRef (plus i h)) (lift n O w))) (tau0_abbr g c0 d v (plus i h)
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
+h (S i)) (plus_comm h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
+h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
+i (S O)) (plus_comm i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: ((\forall (c: C).(\forall (h:
+nat).(\forall (d0: nat).((drop h d0 c d) \to (tau0 g c (lift h d0 v) (lift h
+d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
+(H3: (drop h d0 c0 c)).(lt_le_e i d0 (tau0 g c0 (lift h d0 (TLRef i)) (lift h
+d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
+(Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (tau0 g
+c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
+C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
+(minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
+H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
+(minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
+h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
+(minus d0 (S i)) c1 d)) (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
+i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
+h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
+d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i)
+O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 g
+c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
+(minus d0 (S i)) v)) (\lambda (t: T).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0
+(\lambda (_: nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
+v)))) (tau0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
+x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
+i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
+(le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
+i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
+(le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
+h)) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
+(S i) (\lambda (_: nat).(tau0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
+O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(tau0 g c0
+(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
+c0 (TLRef (plus i h)) (lift n O v))) (tau0_abst g c0 d v (plus i h)
+(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
+h (S i)) (plus_comm h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
+h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
+i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
+i (S O)) (plus_comm i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g (CHead c (Bind b) v) t3
+t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 (CHead c (Bind b) v)) \to (tau0 g c0 (lift h d t3) (lift h
+d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
+(Bind b) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Bind b) v
+t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
+(\lambda (t: T).(tau0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
+d) t3)) t)) (tau0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
+t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
+b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
+h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
+(c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g
+c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (tau0 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
+Appl) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Flat Appl) v
+t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
+t4)) (\lambda (t: T).(tau0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
+(Flat Appl) d) t3)) t)) (tau0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
+d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
+d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
+(THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
+(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c v1
+v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (tau0 g c0 (lift h d v1) (lift h d
+v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g c t3
+t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c0 c) \to (tau0 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
+(Flat Cast) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Flat Cast)
+v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
+d) t4)) (\lambda (t: T).(tau0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
+(s (Flat Cast) d) t3)) t)) (tau0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
+h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
+(s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
+Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
+v1 t3 h d))))))))))))))) e t1 t2 H))))).
+
+theorem tau0_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c
+t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
+T).(ex T (\lambda (t3: T).(tau0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
+(n: nat).(ex_intro T (\lambda (t2: T).(tau0 g c0 (TSort (next g n)) t2))
+(TSort (next g (next g n))) (tau0_sort g c0 (next g n))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
+(CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (tau0 g d v
+w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g d w t2)))).(let H3 \def H2
+in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex T (\lambda (t2: T).(tau0 g
+c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (tau0 g d w
+x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O w) t2)) (lift (S i)
+O x) (tau0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
+H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
+T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g
+d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex
+T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
+(_: (tau0 g d w x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v)
+t2)) (lift (S i) O w) (tau0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
+i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
+T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g (CHead c0 (Bind b)
+v) t2 t3)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g (CHead c0 (Bind b) v)
+t3 t2)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(tau0 g (CHead c0
+(Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3)
+t4))) (\lambda (x: T).(\lambda (H3: (tau0 g (CHead c0 (Bind b) v) t3
+x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3) t4)) (THead
+(Bind b) v x) (tau0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0
+t2 t3)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g c0 t3 t2)))).(let H2
+\def H1 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4:
+T).(tau0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
+(tau0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Appl)
+v t3) t4)) (THead (Flat Appl) v x) (tau0_appl g c0 v t3 x H3)))) H2))))))))
+(\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c0 v1
+v2)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g c0 v2 t2)))).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H3: (ex T
+(\lambda (t2: T).(tau0 g c0 t3 t2)))).(let H4 \def H1 in (ex_ind T (\lambda
+(t4: T).(tau0 g c0 v2 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Flat
+Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (tau0 g c0 v2 x)).(let H6
+\def H3 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4:
+T).(tau0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
+(tau0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Cast)
+v2 t3) t4)) (THead (Flat Cast) x x0) (tau0_cast g c0 v2 x H5 t3 x0 H7))))
+H6)))) H4))))))))))) c t1 t H))))).
-axiom subst1_confluence_lift: \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1 i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1 t2))))))) .
+inductive tau1 (g:G) (c:C) (t1:T): T \to Prop \def
+| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
+| tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c
+t t2) \to (tau1 g c t1 t2)))).
+
+theorem tau1_trans:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c
+t1 t) \to (\forall (t2: T).((tau1 g c t t2) \to (tau1 g c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(tau1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (tau1 g c t t2)).(tau1_ind g
+c t (\lambda (t0: T).(tau1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (tau0 g
+c t t3)).(tau1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (tau1 g
+c t t0)).(\lambda (H2: (tau1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (tau0
+g c t0 t3)).(tau1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))).
+
+theorem tau1_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1:
+T).(\forall (t2: T).((tau1 g (CHead c (Bind b) v) t1 t2) \to (tau1 g c (THead
+(Bind b) v t1) (THead (Bind b) v t2))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H: (tau1 g (CHead c (Bind b) v) t1
+t2)).(tau1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(tau1 g c (THead
+(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (tau0 g
+(CHead c (Bind b) v) t1 t3)).(tau1_tau0 g c (THead (Bind b) v t1) (THead
+(Bind b) v t3) (tau0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_:
+(tau1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (tau1 g c (THead (Bind b) v
+t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g (CHead c
+(Bind b) v) t t3)).(tau1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t)
+H1 (THead (Bind b) v t3) (tau0_bind g b c v t t3 H2))))))) t2 H))))))).
+
+theorem tau1_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((tau1 g c t1 t2) \to (tau1 g c (THead (Flat Appl) v t1) (THead (Flat
+Appl) v t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(tau1
+g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3:
+T).(\lambda (H0: (tau0 g c t1 t3)).(tau1_tau0 g c (THead (Flat Appl) v t1)
+(THead (Flat Appl) v t3) (tau0_appl g c v t1 t3 H0)))) (\lambda (t:
+T).(\lambda (_: (tau1 g c t1 t)).(\lambda (H1: (tau1 g c (THead (Flat Appl) v
+t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g c t
+t3)).(tau1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1
+(THead (Flat Appl) v t3) (tau0_appl g c v t t3 H2))))))) t2 H)))))).
+
+theorem tau1_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau1 g e
+t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
+e) \to (tau1 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (tau1 g e t1 t2)).(tau1_ind g e t1 (\lambda (t: T).(\forall (c:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (tau1 g c (lift h
+d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g e t1
+t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop
+h d c e)).(tau1_tau0 g c (lift h d t1) (lift h d t3) (tau0_lift g e t1 t3 H0
+c h d H1)))))))) (\lambda (t: T).(\lambda (_: (tau1 g e t1 t)).(\lambda (H1:
+((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to
+(tau1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2:
+(tau0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H3: (drop h d c e)).(tau1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3)
+(lift h d t3) (tau0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))).
+
+theorem tau1_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c
+t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(tau1 g c t1 t)).(tau1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2:
+T).(tau0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (tau0 g c t1
+t2)).(tau0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (tau1 g c t1
+t0)).(\lambda (_: (ex T (\lambda (t2: T).(tau0 g c t0 t2)))).(\lambda (t2:
+T).(\lambda (H2: (tau0 g c t0 t2)).(tau0_correct g c t0 t2 H2)))))) t H))))).
+
+theorem tau1_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau1 g d v w)
+\to (tau1 g c (TLRef i) (lift (S i) O w)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w:
+T).(\lambda (H0: (tau1 g d v w)).(tau1_ind g d v (\lambda (t: T).(tau1 g c
+(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (tau0 g d v
+t2)).(tau1_tau0 g c (TLRef i) (lift (S i) O t2) (tau0_abbr g c d v i H t2
+H1)))) (\lambda (t: T).(\lambda (_: (tau1 g d v t)).(\lambda (H2: (tau1 g c
+(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (tau0 g d t
+t2)).(tau1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2)
+(tau0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w
+H0)))))))).
+
+theorem tau1_cast2:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((tau1 g c
+t1 t2) \to (\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T
+(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(\forall (v1:
+T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(tau1 g c
+v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat
+Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g c t1 t3)).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (H1: (tau0 g c v1 v2)).(ex_intro2 T
+(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (tau1_tau0 g c v1 v2 H1)
+(tau1_tau0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (tau0_cast
+g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (tau1 g c t1
+t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to
+(ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead
+(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda
+(H2: (tau0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (tau0 g
+c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T
+(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(tau1 g c v1
+v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast)
+v3 t3)))) (\lambda (x: T).(\lambda (H5: (tau1 g c v1 x)).(\lambda (H6: (tau1
+g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def
+(tau1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4:
+T).(tau0 g c x t4)) (ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3:
+T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda
+(x0: T).(\lambda (H8: (tau0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(tau1 g
+c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat
+Cast) v3 t3))) x0 (tau1_sing g c v1 x H5 x0 H8) (tau1_sing g c (THead (Flat
+Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (tau0_cast
+g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))).
+
+theorem tau1_cnt:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c
+t1 t) \to (ex2 T (\lambda (t2: T).(tau1 g c t1 t2)) (\lambda (t2: T).(cnt
+t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+T).(ex2 T (\lambda (t3: T).(tau1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3))))))
+(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(tau1 g c0
+(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (tau1_tau0 g c0
+(TSort n) (TSort (next g n)) (tau0_sort g c0 n)) (cnt_sort (next g n)))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (tau0
+g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(tau1 g d v t2)) (\lambda
+(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d
+v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef
+i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (tau1 g d v
+x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i)
+t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (tau1_abbr g c0 d v i H0 x
+H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abst) v))).(\lambda (w: T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: (ex2 T
+(\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def
+H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2))
+(ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2)))
+(\lambda (x: T).(\lambda (H4: (tau1 g d v x)).(\lambda (H5: (cnt
+x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2:
+T).(cnt t2)) (lift (S i) O x) (tau1_trans g c0 (TLRef i) (lift (S i) O v)
+(tau1_tau0 g c0 (TLRef i) (lift (S i) O v) (tau0_abst g c0 d v i H0 w H1))
+(lift (S i) O x) (tau1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i
+H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0:
+C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g
+(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t3: T).(tau1 g
+(CHead c0 (Bind b) v) t2 t3)) (\lambda (t2: T).(cnt t2)))).(let H2 \def H1 in
+(ex2_ind T (\lambda (t4: T).(tau1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda
+(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Bind b) v t2)
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g (CHead
+c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4:
+T).(tau1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead
+(Bind b) v x) (tau1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v)))))
+H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t3:
+T).(tau1 g c0 t2 t3)) (\lambda (t2: T).(cnt t2)))).(let H2 \def H1 in
+(ex2_ind T (\lambda (t4: T).(tau1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4))
+(ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda
+(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g c0 t2 x)).(\lambda
+(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v
+t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (tau1_appl g c0 v
+t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0:
+C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (tau0 g c0 v1
+v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(tau1 g c0 v1 t2)) (\lambda (t2:
+T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2
+t3)).(\lambda (H3: (ex2 T (\lambda (t3: T).(tau1 g c0 t2 t3)) (\lambda (t2:
+T).(cnt t2)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(tau1 g c0 t2
+t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead
+(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda
+(H5: (tau1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (tau1_cast2 g c0
+t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(tau1 g
+c0 v1 v3)) (\lambda (v3: T).(tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat
+Cast) v3 x))) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2)
+t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (tau1 g c0 v1
+x0)).(\lambda (H9: (tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0
+x))).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2) t4))
+(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat
+Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))).
-inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
-| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (s k i) v (CHead c k u1) (CHead c k u2))))))))
-| csubst0_fst: \forall (k: K).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (s k i) v (CHead c1 k u) (CHead c2 k u))))))))
-| csubst0_both: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i v c1 c2) \to (csubst0 (s k i) v (CHead c1 k u1) (CHead c2 k u2)))))))))).
+inductive A: Set \def
+| ASort: nat \to (nat \to A)
+| AHead: A \to (A \to A).
-axiom csubst0_snd_bind: \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c: C).(csubst0 (S i) v (CHead c (Bind b) u1) (CHead c (Bind b) u2)))))))) .
+definition asucc:
+ G \to (A \to A)
+\def
+ let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n)
+\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h)
+\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g
+a2))]) in asucc.
-axiom csubst0_fst_bind: \forall (b: B).(\forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (u: T).(csubst0 (S i) v (CHead c1 (Bind b) u) (CHead c2 (Bind b) u)))))))) .
+definition aplus:
+ G \to (A \to (nat \to A))
+\def
+ let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O
+\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus.
-axiom csubst0_both_bind: \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst0 i v c1 c2) \to (csubst0 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))))))))) .
+inductive leq (g:G): A \to (A \to Prop) \def
+| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
+(n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
+| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2
+a4))))))).
+
+theorem leq_gen_sort:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
+g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2:
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H return
+(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort
+h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda
+(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k)
+(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2:
+(eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match
+e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def (f_equal A
+nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 n1) H1)
+in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to ((eq A (ASort h2 n2)
+a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort h2 n2) k)) \to (ex2_3
+nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a2
+(ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0:
+nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))))
+(\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n: nat).((eq A (ASort
+h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g (ASort h2 n2) k)) \to
+(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_:
+nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3:
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3
+n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2
+n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_:
+nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))
+(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda
+(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda
+(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort
+h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq
+nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0
+a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1
+n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead
+a1 a3) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _
+_) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H2) in
+(False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g a3 a4) \to
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_:
+nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2:
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal
+A a2))))))).
+
+theorem leq_gen_head:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g
+(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda
+(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda
+(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H return (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) \to
+((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a (AHead
+a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_:
+A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H2:
+(eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e:
+A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True |
+(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A
+(ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda
+(a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head a0 a3 H0 a4 a5 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H3:
+(eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A (\lambda (e: A).(match e
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a)
+\Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) in ((let H5 \def (f_equal A
+A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2)
+in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a)
+\to ((leq g a6 a3) \to ((leq g a4 a5) \to (ex3_2 A A (\lambda (a7:
+A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_:
+A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8)))))))))
+(\lambda (H6: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3
+a5) a) \to ((leq g a1 a3) \to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7:
+A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_:
+A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))))))))
+(\lambda (H7: (eq A (AHead a3 a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a:
+A).((leq g a1 a3) \to ((leq g a2 a5) \to (ex3_2 A A (\lambda (a6: A).(\lambda
+(a7: A).(eq A a (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1
+a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))))))) (\lambda (H8: (leq
+g a1 a3)).(\lambda (H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6:
+A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) (\lambda (a6:
+A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2
+a7))) a3 a5 (refl_equal A (AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2
+H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead
+a1 a2)) (refl_equal A a))))))).
+
+theorem asucc_gen_sort:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A
+(ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0:
+nat).(eq A a (ASort h0 n0)))))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind
+(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda
+(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0
+n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0
+with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)])
+H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A (ASort
+n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) (\lambda (a0:
+A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda
+(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0)))))))).(\lambda (a1:
+A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) \to (ex_2 nat nat (\lambda
+(h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 n0)))))))).(\lambda (H1: (eq
+A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 \def (eq_ind A (ASort h n)
+(\lambda (ee: A).(match ee return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (asucc g (AHead a0 a1))
+H1) in (False_ind (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A
+(AHead a0 a1) (ASort h0 n0))))) H2))))))) a)))).
+
+theorem asucc_gen_head:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A
+(AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind
+(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3:
+A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3))))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc
+g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g
+(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead
+a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda
+(ee: A).(match ee return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H0) in
+(False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0)))
+(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda
+(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0:
+A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
+a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1)
+n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0:
+A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
+a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2)
+(asucc g a0)) \to (ex2 A (\lambda (a2: A).(eq A a0 (AHead a1 a2))) (\lambda
+(a0: A).(eq A a2 (asucc g a0))))))).(\lambda (a3: A).(\lambda (H0: (((eq A
+(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a0: A).(eq A a3 (AHead a1
+a0))) (\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H1: (eq A (AHead
+a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e:
+A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a _) \Rightarrow a])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in
+((let H3 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a2 | (AHead _ a) \Rightarrow a])) (AHead a1 a2)
+(AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let H5 \def
+(eq_ind_r A a0 (\lambda (a: A).((eq A (AHead a1 a2) (asucc g a)) \to (ex2 A
+(\lambda (a0: A).(eq A a (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g
+a0)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A (\lambda (a5:
+A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A a2 (asucc g
+a5))))) (let H6 \def (eq_ind A a2 (\lambda (a: A).((eq A (AHead a1 a) (asucc
+g a3)) \to (ex2 A (\lambda (a0: A).(eq A a3 (AHead a1 a0))) (\lambda (a0:
+A).(eq A a (asucc g a0)))))) H0 (asucc g a3) H3) in (let H7 \def (eq_ind A a2
+(\lambda (a: A).((eq A (AHead a1 a) (asucc g a1)) \to (ex2 A (\lambda (a0:
+A).(eq A a1 (AHead a1 a0))) (\lambda (a0: A).(eq A a (asucc g a0)))))) H5
+(asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda (a4: A).(ex2 A (\lambda
+(a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A a4 (asucc
+g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead a1 a3) (AHead a1 a4)))
+(\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3 (refl_equal A (AHead a1
+a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4)))) H2))))))) a)))).
+
+theorem aplus_reg_r:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
+(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A
+(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h:
+nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2
+(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n
+h1)) (aplus g a2 (plus n h2)))).(sym_equal A (asucc g (aplus g a2 (plus n
+h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1
+(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus
+g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g
+g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq
+A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
+
+theorem aplus_assoc:
+ \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A
+(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n:
+nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n
+h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n:
+nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus
+g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))))
+(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g
+(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O
+n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n))
+n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda
+(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g
+(aplus g a n1)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0))))
+(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g
+(asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0))))
+(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g
+(asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a
+(plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A
+(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))
+H0))))) (plus n (S n0)) (plus_n_Sm n n0)))) h2)))) h1))).
+
+theorem aplus_asucc:
+ \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
+h) (asucc g (aplus g a h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a
+(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h))))
+(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h)
+(aplus_assoc g a (S O) h)))).
+
+theorem aplus_sort_O_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
+n) (S k)) (aplus g (ASort O (next g n)) k))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc
+g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
+(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n)
+k)) (aplus_asucc g k (ASort O n))))).
+
+theorem aplus_sort_S_S_simpl:
+ \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
+(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind
+A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g
+(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g
+(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
+
+theorem asucc_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
+(asucc g a1) (asucc g a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k))).((match h1 return (\lambda (n: nat).((eq A (aplus g (ASort
+n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow
+(ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O
+\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) with
+[O \Rightarrow (\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort
+h2 n2) k))).((match h2 return (\lambda (n: nat).((eq A (aplus g (ASort O n1)
+k) (aplus g (ASort n n2) k)) \to (leq g (ASort O (next g n1)) (match n with
+[O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))
+with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort O n1) k) (aplus g
+(ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind A (aplus
+g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2))
+k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g
+(ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) (\lambda (a:
+A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) (refl_equal A
+(asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) H2) (aplus g
+(ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g (ASort O
+(next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) | (S n) \Rightarrow (\lambda
+(H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort (S n) n2) k))).(leq_sort
+g O n (next g n1) n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a:
+A).(eq A a (aplus g (ASort n n2) k))) (eq_ind A (aplus g (ASort (S n) n2) (S
+k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus
+g (ASort (S n) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g
+(ASort (S n) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n) n2) k)))
+(aplus g (ASort O n1) k) H2) (aplus g (ASort n n2) k) (aplus_sort_S_S_simpl g
+n2 n k)) (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k))))])
+H1)) | (S n) \Rightarrow (\lambda (H1: (eq A (aplus g (ASort (S n) n1) k)
+(aplus g (ASort h2 n2) k))).((match h2 return (\lambda (n0: nat).((eq A
+(aplus g (ASort (S n) n1) k) (aplus g (ASort n0 n2) k)) \to (leq g (ASort n
+n1) (match n0 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow
+(ASort h n2)])))) with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort (S
+n) n1) k) (aplus g (ASort O n2) k))).(leq_sort g n O n1 (next g n2) k (eq_ind
+A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort n n1) k)
+a)) (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq A a (aplus
+g (ASort O n2) (S k)))) (eq_ind_r A (aplus g (ASort O n2) k) (\lambda (a:
+A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) (refl_equal A
+(asucc g (aplus g (ASort O n2) k))) (aplus g (ASort (S n) n1) k) H2) (aplus g
+(ASort n n1) k) (aplus_sort_S_S_simpl g n1 n k)) (aplus g (ASort O (next g
+n2)) k) (aplus_sort_O_S_simpl g n2 k)))) | (S n0) \Rightarrow (\lambda (H2:
+(eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort (S n0) n2) k))).(leq_sort
+g n n0 n1 n2 k (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq
+A a (aplus g (ASort n0 n2) k))) (eq_ind A (aplus g (ASort (S n0) n2) (S k))
+(\lambda (a: A).(eq A (aplus g (ASort (S n) n1) (S k)) a)) (eq_ind_r A (aplus
+g (ASort (S n0) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g
+(ASort (S n0) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n0) n2)
+k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n0 n2) k)
+(aplus_sort_S_S_simpl g n2 n0 k)) (aplus g (ASort n n1) k)
+(aplus_sort_S_S_simpl g n1 n k))))]) H1))]) H0))))))) (\lambda (a3:
+A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g (asucc g
+a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5
+a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g a3 a4 H0
+(asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
+
+theorem asucc_inj:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc
+g a2)) \to (leq g a1 a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g
+(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0))
+(asucc g (ASort n1 n2)))).((match n return (\lambda (n3: nat).((leq g (asucc
+g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1
+n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc
+g (ASort n1 n2)))).((match n1 return (\lambda (n3: nat).((leq g (asucc g
+(ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3
+n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc
+g (ASort O n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0
+(ASort O (next g n2))) \to (leq g (ASort O n0) (ASort O n2))))))) with
+[(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1)
+(ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g
+n2)))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1]))
+(ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat
+(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g
+n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq
+A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n n1) k)
+(aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda
+(H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq
+A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort O n) k)
+(aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda
+(H6: (eq A (ASort h2 n3) (ASort O (next g n2)))).(let H7 \def (f_equal A nat
+(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort _ n)
+\Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort O (next g
+n2)) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return
+(\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _)
+\Rightarrow h2])) (ASort h2 n3) (ASort O (next g n2)) H6) in (eq_ind nat O
+(\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq A (aplus g (ASort O (next
+g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0) (ASort O n2)))))
+(\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat (next g n2) (\lambda (n:
+nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to
+(leq g (ASort O n0) (ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O
+(next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A
+(aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O
+(next g n2)) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0
+k)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda
+(a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k))
+(aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3
+(sym_eq nat n3 (next g n2) H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq
+nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head
+a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O
+(next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g
+n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead
+a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq
+g (ASort O n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A
+(ASort O (next g n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3)
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S
+n3) n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0
+(ASort n3 n2)) \to (leq g (ASort O n0) (ASort (S n3) n2))))))) with
+[(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1)
+(ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3
+n2))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1]))
+(ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat
+(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g
+n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq
+A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g
+(ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda
+(H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq
+A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g
+(ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6:
+(eq A (ASort h2 n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e:
+A).(match e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n |
+(AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8
+\def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3)
+(ASort n3 n2) H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to
+((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g
+(ASort O n0) (ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat
+n2 (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort
+n3 n) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A
+(aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def
+(eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus
+g (ASort n3 n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g
+n0 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a:
+A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k))
+(aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11))))
+n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1
+(next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2
+H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g
+n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def
+(eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
+(ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort n3 n2))
+\to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort (S n3)
+n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0)))
+(refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0:
+(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1
+return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O \Rightarrow
+(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O
+n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O
+(next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort
+h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3
+n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def
+(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1)
+(ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e
+return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _)
+\Rightarrow h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
+(n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to
+((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort
+(S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0
+(\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A
+(aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3)
+n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g
+n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3]))
+(ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def (f_equal A nat
+(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort O (next g
+n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq
+A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k)) \to (leq g (ASort (S
+n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat
+(next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g
+(ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))) (\lambda (H10:
+(eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2)) k))).(let H
+\def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g
+(ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k))
+(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
+k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in
+(leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9)))
+h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1
+n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda
+(H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4)
+(ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e:
+A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False
+| (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A
+(AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4)
+\to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2
+(refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4)
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort
+(S n4) n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4
+n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1
+h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3) (ASort n3
+n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((let H3 \def
+(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h1 n3)
+(ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e
+return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _)
+\Rightarrow h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda
+(n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A
+(aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0
+(\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g
+(ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0)
+(ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let
+H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat)
+with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4])) (ASort h2 n4)
+(ASort n4 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e
+return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n | (AHead _ _)
+\Rightarrow h2])) (ASort h2 n4) (ASort n4 n2) H6) in (eq_ind nat n4 (\lambda
+(n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort
+n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H9:
+(eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus g (ASort n3
+n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4)
+n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n2)
+k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A
+a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S k))
+(aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a))
+H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in
+(leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2
+(sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3
+H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2:
+(eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort
+n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4)
+(ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3)
+n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3
+n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) H)))) (\lambda (a: A).(\lambda
+(H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0)
+a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g (ASort n n0)) (asucc g
+a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq g (asucc g (ASort n
+n0)) (asucc g (AHead a a0)))).((match n return (\lambda (n1: nat).((((leq g
+(asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) \to
+((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
+a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
+(ASort n1 n0) (AHead a a0)))))) with [O \Rightarrow (\lambda (_: (((leq g
+(asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda
+(_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0)
+a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a
+a0)))).(let H5 \def (match H4 return (\lambda (a1: A).(\lambda (a2:
+A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (ASort O (next g n0))) \to ((eq A a2
+(AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a a0))))))) with
+[(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1)
+(ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g
+a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1]))
+(ASort h1 n1) (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat
+(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g
+n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq
+A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k)
+(aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0)))))) (\lambda
+(H7: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq
+A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort O n) k)
+(aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))))) (\lambda
+(H8: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A
+(ASort h2 n2) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a
+(asucc g a0)) H8) in (False_ind ((eq A (aplus g (ASort O (next g n0)) k)
+(aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))) H9))) n1
+(sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) |
+(leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3)
+(ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a (asucc g
+a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A (AHead
+a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq
+g (ASort O n0) (AHead a a0))))) H6)) H5 H2 H3)))]) in (H5 (refl_equal A
+(ASort O (next g n0))) (refl_equal A (AHead a (asucc g a0)))))))) | (S n1)
+\Rightarrow (\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a))
+\to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S
+n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq
+g (asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H5 \def (match H4
+return (\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A
+a1 (ASort n1 n0)) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S
+n1) n0) (AHead a a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow
+(\lambda (H3: (eq A (ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort
+h2 n2) (AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e:
+A).(match e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n |
+(AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H3) in ((let H6
+\def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1)
+(ASort n1 n0) H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to
+((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1)
+k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))))
+(\lambda (H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort
+h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8:
+(eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2
+n2) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0))
+H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))
+\to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7)))
+h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3)
+\Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5:
+(eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead
+a1 a3) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _
+_) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H4) in
+(False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2)
+\to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0))))) H6)) H5 H2
+H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a (asucc g
+a0))))))))]) H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
+A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
+(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
+(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
+(ASort n n0)))).((match n return (\lambda (n1: nat).((leq g (asucc g (AHead a
+a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) with
+[O \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
+n0)))).(let H3 \def (match H2 return (\lambda (a1: A).(\lambda (a2:
+A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (AHead a (asucc g a0))) \to ((eq A
+a2 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with
+[(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1)
+(AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort O (next g
+n0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort
+h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H5)) H4 H2))) |
+(leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3)
+(AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort O (next g
+n0)))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a]))
+(AHead a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A
+(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g
+a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A
+(AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5 a2) \to ((leq g a3 a4)
+\to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H8: (eq A a3 (asucc g
+a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort O
+(next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0)
+(ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4) (ASort O (next g
+n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort O (next g n0)) H9) in (False_ind ((leq g a a2)
+\to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0) (ASort O n0)))) H10)))
+a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))])
+in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort O (next g
+n0)))))) | (S n1) \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0))
+(asucc g (ASort (S n1) n0)))).(let H3 \def (match H2 return (\lambda (a1:
+A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1 (AHead a (asucc g
+a0))) \to ((eq A a2 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1)
+n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A
+(ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2)
+(ASort n1 n0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match
+e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _
+_) \Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A
+(ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2)))
+| (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3)
+(AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1
+n0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead
+a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in
+(eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4)
+(ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0)
+(ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A
+(asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g
+a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0))))))
+(\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A
+(AHead a2 a4) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
+n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g
+(AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8)))
+a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a
+(asucc g a0))) (refl_equal A (ASort n1 n0)))))]) H1)))) (\lambda (a3:
+A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g
+(AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc g (AHead a
+a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: (leq g (asucc
+g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H4 \def (match H3 return
+(\lambda (a1: A).(\lambda (a2: A).(\lambda (_: (leq ? a1 a2)).((eq A a1
+(AHead a (asucc g a0))) \to ((eq A a2 (AHead a3 (asucc g a4))) \to (leq g
+(AHead a a0) (AHead a3 a4))))))) with [(leq_sort h1 h2 n1 n2 k H4)
+\Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a (asucc g
+a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g a4)))).((let H7
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort h2 n2)
+(AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | (leq_head
+a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a
+(asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (AHead a3 (asucc g
+a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a]))
+(AHead a3 a5) (AHead a (asucc g a0)) H6) in ((let H9 \def (f_equal A A
+(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g
+a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A
+(AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6)
+\to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g
+a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3
+(asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0)
+(AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (AHead a3 (asucc g
+a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e return (\lambda
+(_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a]))
+(AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \def (f_equal A A
+(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a4 | (AHead a _) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc
+g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to
+((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3
+a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4)
+(\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead
+a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g
+(asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6
+(sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq
+A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4
+(refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g
+a4)))))))))) a2)))))) a1)).
+
+theorem aplus_asort_O_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O
+n) h) (ASort O (next_plus g n h)))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
+nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda
+(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall
+(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0
+n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n)
+(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat
+(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next
+g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n))
+(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n
+(ASort O n0)))))) h)).
+
+theorem aplus_asort_le_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h
+k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k:
+nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort
+(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O
+k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n)))
+(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0:
+nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A
+(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k:
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A
+(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda
+(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat
+O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n)
+h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S
+x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S
+_) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus g
+(ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0))))
+(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A
+(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda
+(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort
+(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n
+n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g
+h0 (ASort (S n) n0))))))) k)))) h)).
+
+theorem aplus_asort_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A
+(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n:
+nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus
+g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k))
+(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h)
+(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k)
+(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus
+h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a
+(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O
+(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k
+h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A
+(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k)))))
+(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h
+(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k)
+(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus
+h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h
+(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort
+(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n
+(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h)
+n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h)
+(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h)
+(aplus_asort_le_simpl g h k n H))))))).
+
+theorem aplus_ahead_simpl:
+ \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A
+(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
+\def
+ \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1:
+A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2
+n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2))))
+(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A
+(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1:
+A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda
+(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g
+(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n)
+(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n
+a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2)))))))
+h)).
+
+theorem aplus_asucc_false:
+ \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a)
+h) a) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h:
+nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A
+(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n
+return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0))
+\to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g
+n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0))
+h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g
+(next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2
+\def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus (g:
+G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S i0)
+\Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0) (minus
+h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) (ASort O
+n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n: nat).(eq nat
+(next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in (le_lt_false
+(next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g n0) h)
+(\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n (next_plus g
+(next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1) \Rightarrow
+(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let H1
+\def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A a (ASort (S
+n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
+(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
+A).(match e return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n |
+(AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def
+(\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m
+with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1
+h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0)
+H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let
+rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O
+\Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in
+next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g n0 (minus h
+n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) (S
+n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1))
+(minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda
+(_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
+Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
+g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
+nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
+a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g
+a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc
+g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A
+A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with
+[O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g
+(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g
+a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+
+theorem aplus_inj:
+ \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A
+(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2:
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
+h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A
+(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_:
+(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a:
+A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0:
+(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g
+a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g
+n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq
+nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2:
+nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
+h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq
+A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a:
+A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A
+(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a)
+n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O)))))
+(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a
+n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1:
+(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def
+(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus
+g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def
+(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g
+a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat
+nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).
+
+theorem ahead_inj_snd:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
+(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match
+H return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a
+(AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2 a4)))))) with
+[(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1)
+(AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3 a4))).((let H3
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) (AHead a3
+a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq
+g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1) \Rightarrow (\lambda
+(H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3: (eq A (AHead a5 a7)
+(AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow
+a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def (f_equal A A (\lambda
+(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
+(AHead a _) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in (eq_ind A a1
+(\lambda (a: A).((eq A a6 a2) \to ((eq A (AHead a5 a7) (AHead a3 a4)) \to
+((leq g a a5) \to ((leq g a6 a7) \to (leq g a2 a4)))))) (\lambda (H6: (eq A
+a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a5 a7) (AHead a3 a4)) \to
+((leq g a1 a5) \to ((leq g a a7) \to (leq g a2 a4))))) (\lambda (H7: (eq A
+(AHead a5 a7) (AHead a3 a4))).(let H8 \def (f_equal A A (\lambda (e:
+A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 |
+(AHead _ a) \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in ((let H9 \def
+(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort
+_ _) \Rightarrow a5 | (AHead a _) \Rightarrow a])) (AHead a5 a7) (AHead a3
+a4) H7) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a4) \to ((leq g a1 a) \to
+((leq g a2 a7) \to (leq g a2 a4))))) (\lambda (H10: (eq A a7 a4)).(eq_ind A
+a4 (\lambda (a: A).((leq g a1 a3) \to ((leq g a2 a) \to (leq g a2 a4))))
+(\lambda (_: (leq g a1 a3)).(\lambda (H12: (leq g a2 a4)).H12)) a7 (sym_eq A
+a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8))) a6 (sym_eq A a6 a2 H6))) a0
+(sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2))
+(refl_equal A (AHead a3 a4))))))))).
+
+theorem leq_refl:
+ \forall (g: G).(\forall (a: A).(leq g a a))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0))
+(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A
+(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0
+a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1
+H0))))) a)).
+
+theorem leq_eq:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1
+a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1
+a2)).(eq_ind_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
+
+theorem leq_asucc:
+ \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
+a0)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1:
+A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro
+A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
+(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda
+(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A
+(\lambda (a0: A).(leq g a1 (asucc g a0))))).(let H1 \def H0 in (ex_ind A
+(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g
+(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc
+g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2)))
+(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1))))))
+a)).
+
+theorem leq_sym:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
+a2 a1))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1:
+nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k:
+nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
+k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_:
+(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6:
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3
+H1 a6 a5 H3))))))))) a1 a2 H)))).
+
+theorem leq_trans:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
+a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1:
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g
+(ASort h2 n2) a3)).(let H2 \def (match H1 return (\lambda (a: A).(\lambda
+(a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to ((eq A a0 a3)
+\to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3 k0 H1)
+\Rightarrow (\lambda (H2: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda (H3:
+(eq A (ASort h3 n3) a3)).((let H4 \def (f_equal A nat (\lambda (e: A).(match
+e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H2) in ((let H5 \def (f_equal A
+nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h2 n2) H2)
+in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to ((eq A (ASort h3 n3)
+a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort h3 n3) k0)) \to (leq
+g (ASort h1 n1) a3))))) (\lambda (H6: (eq nat n0 n2)).(eq_ind nat n2 (\lambda
+(n: nat).((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort h2 n) k0) (aplus
+g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a3)))) (\lambda (H7: (eq A
+(ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) (\lambda (a: A).((eq A (aplus g
+(ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a)))
+(\lambda (H8: (eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3)
+k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 n3)) (\lambda (H9: (lt k
+k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) (ASort h2 n2) k k H0 (minus
+k0 k)) in (let H10 \def (eq_ind_r nat (plus (minus k0 k) k) (\lambda (n:
+nat).(eq A (aplus g (ASort h1 n1) n) (aplus g (ASort h2 n2) n))) H_y k0
+(le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 H9)))) in (leq_sort g h1
+h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h2 n2) k0)
+(aplus g (ASort h3 n3) k0) H10 H8))))) (\lambda (H9: (le k0 k)).(let H_y \def
+(aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 k0 H8 (minus k k0)) in (let H10
+\def (eq_ind_r nat (plus (minus k k0) k0) (\lambda (n: nat).(eq A (aplus g
+(ASort h2 n2) n) (aplus g (ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k
+H9)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k)
+(aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) H0 H10))))))) a3 H7)) n0
+(sym_eq nat n0 n2 H6))) h0 (sym_eq nat h0 h2 H5))) H4)) H3 H1))) | (leq_head
+a1 a2 H1 a0 a4 H2) \Rightarrow (\lambda (H3: (eq A (AHead a1 a0) (ASort h2
+n2))).(\lambda (H4: (eq A (AHead a2 a4) a3)).((let H5 \def (eq_ind A (AHead
+a1 a0) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _
+_) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H3) in
+(False_ind ((eq A (AHead a2 a4) a3) \to ((leq g a1 a2) \to ((leq g a0 a4) \to
+(leq g (ASort h1 n1) a3)))) H5)) H4 H1 H2)))]) in (H2 (refl_equal A (ASort h2
+n2)) (refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda
+(_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g
+a3 a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5
+a6)).(\lambda (H3: ((\forall (a3: A).((leq g a6 a3) \to (leq g a5
+a3))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def
+(match H4 return (\lambda (a: A).(\lambda (a1: A).(\lambda (_: (leq ? a
+a1)).((eq A a (AHead a4 a6)) \to ((eq A a1 a0) \to (leq g (AHead a3 a5)
+a0)))))) with [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A
+(ASort h1 n1) (AHead a4 a6))).(\lambda (H6: (eq A (ASort h2 n2) a0)).((let H7
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead a4 a6) H5) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq
+A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3
+a5) a0))) H7)) H6 H4))) | (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda
+(H6: (eq A (AHead a5 a7) (AHead a4 a6))).(\lambda (H7: (eq A (AHead a6 a8)
+a0)).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead
+a5 a7) (AHead a4 a6) H6) in ((let H9 \def (f_equal A A (\lambda (e: A).(match
+e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _)
+\Rightarrow a])) (AHead a5 a7) (AHead a4 a6) H6) in (eq_ind A a4 (\lambda (a:
+A).((eq A a7 a6) \to ((eq A (AHead a6 a8) a0) \to ((leq g a a6) \to ((leq g
+a7 a8) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H10: (eq A a7 a6)).(eq_ind
+A a6 (\lambda (a: A).((eq A (AHead a6 a8) a0) \to ((leq g a4 a6) \to ((leq g
+a a8) \to (leq g (AHead a3 a5) a0))))) (\lambda (H11: (eq A (AHead a6 a8)
+a0)).(eq_ind A (AHead a6 a8) (\lambda (a: A).((leq g a4 a6) \to ((leq g a6
+a8) \to (leq g (AHead a3 a5) a)))) (\lambda (H12: (leq g a4 a6)).(\lambda
+(H13: (leq g a6 a8)).(leq_head g a3 a6 (H1 a6 H12) a5 a8 (H3 a8 H13)))) a0
+H11)) a7 (sym_eq A a7 a6 H10))) a5 (sym_eq A a5 a4 H9))) H8)) H7 H4 H5)))])
+in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A a0))))))))))))) a1 a2 H)))).
+
+theorem leq_ahead_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1)
+\to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n
+n0) a2) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (n1:
+nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) with [O
+\Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O n0))).(let
+H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ?
+a a0)).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O n0)) \to
+P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
+(ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H2: (eq A (ASort h2 n2)
+(ASort O n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _)
+\Rightarrow False])) I (AHead (ASort O n0) a2) H1) in (False_ind ((eq A
+(ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort O n0)
+a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort O n0))).((let H4 \def (f_equal
+A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O
+n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow
+a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in (eq_ind A (ASort O n0)
+(\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort O n0)) \to
+((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3
+a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort O n0)) \to
+((leq g (ASort O n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A
+(AHead a0 a4) (ASort O n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda
+(e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+False | (AHead _ _) \Rightarrow True])) I (ASort O n0) H7) in (False_ind
+((leq g (ASort O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2
+H6))) a1 (sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1
+(refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O n0))))) | (S
+n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S
+n1) n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq
+A a0 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0)
+a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) n0))).((let H3 \def
+(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
+(AHead (ASort (S n1) n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort (S
+n1) n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to
+P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2:
+(eq A (AHead a1 a3) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (AHead
+a0 a4) (ASort (S n1) n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match
+e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a)
+\Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in ((let H5
+\def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3)
+(AHead (ASort (S n1) n0) a2) H2) in (eq_ind A (ASort (S n1) n0) (\lambda (a:
+A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g a
+a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2
+(\lambda (a: A).((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g (ASort (S
+n1) n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4)
+(ASort (S n1) n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e:
+A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False
+| (AHead _ _) \Rightarrow True])) I (ASort (S n1) n0) H7) in (False_ind ((leq
+g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2
+H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) H4)) H3 H0 H1)))]) in (H1
+(refl_equal A (AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort (S n1)
+n0)))))]) H)))))) (\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g
+(AHead a a2) a) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_:
+((\forall (a2: A).((leq g (AHead a0 a2) a0) \to (\forall (P:
+Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2)
+(AHead a a0))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (a1:
+A).(\lambda (a3: A).(\lambda (_: (leq ? a1 a3)).((eq A a1 (AHead (AHead a a0)
+a2)) \to ((eq A a3 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2)
+\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0)
+a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind
+A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead
+(AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to
+((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4
+H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead
+a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a
+a0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead
+a1 a4) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e:
+A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a _) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in
+(eq_ind A (AHead a a0) (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3
+a5) (AHead a a0)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda
+(H8: (eq A a4 a2)).(eq_ind A a2 (\lambda (a2: A).((eq A (AHead a3 a5) (AHead
+a a0)) \to ((leq g (AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda
+(H9: (eq A (AHead a3 a5) (AHead a a0))).(let H10 \def (f_equal A A (\lambda
+(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 |
+(AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a a0) H9) in ((let H11 \def
+(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort
+_ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a a0)
+H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 a0) \to ((leq g (AHead a a0)
+a6) \to ((leq g a2 a5) \to P)))) (\lambda (H12: (eq A a5 a0)).(eq_ind A a0
+(\lambda (a6: A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda
+(H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 a0)).(H a0 H13 P))) a5
+(sym_eq A a5 a0 H12))) a3 (sym_eq A a3 a H11))) H10))) a4 (sym_eq A a4 a2
+H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2
+(refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a a0)))))))))))
+a1)).
+
+theorem leq_ahead_asucc_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2)
+(asucc g a1)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2:
+A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead
+(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).((match n return (\lambda
+(n1: nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) with [O
+\Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O (next g
+n0)))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0:
+A).(\lambda (_: (leq ? a a0)).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0
+(ASort O (next g n0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0)
+\Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort O n0)
+a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H3 \def
+(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
+(AHead (ASort O n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort O (next
+g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to
+P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2:
+(eq A (AHead a1 a3) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (AHead a0
+a4) (ASort O (next g n0)))).((let H4 \def (f_equal A A (\lambda (e: A).(match
+e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a)
+\Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in ((let H5 \def
+(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort
+_ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead
+(ASort O n0) a2) H2) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a3 a2)
+\to ((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g a a0) \to ((leq g
+a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a:
+A).((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g (ASort O n0) a0)
+\to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort O (next
+g n0)))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort O (next g n0)) H7) in (False_ind ((leq g (ASort
+O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1
+(sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A
+(AHead (ASort O n0) a2)) (refl_equal A (ASort O (next g n0)))))) | (S n1)
+\Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1
+n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda
+(_: (leq ? a a0)).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort
+n1 n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda
+(H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) a2))).(\lambda (H2: (eq A
+(ASort h2 n2) (ASort n1 n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda
+(e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow
+True | (AHead _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H1) in
+(False_ind ((eq A (ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1
+n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0
+a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort (S n1)
+n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort n1 n0))).((let H4 \def
+(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort
+_ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead
+(ASort (S n1) n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e:
+A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 |
+(AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2)
+in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A
+(AHead a0 a4) (ASort n1 n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P)))))
+(\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4)
+(ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a0) \to ((leq g a a4) \to P))))
+(\lambda (H7: (eq A (AHead a0 a4) (ASort n1 n0))).(let H8 \def (eq_ind A
+(AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: A).Prop) with
+[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1
+n0) H7) in (False_ind ((leq g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to
+P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5)))
+H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2))
+(refl_equal A (ASort n1 n0)))))]) H)))))) (\lambda (a: A).(\lambda (_:
+((\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P:
+Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead
+a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda
+(H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P:
+Prop).(let H2 \def (match H1 return (\lambda (a1: A).(\lambda (a3:
+A).(\lambda (_: (leq ? a1 a3)).((eq A a1 (AHead (AHead a a0) a2)) \to ((eq A
+a3 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2)
+\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0)
+a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5
+\def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_:
+A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2)
+(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort
+h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow
+(\lambda (H4: (eq A (AHead a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq
+A (AHead a3 a5) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda
+(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 |
+(AHead _ a) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in
+((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A)
+with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a4)
+(AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda (a6: A).((eq A
+a4 a2) \to ((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g a6 a3)
+\to ((leq g a4 a5) \to P))))) (\lambda (H8: (eq A a4 a2)).(eq_ind A a2
+(\lambda (a2: A).((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g
+(AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda (H9: (eq A (AHead a3
+a5) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A (\lambda (e:
+A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 |
+(AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H9) in
+((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead
+a3 a5) (AHead a (asucc g a0)) H9) in (eq_ind A a (\lambda (a6: A).((eq A a5
+(asucc g a0)) \to ((leq g (AHead a a0) a6) \to ((leq g a2 a5) \to P))))
+(\lambda (H12: (eq A a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a6:
+A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda (H13: (leq g
+(AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g a0))).(leq_ahead_false g a
+a0 H13 P))) a5 (sym_eq A a5 (asucc g a0) H12))) a3 (sym_eq A a3 a H11)))
+H10))) a4 (sym_eq A a4 a2 H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2
+H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a
+(asucc g a0)))))))))))) a1)).
+
+theorem leq_asucc_false:
+ \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0)
+a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).((match n return
+(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g
+n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) with [O
+\Rightarrow (\lambda (H0: (leq g (ASort O (next g n0)) (ASort O n0))).(let H1
+\def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a
+a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort O n0)) \to P)))))
+with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1
+n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O
+n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1]))
+(ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat
+(\lambda (e: A).(match e return (\lambda (_: A).nat) with [(ASort n _)
+\Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g
+n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq
+A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g
+(ASort h2 n2) k)) \to P)))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind
+nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n2) (ASort O n0)) \to ((eq
+A (aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to P))) (\lambda (H6:
+(eq A (ASort h2 n2) (ASort O n0))).(let H7 \def (f_equal A nat (\lambda (e:
+A).(match e return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n |
+(AHead _ _) \Rightarrow n2])) (ASort h2 n2) (ASort O n0) H6) in ((let H8 \def
+(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2)
+(ASort O n0) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n2 n0) \to ((eq
+A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n2) k)) \to P)))
+(\lambda (H9: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus
+g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to P)) (\lambda (H10:
+(eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n0) k))).(let H
+\def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a
+(aplus g (ASort O n0) k))) H10 (aplus g (ASort O n0) (S k))
+(aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k) k (ASort O
+n0) H) in (le_Sx_x k (eq_ind_r nat k (\lambda (n: nat).(le n k)) (le_n k) (S
+k) H_y) P)))) n2 (sym_eq nat n2 n0 H9))) h2 (sym_eq nat h2 O H8))) H7))) n1
+(sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) |
+(leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3)
+(ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O
+n0))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead
+a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H4)) H3
+H0 H1)))]) in (H1 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O
+n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1)
+n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).(\lambda
+(_: (leq ? a a0)).((eq A a (ASort n1 n0)) \to ((eq A a0 (ASort (S n1) n0))
+\to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A
+(ASort h1 n1) (ASort n1 n0))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1)
+n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1]))
+(ASort h1 n1) (ASort n1 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e:
+A).(match e return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n |
+(AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0) H1) in (eq_ind nat
+n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n2) (ASort (S n1)
+n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to P))))
+(\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort
+h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort
+h2 n2) k)) \to P))) (\lambda (H6: (eq A (ASort h2 n2) (ASort (S n1)
+n0))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda
+(_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n2]))
+(ASort h2 n2) (ASort (S n1) n0) H6) in ((let H8 \def (f_equal A nat (\lambda
+(e: A).(match e return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n |
+(AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort (S n1) n0) H6) in (eq_ind
+nat (S n1) (\lambda (n: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort n1
+n0) k) (aplus g (ASort n n2) k)) \to P))) (\lambda (H9: (eq nat n2
+n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus g (ASort n1 n0) k) (aplus
+g (ASort (S n1) n) k)) \to P)) (\lambda (H10: (eq A (aplus g (ASort n1 n0) k)
+(aplus g (ASort (S n1) n0) k))).(let H \def (eq_ind_r A (aplus g (ASort n1
+n0) k) (\lambda (a: A).(eq A a (aplus g (ASort (S n1) n0) k))) H10 (aplus g
+(ASort (S n1) n0) (S k)) (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def
+(aplus_inj g (S k) k (ASort (S n1) n0) H) in (le_Sx_x k (eq_ind_r nat k
+(\lambda (n: nat).(le n k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0
+H9))) h2 (sym_eq nat h2 (S n1) H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1
+(sym_eq nat h1 n1 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1)
+\Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H3:
+(eq A (AHead a2 a4) (ASort (S n1) n0))).((let H4 \def (eq_ind A (AHead a1 a3)
+(\lambda (e: A).(match e return (\lambda (_: A).Prop) with [(ASort _ _)
+\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H2) in
+(False_ind ((eq A (AHead a2 a4) (ASort (S n1) n0)) \to ((leq g a1 a2) \to
+((leq g a3 a4) \to P))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (ASort n1
+n0)) (refl_equal A (ASort (S n1) n0)))))]) H))))) (\lambda (a0: A).(\lambda
+(_: (((leq g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1:
+A).(\lambda (H0: (((leq g (asucc g a1) a1) \to (\forall (P:
+Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead a0
+a1))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (a:
+A).(\lambda (a2: A).(\lambda (_: (leq ? a a2)).((eq A a (AHead a0 (asucc g
+a1))) \to ((eq A a2 (AHead a0 a1)) \to P))))) with [(leq_sort h1 h2 n1 n2 k
+H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a0 (asucc g
+a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0 a1))).((let H5 \def
+(eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I
+(AHead a0 (asucc g a1)) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a0 a1))
+\to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5))
+H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A
+(AHead a1 a3) (AHead a0 (asucc g a1)))).(\lambda (H5: (eq A (AHead a2 a4)
+(AHead a0 a1))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow
+a])) (AHead a1 a3) (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A
+(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a0 (asucc
+g a1)) H4) in (eq_ind A a0 (\lambda (a: A).((eq A a3 (asucc g a1)) \to ((eq A
+(AHead a2 a4) (AHead a0 a1)) \to ((leq g a a2) \to ((leq g a3 a4) \to P)))))
+(\lambda (H8: (eq A a3 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a:
+A).((eq A (AHead a2 a4) (AHead a0 a1)) \to ((leq g a0 a2) \to ((leq g a a4)
+\to P)))) (\lambda (H9: (eq A (AHead a2 a4) (AHead a0 a1))).(let H10 \def
+(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort
+_ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a2 a4) (AHead a0
+a1) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead a _) \Rightarrow
+a])) (AHead a2 a4) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda (a: A).((eq A
+a4 a1) \to ((leq g a0 a) \to ((leq g (asucc g a1) a4) \to P)))) (\lambda
+(H12: (eq A a4 a1)).(eq_ind A a1 (\lambda (a: A).((leq g a0 a0) \to ((leq g
+(asucc g a1) a) \to P))) (\lambda (_: (leq g a0 a0)).(\lambda (H14: (leq g
+(asucc g a1) a1)).(H0 H14 P))) a4 (sym_eq A a4 a1 H12))) a2 (sym_eq A a2 a0
+H11))) H10))) a3 (sym_eq A a3 (asucc g a1) H8))) a1 (sym_eq A a1 a0 H7)))
+H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0 (asucc g a1))) (refl_equal
+A (AHead a0 a1)))))))))) a)).
+
+definition lweight:
+ A \to nat
+\def
+ let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow
+O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in
+lweight.
+
+definition llt:
+ A \to (A \to Prop)
+\def
+ \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
+
+theorem lweight_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
+(lweight a1) (lweight a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
+nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g
+(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
+a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
+(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
+a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
+(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
+H)))).
+
+theorem llt_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(a3: A).((llt a1 a3) \to (llt a2 a3))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
+\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
+(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
+
+theorem llt_trans:
+ \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
+a3) \to (llt a1 a3)))))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
+a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
+(lweight a1) (lweight a2) (lweight a3) H H0))))).
+
+theorem llt_head_sx:
+ \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a1)) (S (plus (lweight
+a1) (lweight a2))) (le_n_S (S (lweight a1)) (S (plus (lweight a1) (lweight
+a2))) (le_n_S (lweight a1) (plus (lweight a1) (lweight a2)) (le_plus_l
+(lweight a1) (lweight a2)))))).
+
+theorem llt_head_dx:
+ \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a2)) (S (plus (lweight
+a1) (lweight a2))) (le_n_S (S (lweight a2)) (S (plus (lweight a1) (lweight
+a2))) (le_n_S (lweight a2) (plus (lweight a1) (lweight a2)) (le_plus_r
+(lweight a1) (lweight a2)))))).
+
+theorem llt_wf__q_ind:
+ \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P: ((A \to
+Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P
+a))))) P n))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
+n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
+a)))))).
+
+theorem llt_wf_ind:
+ \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
+a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
+\def
+ let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
+A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
+Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
+(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
+(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
+A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
+\to (Q (\lambda (a: A).(P a)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
+(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n: nat).(\forall
+(m: nat).((lt m n) \to (\forall (a: A).((eq nat (lweight a) m) \to (P a))))))
+H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt (lweight a1)
+(lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
+a1))))))))))))) a)))).
-axiom csubst0_gen_sort: \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 i v (CSort n) x) \to (\forall (P: Prop).P))))) .
+inductive aprem: nat \to (A \to (A \to Prop)) \def
+| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
+| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2
+a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
+
+theorem aprem_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
+(i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g
+b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall
+(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda
+(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g
+(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2:
+A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H2 \def (match H1 return
+(\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem n a
+a0)).((eq nat n i) \to ((eq A a (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1)
+b1)))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H1: (eq nat O
+i)).(\lambda (H2: (eq A (AHead a0 a3) (ASort h2 n2))).(\lambda (H3: (eq A a0
+b2)).(eq_ind nat O (\lambda (n: nat).((eq A (AHead a0 a3) (ASort h2 n2)) \to
+((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem n (ASort h1 n1) b1)))))) (\lambda (H4: (eq A (AHead a0 a3) (ASort
+h2 n2))).(let H5 \def (eq_ind A (AHead a0 a3) (\lambda (e: A).(match e return
+(\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort h2 n2) H4) in (False_ind ((eq A a0 b2) \to (ex2
+A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (ASort h1 n1)
+b1)))) H5))) i H1 H2 H3)))) | (aprem_succ a0 a i0 H1 a3) \Rightarrow (\lambda
+(H2: (eq nat (S i0) i)).(\lambda (H3: (eq A (AHead a3 a0) (ASort h2
+n2))).(\lambda (H4: (eq A a b2)).(eq_ind nat (S i0) (\lambda (n: nat).((eq A
+(AHead a3 a0) (ASort h2 n2)) \to ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1)
+b1))))))) (\lambda (H5: (eq A (AHead a3 a0) (ASort h2 n2))).(let H6 \def
+(eq_ind A (AHead a3 a0) (\lambda (e: A).(match e return (\lambda (_: A).Prop)
+with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
+(ASort h2 n2) H5) in (False_ind ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (ASort h1 n1)
+b1))))) H6))) i H2 H3 H4 H1))))]) in (H2 (refl_equal nat i) (refl_equal A
+(ASort h2 n2)) (refl_equal A b2)))))))))))) (\lambda (a0: A).(\lambda (a3:
+A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall (i: nat).(\forall (b2:
+A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq
+g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall (b2: A).((aprem i a5 b2)
+\to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a4
+b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H4: (aprem i (AHead
+a3 a5) b2)).((match i return (\lambda (n: nat).((aprem n (AHead a3 a5) b2)
+\to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (AHead
+a0 a4) b1))))) with [O \Rightarrow (\lambda (H5: (aprem O (AHead a3 a5)
+b2)).(let H6 \def (match H5 return (\lambda (n: nat).(\lambda (a: A).(\lambda
+(a1: A).(\lambda (_: (aprem n a a1)).((eq nat n O) \to ((eq A a (AHead a3
+a5)) \to ((eq A a1 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem O (AHead a0 a4) b1)))))))))) with [(aprem_zero a6 a7)
+\Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H5: (eq A (AHead a6 a7)
+(AHead a3 a5))).(\lambda (H6: (eq A a6 b2)).((let H7 \def (f_equal A A
+(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H5)
+in ((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a6 | (AHead a _) \Rightarrow a])) (AHead
+a6 a7) (AHead a3 a5) H5) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a5) \to
+((eq A a b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem O (AHead a0 a4) b1)))))) (\lambda (H9: (eq A a7 a5)).(eq_ind A a5
+(\lambda (_: A).((eq A a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2))
+(\lambda (b1: A).(aprem O (AHead a0 a4) b1))))) (\lambda (H10: (eq A a3
+b2)).(eq_ind A b2 (\lambda (_: A).(ex2 A (\lambda (b1: A).(leq g b1 b2))
+(\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (eq_ind A a3 (\lambda (a:
+A).(ex2 A (\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0
+a4) b1)))) (ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1:
+A).(aprem O (AHead a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H10) a3 (sym_eq A
+a3 b2 H10))) a7 (sym_eq A a7 a5 H9))) a6 (sym_eq A a6 a3 H8))) H7)) H6)))) |
+(aprem_succ a6 a i H4 a7) \Rightarrow (\lambda (H5: (eq nat (S i)
+O)).(\lambda (H6: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H7: (eq A a
+b2)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H5) in (False_ind ((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2)
+\to ((aprem i a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1:
+A).(aprem O (AHead a0 a4) b1)))))) H8)) H6 H7 H4))))]) in (H6 (refl_equal nat
+O) (refl_equal A (AHead a3 a5)) (refl_equal A b2)))) | (S n) \Rightarrow
+(\lambda (H5: (aprem (S n) (AHead a3 a5) b2)).(let H6 \def (match H5 return
+(\lambda (n0: nat).(\lambda (a: A).(\lambda (a1: A).(\lambda (_: (aprem n0 a
+a1)).((eq nat n0 (S n)) \to ((eq A a (AHead a3 a5)) \to ((eq A a1 b2) \to
+(ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead
+a0 a4) b1)))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (H4: (eq
+nat O (S n))).(\lambda (H5: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H6:
+(eq A a6 b2)).((let H7 \def (eq_ind nat O (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S n) H4) in (False_ind ((eq A (AHead a6 a7) (AHead a3 a5)) \to ((eq A a6
+b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n)
+(AHead a0 a4) b1))))) H7)) H5 H6)))) | (aprem_succ a6 a i H4 a7) \Rightarrow
+(\lambda (H5: (eq nat (S i) (S n))).(\lambda (H6: (eq A (AHead a7 a6) (AHead
+a3 a5))).(\lambda (H7: (eq A a b2)).((let H8 \def (f_equal nat nat (\lambda
+(e: nat).(match e return (\lambda (_: nat).nat) with [O \Rightarrow i | (S n)
+\Rightarrow n])) (S i) (S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A
+(AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem n0 a6 a) \to (ex2 A
+(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4)
+b1))))))) (\lambda (H9: (eq A (AHead a7 a6) (AHead a3 a5))).(let H10 \def
+(f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort
+_ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a7 a6) (AHead a3
+a5) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) \Rightarrow
+a])) (AHead a7 a6) (AHead a3 a5) H9) in (eq_ind A a3 (\lambda (_: A).((eq A
+a6 a5) \to ((eq A a b2) \to ((aprem n a6 a) \to (ex2 A (\lambda (b1: A).(leq
+g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))) (\lambda
+(H12: (eq A a6 a5)).(eq_ind A a5 (\lambda (a1: A).((eq A a b2) \to ((aprem n
+a1 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S
+n) (AHead a0 a4) b1)))))) (\lambda (H13: (eq A a b2)).(eq_ind A b2 (\lambda
+(a1: A).((aprem n a5 a1) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda
+(b1: A).(aprem (S n) (AHead a0 a4) b1))))) (\lambda (H14: (aprem n a5
+b2)).(let H_x \def (H3 n b2 H14) in (let H3 \def H_x in (ex2_ind A (\lambda
+(b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n a4 b1)) (ex2 A (\lambda (b1:
+A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))) (\lambda
+(x: A).(\lambda (H15: (leq g x b2)).(\lambda (H16: (aprem n a4 x)).(ex_intro2
+A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4)
+b1)) x H15 (aprem_succ a4 x n H16 a0))))) H3)))) a (sym_eq A a b2 H13))) a6
+(sym_eq A a6 a5 H12))) a7 (sym_eq A a7 a3 H11))) H10))) i (sym_eq nat i n
+H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) (refl_equal A (AHead a3
+a5)) (refl_equal A b2))))]) H4)))))))))))) a1 a2 H)))).
+
+theorem aprem_asucc:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i
+a1 a2) \to (aprem i (asucc g a1) a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda
+(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda
+(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3:
+A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda
+(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0)
+a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2
+H))))).
+
+definition gz:
+ G
+\def
+ mk_G S lt_n_Sn.
-axiom csubst0_gen_head: \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall (v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))))) .
+inductive leqz: A \to (A \to Prop) \def
+| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
+(n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort
+h2 n2))))))
+| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3:
+A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
+
+theorem aplus_gz_le:
+ \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A
+(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n))))))
+\def
+ \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0:
+nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h)
+n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y
+\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n)
+(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0:
+nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A
+(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h:
+nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A
+(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O
+\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n:
+nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n))
+k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O
+(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0
+n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n)))
+(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0:
+nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S
+(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0))
+(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz
+(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n:
+nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz
+(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S
+k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda
+(H0: (le (S n) (S k0))).(ex2_ind nat (\lambda (n1: nat).(eq nat (S k0) (S
+n1))) (\lambda (n1: nat).(le n n1)) (eq A (asucc gz (aplus gz (ASort (S n)
+n0) k0)) (ASort O (plus (minus k0 n) n0))) (\lambda (x: nat).(\lambda (H1:
+(eq nat (S k0) (S x))).(\lambda (H2: (le n x)).(let H3 \def (f_equal nat nat
+(\lambda (e: nat).(match e return (\lambda (_: nat).nat) with [O \Rightarrow
+k0 | (S n) \Rightarrow n])) (S k0) (S x) H1) in (let H4 \def (eq_ind_r nat x
+(\lambda (n0: nat).(le n n0)) H2 k0 H3) in (eq_ind A (aplus gz (ASort n n0)
+k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) k0)) a))
+(eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: A).(eq A a
+(aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) k0)) (asucc
+gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S n) n0)))
+(ASort O (plus (minus k0 n) n0)) (IH n n0 H4))))))) (le_gen_S n (S k0)
+H0)))))) h)))) k).
+
+theorem aplus_gz_ge:
+ \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A
+(aplus gz (ASort h n) k) (ASort (minus h k) n)))))
+\def
+ \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h:
+nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0)
+n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0:
+nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O)
+(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0
+h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda
+(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz
+(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le
+(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0:
+nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n))
+(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0
+x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda
+(_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x)
+H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n))
+H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) n0)
+\to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0))
+n))))).(\lambda (H0: (le (S k0) (S n0))).(ex2_ind nat (\lambda (n1: nat).(eq
+nat (S n0) (S n1))) (\lambda (n1: nat).(le k0 n1)) (eq A (asucc gz (aplus gz
+(ASort (S n0) n) k0)) (ASort (minus n0 k0) n)) (\lambda (x: nat).(\lambda
+(H1: (eq nat (S n0) (S x))).(\lambda (H2: (le k0 x)).(let H3 \def (f_equal
+nat nat (\lambda (e: nat).(match e return (\lambda (_: nat).nat) with [O
+\Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x) H1) in (let H4 \def
+(eq_ind_r nat x (\lambda (n: nat).(le k0 n)) H2 n0 H3) in (eq_ind A (aplus gz
+(ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n0) n)
+k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) k0) (\lambda (a:
+A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus gz (ASort n0 n)
+k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc gz k0 (ASort (S
+n0) n))) (ASort (minus n0 k0) n) (IH n0 H4))))))) (le_gen_S k0 (S n0) H0)))))
+h)))) k)).
+
+theorem next_plus_gz:
+ \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n)))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat
+(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0:
+nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat
+S (next_plus gz n n0) (plus n0 n) H))) h)).
+
+theorem leqz_leq:
+ \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz
+(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda
+(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k
+h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2
+(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def
+(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort
+h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1
+(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k)
+(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2)
+(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def
+(f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat)
+on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O |
+(S k) \Rightarrow (match m with [O \Rightarrow (S k) | (S l) \Rightarrow
+(minus k l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2
+k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e return
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
+\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in
+(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n:
+nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n:
+nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal
+nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1))
+(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2
+k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a
+(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1
+(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort
+h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus
+(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat
+(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2)
+n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind
+A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee return (\lambda
+(_: A).Prop) with [(ASort n _) \Rightarrow (match n return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _)
+\Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in (False_ind
+(leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1
+k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k
+h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A
+a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1))
+(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2)
+k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort
+(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in
+(let H5 \def (sym_equal A (ASort O (plus (minus k h1) n1)) (ASort (minus h2
+k) n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A
+(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k)))
+(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k)))
+n2) (\lambda (ee: A).(match ee return (\lambda (_: A).Prop) with [(ASort n _)
+\Rightarrow (match n return (\lambda (_: nat).Prop) with [O \Rightarrow False
+| (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow False])) I (ASort O
+(plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 n1) (ASort h2 n2))
+H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A (aplus gz (ASort h1
+n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus
+(minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus
+gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1))
+a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5
+\def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: A).nat) with
+[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec plus (n: nat)
+on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m |
+(S p) \Rightarrow (S (plus p m))])) in plus) (minus k h1) n1)])) (ASort O
+(plus (minus k h1) n1)) (ASort O (plus (minus k h2) n2)) H4) in (let H_y \def
+(plus_plus k h1 h2 n1 n2 H1 H2 H5) in (leqz_sort h1 h2 n1 n2
+H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leq gz a0
+a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda
+(_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 a5)).(leqz_head a0 a3 H1 a4 a5
+H3))))))))) a1 a2 H))).
+
+theorem leq_leqz:
+ \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2)))
+\def
+ \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind
+(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus
+h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A
+(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1)))
+(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A
+(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2)))
+(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus
+(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus
+h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2))
+(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n:
+nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus
+h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq
+A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz
+n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1
+h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n:
+nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus
+h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal
+nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2
+n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2)
+(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2)
+(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2)
+(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus
+(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2))
+(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1
+h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0:
+A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0
+a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda
+(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))).
-axiom csubst0_drop_gt: \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (drop n O c2 e))))))))) .
+inductive arity (g:G): C \to (T \to (A \to Prop)) \def
+| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O
+n)))
+| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a)
+\to (arity g c (TLRef i) a)))))))
+| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u
+(asucc g a)) \to (arity g c (TLRef i) a)))))))
+| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c:
+C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t:
+T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c
+(THead (Bind b) u t) a2)))))))))
+| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
+(asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind
+Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
+| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u
+a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to
+(arity g c (THead (Flat Appl) u t) a2)))))))
+| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u
+(asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat
+Cast) u t) a))))))
+| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t
+a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
+
+theorem arity_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c
+(TSort n) a) \to (leq g a (ASort O n))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda
+(H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g
+c t a)) (leq g a (ASort O n)) (\lambda (y: T).(\lambda (H0: (arity g c y
+a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0: A).((eq T t
+(TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_: C).(\lambda (n0:
+nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def (f_equal T nat
+(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort n)
+\Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0]))
+(TSort n0) (TSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort
+O n1) (ASort O n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u
+a0)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda
+(H4: (eq T (TLRef i) (TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda
+(ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5))))))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl
+i c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u
+(asucc g a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0)
+(ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort n))).(let H5 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n))
+H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda
+(_: (((eq T u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t
+a2)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda
+(H6: (eq T (THead (Bind b) u t) (TSort n))).(let H7 \def (eq_ind T (THead
+(Bind b) u t) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H6) in (False_ind (leq g a2 (ASort O n))
+H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u (TSort n)) \to (leq g
+(asucc g a1) (ASort O n))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TSort n))
+\to (leq g a2 (ASort O n))))).(\lambda (H5: (eq T (THead (Bind Abst) u t)
+(TSort n))).(let H6 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H5) in (False_ind (leq g (AHead a1 a2) (ASort O n)) H6)))))))))))) (\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: (((eq T u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda
+(t: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda
+(_: (((eq T t (TSort n)) \to (leq g (AHead a1 a2) (ASort O n))))).(\lambda
+(H5: (eq T (THead (Flat Appl) u t) (TSort n))).(let H6 \def (eq_ind T (THead
+(Flat Appl) u t) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H5) in (False_ind (leq g a2 (ASort O n))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g
+(asucc g a0) (ASort O n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t
+a0)).(\lambda (_: (((eq T t (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda
+(H5: (eq T (THead (Flat Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead
+(Flat Cast) u t) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H5) in (False_ind (leq g a0 (ASort O n))
+H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
+(arity g c0 t a1)).(\lambda (H2: (((eq T t (TSort n)) \to (leq g a1 (ASort O
+n))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t
+(TSort n))).(let H5 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H4) in
+(let H6 \def (eq_ind T t (\lambda (t: T).((eq T t (TSort n)) \to (leq g a1
+(ASort O n)))) H2 (TSort n) H5) in (let H7 \def (eq_ind T t (\lambda (t:
+T).(arity g c0 t a1)) H1 (TSort n) H5) in (leq_trans g a2 a1 (leq_sym g a1 a2
+H3) (ASort O n) (H6 (refl_equal T (TSort n))))))))))))))) c y a H0))) H))))).
+
+theorem arity_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c
+(TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a))))
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda
+(H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g
+c t a)) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))) (\lambda (y:
+T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t:
+T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a0)))))))))) (\lambda (c0: C).(\lambda (n:
+nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def (eq_ind T (TSort
+n) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1:
+(getl i0 c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g
+d u a0)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0:
+C).(\lambda (u: T).(getl i d (CHead d0 (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda
+(u: T).(getl i d (CHead d0 (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef
+i))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_:
+T).nat) with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _
+_) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0
+(\lambda (n: nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 i H5) in (or_introl
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr)
+u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))))
+(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
+Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))) d u H6
+H2))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i0: nat).(\lambda (H1: (getl i0 c0 (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (H2: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TLRef
+i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u: T).(getl i d (CHead d0
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g
+a0))))) (ex2_2 C T (\lambda (d0: C).(\lambda (u: T).(getl i d (CHead d0 (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (asucc g
+a0)))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal
+T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
+(TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n:
+nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in (or_intror (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T
+(\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))) d u H6
+H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead
+c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
+(CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda
+(u: T).(arity g d u (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind b)
+u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i)
+H6) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))
+H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u (TLRef i)) \to (or
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1))))) (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (asucc g
+a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
+(Bind Abst) u) t a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d
+(Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2
+C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead
+d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g
+a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) u t) (TLRef i))).(let H6
+\def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
+(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
+d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1
+a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead
+a1 a2))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda
+(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (t: T).(\lambda
+(a2: A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t
+(TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(AHead a1 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g (AHead a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t)
+(TLRef i))).(let H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i)
+H5) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0
+(CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0
+a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
+Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a2))))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c0 u (asucc g a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2
+C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a0))))) (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g (asucc g
+a0)))))))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_:
+(((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl
+i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a0)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g
+a0))))))))).(\lambda (H5: (eq T (THead (Flat Cast) u t) (TLRef i))).(let H6
+\def (eq_ind T (THead (Flat Cast) u t) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
+(False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
+d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0))))
+(ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst)
+u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))
+H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
+(arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2:
+A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5
+\def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind
+T t (\lambda (t: T).((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind
+T t (\lambda (t: T).(arity g c0 t a1)) H1 (TLRef i) H5) in (let H8 \def (H6
+(refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
+Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2))))))
+(\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
+(Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
+u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda
+(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11:
+(arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))
+x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C
+T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
+(\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11:
+(arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
+(CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
+(asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2)
+(asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
+
+theorem arity_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c:
+C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind
+b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
+A).(arity g (CHead c (Bind b) u) t a2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda
+(c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity
+g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
+T).(arity g c t0 a2)) (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
+A).(arity g (CHead c (Bind b) u) t a2))) (\lambda (y: T).(\lambda (H1: (arity
+g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
+A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a))))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n) (THead (Bind b) u
+t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t)
+H2) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0
+a)).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g d u a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t
+a)))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
+u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_:
+(((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
+a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g
+a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Bind b) u t) H5) in (False_ind (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a))) H6))))))))))) (\lambda (b0: B).(\lambda (H2: (not (eq B b0
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3:
+(arity g c0 u0 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
+b) u) t a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g
+(CHead c0 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u
+t)) \to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u0) u a1))
+(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t
+a0)))))).(\lambda (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u
+t))).(let H8 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_:
+T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow
+b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead (Bind b) u t)
+H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t _) \Rightarrow t])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in
+((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _
+t) \Rightarrow t])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in
+(\lambda (H11: (eq T u0 u)).(\lambda (H12: (eq B b0 b)).(let H13 \def (eq_ind
+T t0 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g (CHead c0 (Bind b0) u0) u a1)) (\lambda (_: A).(arity g
+(CHead (CHead c0 (Bind b0) u0) (Bind b) u) t a0))))) H6 t H10) in (let H14
+\def (eq_ind T t0 (\lambda (t: T).(arity g (CHead c0 (Bind b0) u0) t a0)) H5
+t H10) in (let H15 \def (eq_ind T u0 (\lambda (t0: T).((eq T t (THead (Bind
+b) u t)) \to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) t0) u a1))
+(\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) t0) (Bind b) u) t a0)))))
+H13 u H11) in (let H16 \def (eq_ind T u0 (\lambda (t0: T).(arity g (CHead c0
+(Bind b0) t0) t a0)) H14 u H11) in (let H17 \def (eq_ind T u0 (\lambda (t0:
+T).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1))))) H4 u H11) in
+(let H18 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t a1)) H3 u H11) in
+(let H19 \def (eq_ind B b0 (\lambda (b0: B).((eq T t (THead (Bind b) u t))
+\to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u) u a1)) (\lambda
+(_: A).(arity g (CHead (CHead c0 (Bind b0) u) (Bind b) u) t a0))))) H15 b
+H12) in (let H20 \def (eq_ind B b0 (\lambda (b: B).(arity g (CHead c0 (Bind
+b) u) t a0)) H16 b H12) in (let H21 \def (eq_ind B b0 (\lambda (b: B).(not
+(eq B b Abst))) H2 b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20)))))))))))))
+H9)) H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
+(_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0:
+T).(\lambda (a0: A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0
+a0)).(\lambda (H5: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g (CHead c0 (Bind Abst) u0) u a1)) (\lambda (_: A).(arity g (CHead
+(CHead c0 (Bind Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead
+(Bind Abst) u0 t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda
+(e: T).(match e return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
+(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H8 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal
+T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0
+u)).(\lambda (H11: (eq B Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t0:
+T).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g
+(CHead c0 (Bind Abst) u0) u a1)) (\lambda (_: A).(arity g (CHead (CHead c0
+(Bind Abst) u0) (Bind b) u) t a0))))) H5 t H9) in (let H13 \def (eq_ind T t0
+(\lambda (t: T).(arity g (CHead c0 (Bind Abst) u0) t a0)) H4 t H9) in (let
+H14 \def (eq_ind T u0 (\lambda (t0: T).((eq T t (THead (Bind b) u t)) \to
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) t0) u a1)) (\lambda
+(_: A).(arity g (CHead (CHead c0 (Bind Abst) t0) (Bind b) u) t a0))))) H12 u
+H10) in (let H15 \def (eq_ind T u0 (\lambda (t0: T).(arity g (CHead c0 (Bind
+Abst) t0) t a0)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t0:
+T).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u
+H10) in (let H17 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t (asucc g
+a1))) H2 u H10) in (let H18 \def (eq_ind_r B b (\lambda (b: B).((eq T t
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind
+Abst) u) u a1)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u)
+(Bind b) u) t a0))))) H14 Abst H11) in (let H19 \def (eq_ind_r B b (\lambda
+(b: B).((eq T u (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0
+u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1)))))) H16
+Abst H11) in (let H20 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst)))
+H Abst H11) in (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: A).(arity
+g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (AHead a1
+a0))))) (let H21 \def (match (H20 (refl_equal B Abst)) return (\lambda (_:
+False).(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g
+(CHead c0 (Bind Abst) u) t (AHead a1 a0))))) with []) in H21) b
+H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda (u0:
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
+(_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: T).(\lambda
+(a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: (((eq T t0
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
+(_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 a0))))))).(\lambda (H6: (eq
+T (THead (Flat Appl) u0 t0) (THead (Bind b) u t))).(let H7 \def (eq_ind T
+(THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t)
+H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t a0))) H7)))))))))))) (\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: (arity g c0 u0 (asucc g
+a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t (asucc
+g a))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (_:
+(((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u
+a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a)))))).(\lambda (H6:
+(eq T (THead (Flat Cast) u0 t0) (THead (Bind b) u t))).(let H7 \def (eq_ind T
+(THead (Flat Cast) u0 t0) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t)
+H6) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t a))) H7))))))))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (H2: (arity g c0 t0
+a1)).(\lambda (H3: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
+a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1 a0)).(\lambda (H5: (eq T t0
+(THead (Bind b) u t))).(let H6 \def (f_equal T T (\lambda (e: T).e) t0 (THead
+(Bind b) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t0: T).((eq T t0
+(THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda
+(_: A).(arity g (CHead c0 (Bind b) u) t a1))))) H3 (THead (Bind b) u t) H6)
+in (let H8 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t a1)) H2 (THead
+(Bind b) u t) H6) in (let H9 \def (H7 (refl_equal T (THead (Bind b) u t))) in
+(ex2_ind A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t a1)) (ex2 A (\lambda (a3: A).(arity g c0 u a3))
+(\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) (\lambda (x:
+A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11: (arity g (CHead c0 (Bind
+b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
+A).(arity g (CHead c0 (Bind b) u) t a0)) x H10 (arity_repl g (CHead c0 (Bind
+b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y a2 H1))) H0)))))))).
+
+theorem arity_gen_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
+A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) u) t a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
+A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead
+(Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c (Bind Abst) u) t a2)))) (\lambda (y: T).(\lambda (H0: (arity g c y
+a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: A).((eq T t0
+(THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq
+A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Bind Abst) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead
+(Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2:
+A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity
+g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0
+(Bind Abst) u) t a2)))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
+u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T
+u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2:
+A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g d u
+(asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead d (Bind
+Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead (Bind Abst) u
+t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u
+t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
+u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_:
+(((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda
+(a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
+(Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
+ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u
+t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
+(AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2:
+(arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
+(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0
+a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a1)))) (\lambda
+(_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u)
+t a2))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u
+t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_:
+T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow b])])) (THead (Bind b) u0 t0) (THead (Bind Abst) u t)
+H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t _) \Rightarrow t])) (THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in
+((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in
+(\lambda (H10: (eq T u0 u)).(\lambda (H11: (eq B b Abst)).(let H12 \def
+(eq_ind T t0 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A
+A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a1)))) (\lambda
+(_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u)
+t a2)))))) H5 t H9) in (let H13 \def (eq_ind T t0 (\lambda (t: T).(arity g
+(CHead c0 (Bind b) u0) t a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda
+(t0: T).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
+A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g (CHead c0 (Bind b) t0) u (asucc g a1)))) (\lambda (_: A).(\lambda
+(a2: A).(arity g (CHead (CHead c0 (Bind b) t0) (Bind Abst) u) t a2)))))) H12
+u H10) in (let H15 \def (eq_ind T u0 (\lambda (t0: T).(arity g (CHead c0
+(Bind b) t0) t a2)) H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t0:
+T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a2:
+A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))))) H3 u H10) in (let H17 \def (eq_ind T u0
+(\lambda (t: T).(arity g c0 t a1)) H2 u H10) in (let H18 \def (eq_ind B b
+(\lambda (b: B).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
+(a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: A).(\lambda
+(_: A).(arity g (CHead c0 (Bind b) u) u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind b) u) (Bind Abst) u) t
+a2)))))) H14 Abst H11) in (let H19 \def (eq_ind B b (\lambda (b: B).(arity g
+(CHead c0 (Bind b) u) t a2)) H15 Abst H11) in (let H20 \def (eq_ind B b
+(\lambda (b: B).(not (eq B b Abst))) H1 Abst H11) in (let H21 \def (match
+(H20 (refl_equal B Abst)) return (\lambda (_: False).(ex3_2 A A (\lambda (a1:
+A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2))))) with []) in H21))))))))))))) H8))
+H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
+(H1: (arity g c0 u0 (asucc g a1))).(\lambda (H2: (((eq T u0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g
+a1) (AHead a2 a3)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
+a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0
+(Bind Abst) u0) t0 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to
+(ex3_2 A A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3))))
+(\lambda (a1: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc
+g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind
+Abst) u0) (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Bind Abst)
+u0 t0) (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef
+_) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) u0 t0)
+(THead (Bind Abst) u t) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) u0 t0)
+(THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def
+(eq_ind T t0 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A
+A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a1))))
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind Abst) u0)
+(Bind Abst) u) t a2)))))) H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t:
+T).(arity g (CHead c0 (Bind Abst) u0) t a2)) H3 t H7) in (let H11 \def
+(eq_ind T u0 (\lambda (t0: T).((eq T t (THead (Bind Abst) u t)) \to (ex3_2 A
+A (\lambda (a1: A).(\lambda (a3: A).(eq A a2 (AHead a1 a3)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t0) u (asucc g a1))))
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead (CHead c0 (Bind Abst) t0)
+(Bind Abst) u) t a2)))))) H9 u H8) in (let H12 \def (eq_ind T u0 (\lambda
+(t0: T).(arity g (CHead c0 (Bind Abst) t0) t a2)) H10 u H8) in (let H13 \def
+(eq_ind T u0 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A
+A (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 a3))))
+(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))))) H2 u H8) in
+(let H14 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t (asucc g a1))) H1 u
+H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2)
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
+a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12))))))))) H6))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
+u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (_:
+(((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda
+(a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2))))))).(\lambda (H5: (eq T (THead (Flat Appl)
+u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T (THead (Flat Appl) u0
+t0) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind
+(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) H6))))))))))))
+(\lambda (c0: C).(\lambda (u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0
+u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2
+A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
+(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda
+(t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind
+Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead
+a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
+(\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
+a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u
+t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
+ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a1:
+A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
+A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
+(CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda (c0: C).(\lambda
+(t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2:
+(((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a2: A).(\lambda
+(a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a1: A).(\lambda (_: A).(arity g
+c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0
+(Bind Abst) u) t a2))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
+a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u t))).(let H5 \def (f_equal T
+T (\lambda (e: T).e) t0 (THead (Bind Abst) u t) H4) in (let H6 \def (eq_ind T
+t0 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
+(\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a1:
+A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
+(a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))))) H2 (THead (Bind Abst) u
+t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t a1)) H1
+(THead (Bind Abst) u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Bind
+Abst) u t))) in (ex3_2_ind A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
+(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
+a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
+a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x0:
+A).(\lambda (x1: A).(\lambda (H9: (eq A a1 (AHead x0 x1))).(\lambda (H10:
+(arity g c0 u (asucc g x0))).(\lambda (H11: (arity g (CHead c0 (Bind Abst) u)
+t x1)).(let H12 \def (eq_ind A a1 (\lambda (a: A).(leq g a a2)) H3 (AHead x0
+x1) H9) in (let H13 \def (eq_ind A a1 (\lambda (a: A).(arity g c0 (THead
+(Bind Abst) u t) a)) H7 (AHead x0 x1) H9) in (let H_x \def (leq_gen_head g x0
+x1 a2 H12) in (let H14 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda
+(a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g x0
+a3))) (\lambda (_: A).(\lambda (a4: A).(leq g x1 a4))) (ex3_2 A A (\lambda
+(a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda
+(_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity
+g (CHead c0 (Bind Abst) u) t a4)))) (\lambda (x2: A).(\lambda (x3:
+A).(\lambda (H15: (eq A a2 (AHead x2 x3))).(\lambda (H16: (leq g x0
+x2)).(\lambda (H17: (leq g x1 x3)).(eq_ind_r A (AHead x2 x3) (\lambda (a0:
+A).(ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a0 (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro
+A A (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead x2 x3) (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))) x2 x3
+(refl_equal A (AHead x2 x3)) (arity_repl g c0 u (asucc g x0) H10 (asucc g x2)
+(asucc_repl g x0 x2 H16)) (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3
+H17)) a2 H15)))))) H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
+
+theorem arity_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2:
+A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity
+g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2:
+A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead
+(Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (ex2 A (\lambda (a1:
+A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))) (\lambda
+(y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
+(t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1
+a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead
+(Flat Appl) u t) H1) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1))
+(\lambda (a1: A).(arity g c0 t (AHead a1 (ASort O n))))) H2))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl
+i c0 (CHead d (Bind Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0
+a)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (AHead a1
+a))))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1
+a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst)
+u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda (_:
+(((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
+a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda (H4:
+(eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) u t) H4) in (False_ind (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a))))
+H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0
+a1)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2
+a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
+(Bind b) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u0) u a1)) (\lambda (a1:
+A).(arity g (CHead c0 (Bind b) u0) t (AHead a1 a0))))))).(\lambda (H6: (eq T
+(THead (Bind b) u0 t0) (THead (Flat Appl) u t))).(let H7 \def (eq_ind T
+(THead (Bind b) u0 t0) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u
+t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
+A).(arity g c0 t (AHead a3 a0)))) H7)))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda
+(_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
+c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 (asucc g
+a1)))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (_: (arity g (CHead c0
+(Bind Abst) u0) t0 a0)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u0) u a1)) (\lambda
+(a1: A).(arity g (CHead c0 (Bind Abst) u0) t (AHead a1 a0))))))).(\lambda
+(H5: (eq T (THead (Bind Abst) u0 t0) (THead (Flat Appl) u t))).(let H6 \def
+(eq_ind T (THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0
+u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 (AHead a1 a0)))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
+(H1: (arity g c0 u0 a1)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u t))
+\to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t
+(AHead a2 a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H3: (arity g
+c0 t0 (AHead a1 a0))).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u t)) \to
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t
+(AHead a2 (AHead a1 a0)))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0)
+(THead (Flat Appl) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) u0 t0)
+(THead (Flat Appl) u t) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) u0 t0)
+(THead (Flat Appl) u t) H5) in (\lambda (H8: (eq T u0 u)).(let H9 \def
+(eq_ind T t0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2
+(AHead a1 a0))))))) H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t:
+T).(arity g c0 t (AHead a1 a0))) H3 t H7) in (let H11 \def (eq_ind T u0
+(\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 a1)))))) H2 u
+H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t a1)) H1 u H8)
+in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g
+c0 t (AHead a3 a0))) a1 H12 H10))))))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: (arity g c0 u0 (asucc g
+a))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 (asucc g
+a)))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (_: (((eq
+T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0 u a1))
+(\lambda (a1: A).(arity g c0 t (AHead a1 a))))))).(\lambda (H5: (eq T (THead
+(Flat Cast) u0 t0) (THead (Flat Appl) u t))).(let H6 \def (eq_ind T (THead
+(Flat Cast) u0 t0) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
+(Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1))
+(\lambda (a1: A).(arity g c0 t (AHead a1 a)))) H6))))))))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0
+a1)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
+(a1: A).(arity g c0 u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2
+a1))))))).(\lambda (a0: A).(\lambda (H3: (leq g a1 a0)).(\lambda (H4: (eq T
+t0 (THead (Flat Appl) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0
+(THead (Flat Appl) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t0:
+T).((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g c0
+u a1)) (\lambda (a2: A).(arity g c0 t (AHead a2 a1)))))) H2 (THead (Flat
+Appl) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t
+a1)) H1 (THead (Flat Appl) u t) H5) in (let H8 \def (H6 (refl_equal T (THead
+(Flat Appl) u t))) in (ex2_ind A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
+(a3: A).(arity g c0 t (AHead a3 a1))) (ex2 A (\lambda (a3: A).(arity g c0 u
+a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0)))) (\lambda (x: A).(\lambda
+(H9: (arity g c0 u x)).(\lambda (H10: (arity g c0 t (AHead x a1))).(ex_intro2
+A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead
+a3 a0))) x H9 (arity_repl g c0 t (AHead x a1) H10 (AHead x a0) (leq_head g x
+x (leq_refl g x) a1 a0 H3)))))) H8))))))))))))) c y a2 H0))) H)))))).
+
+theorem arity_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
+A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a))
+(arity g c t a)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
+A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead
+(Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (land (arity g c u (asucc
+g a)) (arity g c t a)) (\lambda (y: T).(\lambda (H0: (arity g c y
+a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0: A).((eq T t0
+(THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t
+a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n)
+(THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead
+(Flat Cast) u t) H1) in (False_ind (land (arity g c0 u (asucc g (ASort O n)))
+(arity g c0 t (ASort O n))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr)
+u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T
+u0 (THead (Flat Cast) u t)) \to (land (arity g d u (asucc g a0)) (arity g d t
+a0))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u t))).(let H5 \def
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Flat Cast) u t) H4) in (False_ind (land
+(arity g c0 u (asucc g a0)) (arity g c0 t a0)) H5))))))))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
+(CHead d (Bind Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0 (asucc
+g a0))).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g d
+u (asucc g (asucc g a0))) (arity g d t (asucc g a0)))))).(\lambda (H4: (eq T
+(TLRef i) (THead (Flat Cast) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda
+(ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Flat Cast) u t) H4) in (False_ind (land (arity g c0 u (asucc g a0))
+(arity g c0 t a0)) H5))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
+Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_:
+(arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to
+(land (arity g c0 u (asucc g a1)) (arity g c0 t a1))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0
+a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
+(CHead c0 (Bind b) u0) u (asucc g a2)) (arity g (CHead c0 (Bind b) u0) t
+a2))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Flat Cast) u
+t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Cast) u t) H6) in (False_ind (land (arity g c0 u (asucc g a2))
+(arity g c0 t a2)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0:
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_:
+(((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g
+a1))) (arity g c0 t (asucc g a1)))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq
+T t0 (THead (Flat Cast) u t)) \to (land (arity g (CHead c0 (Bind Abst) u0) u
+(asucc g a2)) (arity g (CHead c0 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq
+T (THead (Bind Abst) u0 t0) (THead (Flat Cast) u t))).(let H6 \def (eq_ind T
+(THead (Bind Abst) u0 t0) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u
+t) H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0
+t (AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead
+a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
+c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5:
+(eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def
+(eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow
+False])])])) I (THead (Flat Cast) u t) H5) in (False_ind (land (arity g c0 u
+(asucc g a2)) (arity g c0 t a2)) H6)))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (a0: A).(\lambda (H1: (arity g c0 u0 (asucc g a0))).(\lambda
+(H2: (((eq T u0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g
+(asucc g a0))) (arity g c0 t (asucc g a0)))))).(\lambda (t0: T).(\lambda (H3:
+(arity g c0 t0 a0)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) u t)) \to
+(land (arity g c0 u (asucc g a0)) (arity g c0 t a0))))).(\lambda (H5: (eq T
+(THead (Flat Cast) u0 t0) (THead (Flat Cast) u t))).(let H6 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in ((let H7 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in (\lambda (H8: (eq T
+u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t0: T).((eq T t0 (THead (Flat
+Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t a0)))) H4 t
+H7) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(arity g c0 t a0)) H3 t H7)
+in (let H11 \def (eq_ind T u0 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u
+t)) \to (land (arity g c0 u (asucc g (asucc g a0))) (arity g c0 t (asucc g
+a0))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).(arity g c0 t
+(asucc g a0))) H1 u H8) in (conj (arity g c0 u (asucc g a0)) (arity g c0 t
+a0) H12 H10))))))) H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
+(a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead
+(Flat Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
+a1))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0
+(THead (Flat Cast) u t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0
+(THead (Flat Cast) u t) H4) in (let H6 \def (eq_ind T t0 (\lambda (t0:
+T).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a1))
+(arity g c0 t a1)))) H2 (THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T
+t0 (\lambda (t: T).(arity g c0 t a1)) H1 (THead (Flat Cast) u t) H5) in (let
+H8 \def (H6 (refl_equal T (THead (Flat Cast) u t))) in (and_ind (arity g c0 u
+(asucc g a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0
+t a2)) (\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0
+t a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u
+(asucc g a1) H9 (asucc g a2) (asucc_repl g a1 a2 H3)) (arity_repl g c0 t a1
+H10 a2 H3)))) H8))))))))))))) c y a H0))) H)))))).
+
+theorem arity_gen_appls:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall
+(a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a:
+A).(arity g c t a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads
+(Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda
+(a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c
+t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall
+(a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a:
+A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g
+c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g
+c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1
+a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_:
+(arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x
+a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
+(\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a)))
+(\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a:
+A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
+
+theorem node_inh:
+ \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c:
+C).(\lambda (t: T).(arity g c t (ASort k n)))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0:
+nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0 n))))))
+(ex_2_intro C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort O n))))
+(CSort O) (TSort n) (arity_sort g (CSort O) n)) (\lambda (n0: nat).(\lambda
+(H: (ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort n0
+n)))))).(let H0 \def H in (ex_2_ind C T (\lambda (c: C).(\lambda (t:
+T).(arity g c t (ASort n0 n)))) (ex_2 C T (\lambda (c: C).(\lambda (t:
+T).(arity g c t (ASort (S n0) n))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H1: (arity g x0 x1 (ASort n0 n))).(ex_2_intro C T (\lambda (c:
+C).(\lambda (t: T).(arity g c t (ASort (S n0) n)))) (CHead x0 (Bind Abst) x1)
+(TLRef O) (arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0
+x1) (ASort (S n0) n) H1))))) H0)))) k))).
+
+theorem arity_gen_cvoid_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to
+(\forall (P: Prop).P))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
+A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d
+(Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to
+(\forall (P: Prop).P))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d
+(Bind Void) u))).(\lambda (w: T).(\lambda (v: T).(\lambda (H1: (subst0 i w
+(TSort n) v)).(\lambda (P: Prop).(subst0_gen_sort w v i n H1 P)))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_:
+(arity g d u a0)).(\lambda (_: ((\forall (d0: C).(\forall (u0: T).(\forall
+(i: nat).((getl i d (CHead d0 (Bind Void) u0)) \to (\forall (w: T).(\forall
+(v: T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0
+(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w
+(TLRef i) v)).(\lambda (P: Prop).(and_ind (eq nat i i0) (eq T v (lift (S i) O
+w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
+w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0
+(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c: C).(getl i c0 c)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0
+(CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind Void) u0) H7)) in (let H9 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
+d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead d0 (Bind
+Void) u0) H7)) in (False_ind P H9)))))) (subst0_gen_lref w v i0 i
+H4)))))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_:
+((\forall (d0: C).(\forall (u0: T).(\forall (i: nat).((getl i d (CHead d0
+(Bind Void) u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to
+(\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda
+(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P:
+Prop).(and_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
+nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat
+i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let
+H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c: C).(getl i c0 c)) H0
+(CHead d0 (Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
+d0 (Bind Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Void) u0)
+(getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead d0 (Bind Void) u0) H7)) in
+(False_ind P H9)))))) (subst0_gen_lref w v i0 i H4)))))))))))))))))) (\lambda
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (H2: ((\forall
+(d: C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void)
+u0)) \to (\forall (w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g
+(CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d: C).(\forall (u0:
+T).(\forall (i: nat).((getl i (CHead c0 (Bind b) u) (CHead d (Bind Void) u0))
+\to (\forall (w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H5: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H6: (subst0 i w (THead (Bind b) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Bind b) i) w t0 t2)))) P (\lambda (H7: (ex2 T (\lambda
+(u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda (u2: T).(subst0 i w u
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind b) u2 t0))) (\lambda
+(u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead
+(Bind b) x t0))).(\lambda (H9: (subst0 i w u x)).(H2 d u0 i H5 w x H9 P))))
+H7)) (\lambda (H7: (ex2 T (\lambda (t2: T).(eq T v (THead (Bind b) u t2)))
+(\lambda (t2: T).(subst0 (s (Bind b) i) w t0 t2)))).(ex2_ind T (\lambda (t2:
+T).(eq T v (THead (Bind b) u t2))) (\lambda (t2: T).(subst0 (s (Bind b) i) w
+t0 t2)) P (\lambda (x: T).(\lambda (_: (eq T v (THead (Bind b) u
+x))).(\lambda (H9: (subst0 (s (Bind b) i) w t0 x)).(H4 d u0 (S i)
+(getl_clear_bind b (CHead c0 (Bind b) u) c0 u (clear_bind b c0 u) (CHead d
+(Bind Void) u0) i H5) w x H9 P)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Bind b) x0 x1))).(\lambda (H9: (subst0 i w u
+x0)).(\lambda (_: (subst0 (s (Bind b) i) w t0 x1)).(H2 d u0 i H5 w x0 H9
+P)))))) H7)) (subst0_gen_head (Bind b) w u t0 v i H6)))))))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+(asucc g a1))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (d: C).(\forall (u0: T).(\forall (i: nat).((getl
+i (CHead c0 (Bind Abst) u) (CHead d (Bind Void) u0)) \to (\forall (w:
+T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H5: (subst0 i w (THead (Bind Abst) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)))) P (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T v (THead (Bind Abst) u2 t0))) (\lambda (u2:
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Bind
+Abst) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
+(_: (eq T v (THead (Bind Abst) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
+(THead (Bind Abst) u t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Bind Abst) u t2)))
+(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Bind Abst) u x))).(\lambda (H8: (subst0 (s
+(Bind Abst) i) w t0 x)).(H3 d u0 (S i) (getl_clear_bind Abst (CHead c0 (Bind
+Abst) u) c0 u (clear_bind Abst c0 u) (CHead d (Bind Void) u0) i H4) w x H8
+P)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda
+(_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) w t0 t2))) P (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (_: (eq T v (THead (Bind Abst) x0 x1))).(\lambda
+(H8: (subst0 i w u x0)).(\lambda (_: (subst0 (s (Bind Abst) i) w t0 x1)).(H1
+d u0 i H4 w x0 H8 P)))))) H6)) (subst0_gen_head (Bind Abst) w u t0 v i
+H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: ((\forall (d: C).(\forall
+(u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall
+(w: T).(\forall (v: T).((subst0 i w u v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0
+t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d: C).(\forall (u: T).(\forall
+(i: nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall
+(v: T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c0 (CHead d (Bind
+Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H5: (subst0 i w (THead
+(Flat Appl) u t0) v)).(\lambda (P: Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq
+T v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2))) (ex2 T
+(\lambda (t2: T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) w t0 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T v (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w
+u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0
+t2)))) P (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Appl) u2
+t0))) (\lambda (u2: T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T
+v (THead (Flat Appl) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda
+(x: T).(\lambda (_: (eq T v (THead (Flat Appl) x t0))).(\lambda (H8: (subst0
+i w u x)).(H1 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2:
+T).(eq T v (THead (Flat Appl) u t2))) (\lambda (t2: T).(subst0 (s (Flat Appl)
+i) w t0 t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Appl) u t2)))
+(\lambda (t2: T).(subst0 (s (Flat Appl) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Flat Appl) u x))).(\lambda (H8: (subst0 (s
+(Flat Appl) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Flat Appl) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Appl) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i w
+u x0)).(\lambda (_: (subst0 (s (Flat Appl) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
+P)))))) H6)) (subst0_gen_head (Flat Appl) w u t0 v i H5)))))))))))))))))))
+(\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u
+(asucc g a0))).(\lambda (H1: ((\forall (d: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u0)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (t0:
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (d: C).(\forall
+(u: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall
+(w: T).(\forall (v: T).((subst0 i w t0 v) \to (\forall (P:
+Prop).P)))))))))).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
+(H4: (getl i c0 (CHead d (Bind Void) u0))).(\lambda (w: T).(\lambda (v:
+T).(\lambda (H5: (subst0 i w (THead (Flat Cast) u t0) v)).(\lambda (P:
+Prop).(or3_ind (ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0)))
+(\lambda (u2: T).(subst0 i w u u2))) (ex2 T (\lambda (t2: T).(eq T v (THead
+(Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)))) P (\lambda (H6:
+(ex2 T (\lambda (u2: T).(eq T v (THead (Flat Cast) u2 t0))) (\lambda (u2:
+T).(subst0 i w u u2)))).(ex2_ind T (\lambda (u2: T).(eq T v (THead (Flat
+Cast) u2 t0))) (\lambda (u2: T).(subst0 i w u u2)) P (\lambda (x: T).(\lambda
+(_: (eq T v (THead (Flat Cast) x t0))).(\lambda (H8: (subst0 i w u x)).(H1 d
+u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T v
+(THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0
+t2)))).(ex2_ind T (\lambda (t2: T).(eq T v (THead (Flat Cast) u t2)))
+(\lambda (t2: T).(subst0 (s (Flat Cast) i) w t0 t2)) P (\lambda (x:
+T).(\lambda (_: (eq T v (THead (Flat Cast) u x))).(\lambda (H8: (subst0 (s
+(Flat Cast) i) w t0 x)).(H3 d u0 i H4 w x H8 P)))) H6)) (\lambda (H6: (ex3_2
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda
+(t2: T).(subst0 (s (Flat Cast) i) w t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i w u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Cast) i) w t0 t2))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (eq T v (THead (Flat Cast) x0 x1))).(\lambda (H8: (subst0 i w
+u x0)).(\lambda (_: (subst0 (s (Flat Cast) i) w t0 x1)).(H1 d u0 i H4 w x0 H8
+P)))))) H6)) (subst0_gen_head (Flat Cast) w u t0 v i H5))))))))))))))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
+t0 a1)).(\lambda (H1: ((\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c0 (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v:
+T).((subst0 i w t0 v) \to (\forall (P: Prop).P)))))))))).(\lambda (a2:
+A).(\lambda (_: (leq g a1 a2)).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H3: (getl i c0 (CHead d (Bind Void) u))).(\lambda (w:
+T).(\lambda (v: T).(\lambda (H4: (subst0 i w t0 v)).(\lambda (P: Prop).(H1 d
+u i H3 w v H4 P)))))))))))))))) c t a H))))).
+
+theorem arity_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d
+(Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Void) u))).(let H_x \def (dnf_dec u t i) in
+(let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 i u t (lift (S O) i
+v)) (eq T t (lift (S O) i v)))) (ex T (\lambda (v: T).(eq T t (lift (S O) i
+v)))) (\lambda (x: T).(\lambda (H2: (or (subst0 i u t (lift (S O) i x)) (eq T
+t (lift (S O) i x)))).(or_ind (subst0 i u t (lift (S O) i x)) (eq T t (lift
+(S O) i x)) (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))) (\lambda (H3:
+(subst0 i u t (lift (S O) i x))).(arity_gen_cvoid_subst0 g c t a H d u i H0 u
+(lift (S O) i x) H3 (ex T (\lambda (v: T).(eq T t (lift (S O) i v))))))
+(\lambda (H3: (eq T t (lift (S O) i x))).(let H4 \def (eq_ind T t (\lambda
+(t: T).(arity g c t a)) H (lift (S O) i x) H3) in (eq_ind_r T (lift (S O) i
+x) (\lambda (t0: T).(ex T (\lambda (v: T).(eq T t0 (lift (S O) i v)))))
+(ex_intro T (\lambda (v: T).(eq T (lift (S O) i x) (lift (S O) i v))) x
+(refl_equal T (lift (S O) i x))) t H3))) H2))) H1))))))))))).
+
+theorem arity_gen_lift:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h:
+nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2:
+C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T
+(lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\forall (c2: C).((drop h d
+c1 c2) \to (arity g c2 t a))) (\lambda (y: T).(\lambda (H0: (arity g c1 y
+a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0)) \to (\forall (c2:
+C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat d (\lambda (n:
+nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2: C).((drop h n
+c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c: C).(\lambda (t0:
+T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x
+x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 a0)))))))))
+(\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0: T).(\lambda
+(H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda (_: (drop h x
+c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0 (ASort O n)))
+(arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1))))))))) (\lambda (c:
+C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c
+(CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u
+a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x
+x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
+(lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
+i)))).(and_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
+(Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
+\def (eq_ind T u (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t
+(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
+a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u
+(\lambda (t: T).(arity g d0 t a0)) H2 (lift h (minus x (S i)) x1) H11) in
+(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
+(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
+(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
+(eq T x0 (TLRef (minus i h))))).(and_ind (le (plus x h) i) (eq T x0 (TLRef
+(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
+(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
+(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0
+H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst)
+u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda
+(H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
+(c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda
+(x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x
+x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
+(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
+T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
+(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
+i)))).(and_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
+(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
+nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
+in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
+i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst)
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
+(arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
+(eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
+(Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
+\def (eq_ind T u (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t
+(lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
+(asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def
+(eq_ind T u (\lambda (t: T).(arity g d0 t (asucc g a0))) H2 (lift h (minus x
+(S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
+(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
+Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
+(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(and_ind (le (plus x
+h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
+(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
+(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
+u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5
+H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1:
+(not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall
+(x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
+(arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4:
+(arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x:
+nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h
+x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x:
+nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x
+x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda
+(y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda
+(x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u
+(lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T
+(THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def
+(eq_ind T t0 (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t
+(lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c (Bind b) u) c2) \to
+(arity g c2 x0 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T
+t0 (\lambda (t: T).(arity g (CHead c (Bind b) u) t a2)) H4 (lift h (S x) x2)
+H10) in (let H13 \def (eq_ind T u (\lambda (t: T).(arity g (CHead c (Bind b)
+t) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def (eq_ind T u
+(\lambda (t: T).(\forall (x0: nat).(\forall (x1: T).((eq T (lift h (S x) x2)
+(lift h x0 x1)) \to (\forall (c2: C).((drop h x0 (CHead c (Bind b) t) c2) \to
+(arity g c2 x1 a2))))))) H11 (lift h x x1) H9) in (let H15 \def (eq_ind T u
+(\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift h x x0))
+\to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 a1))))))) H3 (lift h
+x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t: T).(arity g c t a1)) H2
+(lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1 (H15 x x1 (refl_equal T
+(lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal T (lift h (S x) x2))
+(CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b x1))))))))) x0 H8))))))
+(lift_gen_bind b u t0 x0 h x H6)))))))))))))))))) (\lambda (c: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u (asucc g a1))).(\lambda
+(H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
+(c2: C).((drop h x c c2) \to (arity g c2 x0 (asucc g a1))))))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c (Bind Abst) u) t0
+a2)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x
+x0)) \to (\forall (c2: C).((drop h x (CHead c (Bind Abst) u) c2) \to (arity g
+c2 x0 a2)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T
+(THead (Bind Abst) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop
+h x c c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead
+(Bind Abst) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S x) z)))) (arity g
+c2 x0 (AHead a1 a2)) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0
+(THead (Bind Abst) x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda
+(H9: (eq T t0 (lift h (S x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2)
+(\lambda (t1: T).(arity g c2 t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0
+(\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift h x x0))
+\to (\forall (c2: C).((drop h x (CHead c (Bind Abst) u) c2) \to (arity g c2
+x0 a2))))))) H4 (lift h (S x) x2) H9) in (let H11 \def (eq_ind T t0 (\lambda
+(t: T).(arity g (CHead c (Bind Abst) u) t a2)) H3 (lift h (S x) x2) H9) in
+(let H12 \def (eq_ind T u (\lambda (t: T).(arity g (CHead c (Bind Abst) t)
+(lift h (S x) x2) a2)) H11 (lift h x x1) H8) in (let H13 \def (eq_ind T u
+(\lambda (t: T).(\forall (x0: nat).(\forall (x1: T).((eq T (lift h (S x) x2)
+(lift h x0 x1)) \to (\forall (c2: C).((drop h x0 (CHead c (Bind Abst) t) c2)
+\to (arity g c2 x1 a2))))))) H10 (lift h x x1) H8) in (let H14 \def (eq_ind T
+u (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift h x x0))
+\to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 (asucc g a1))))))))
+H2 (lift h x x1) H8) in (let H15 \def (eq_ind T u (\lambda (t: T).(arity g c
+t (asucc g a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1
+(refl_equal T (lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h
+(S x) x2)) (CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst
+x1))))))))) x0 H7)))))) (lift_gen_bind Abst u t0 x0 h x H5))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u
+a1)).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x
+x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g c t0
+(AHead a1 a2))).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T t0
+(lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+(AHead a1 a2))))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T
+(THead (Flat Appl) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop
+h x c c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead
+(Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2
+x0 a2) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat
+Appl) x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0
+(lift h x x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity
+g c2 t1 a2)) (let H10 \def (eq_ind T t0 (\lambda (t: T).(\forall (x:
+nat).(\forall (x0: T).((eq T t (lift h x x0)) \to (\forall (c2: C).((drop h x
+c c2) \to (arity g c2 x0 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let
+H11 \def (eq_ind T t0 (\lambda (t: T).(arity g c t (AHead a1 a2))) H3 (lift h
+x x2) H9) in (let H12 \def (eq_ind T u (\lambda (t: T).(\forall (x:
+nat).(\forall (x0: T).((eq T t (lift h x x0)) \to (\forall (c2: C).((drop h x
+c c2) \to (arity g c2 x0 a1))))))) H2 (lift h x x1) H8) in (let H13 \def
+(eq_ind T u (\lambda (t: T).(arity g c t a1)) H1 (lift h x x1) H8) in
+(arity_appl g c2 x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2
+(H10 x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat
+Appl u t0 x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda
+(a0: A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x:
+nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x
+c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3:
+(arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T
+t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead
+(Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c
+c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0)
+(\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
+x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h
+x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1
+a0)) (let H10 \def (eq_ind T t0 (\lambda (t: T).(\forall (x: nat).(\forall
+(x0: T).((eq T t (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
+(arity g c2 x0 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0
+(\lambda (t: T).(arity g c t a0)) H3 (lift h x x2) H9) in (let H12 \def
+(eq_ind T u (\lambda (t: T).(\forall (x: nat).(\forall (x0: T).((eq T t (lift
+h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0 (asucc g
+a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t:
+T).(arity g c t (asucc g a0))) H1 (lift h x x1) H8) in (arity_cast g c2 x1 a0
+(H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10 x x2 (refl_equal T
+(lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast u t0 x0 h x
+H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq
+T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
+a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda (x:
+nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x x0))).(\lambda (c2:
+C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1 (H2 x x0 H4 c2 H5)
+a2 H3))))))))))))) c1 y a H0))))) H))))))).
+
+theorem arity_lift:
+ \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2
+t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1
+c2) \to (arity g c1 (lift h d t) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c2 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to
+(arity g c1 (lift h d t0) a0)))))))) (\lambda (c: C).(\lambda (n:
+nat).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop
+h d c1 c)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c1 t0 (ASort O
+n))) (arity_sort g c1 n) (lift h d (TSort n)) (lift_sort n h d))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H1:
+(arity g d u a0)).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall
+(d0: nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) a0))))))).(\lambda
+(c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3: (drop h d0 c1
+c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0) (\lambda (H4: (lt i
+d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1 t0 a0)) (let H5 \def
+(drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c1 c h H3 (CHead d
+(Bind Abbr) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (arity
+g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H6: (drop i O
+c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0 x1)).(\lambda (H8: (clear x1
+(CHead d (Bind Abbr) u))).(let H9 \def (eq_ind nat (minus d0 i) (\lambda (n:
+nat).(drop h n x0 x1)) H7 (S (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let
+H10 \def (drop_clear_S x1 x0 h (minus d0 (S i)) H9 Abbr d u H8) in (ex2_ind C
+(\lambda (c3: C).(clear x0 (CHead c3 (Bind Abbr) (lift h (minus d0 (S i))
+u)))) (\lambda (c3: C).(drop h (minus d0 (S i)) c3 d)) (arity g c1 (TLRef i)
+a0) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h
+(minus d0 (S i)) u)))).(\lambda (H12: (drop h (minus d0 (S i)) x
+d)).(arity_abbr g c1 x (lift h (minus d0 (S i)) u) i (getl_intro i c1 (CHead
+x (Bind Abbr) (lift h (minus d0 (S i)) u)) x0 H6 H11) a0 (H2 x h (minus d0 (S
+i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
+H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i h)) (\lambda (t0:
+T).(arity g c1 t0 a0)) (arity_abbr g c1 d u (plus i h) (drop_getl_trans_ge i
+c1 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) a0 H1) (lift h d0 (TLRef i))
+(lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c1: C).(\forall (h: nat).(\forall (d0:
+nat).((drop h d0 c1 d) \to (arity g c1 (lift h d0 u) (asucc g
+a0)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
+(H3: (drop h d0 c1 c)).(lt_le_e i d0 (arity g c1 (lift h d0 (TLRef i)) a0)
+(\lambda (H4: (lt i d0)).(eq_ind_r T (TLRef i) (\lambda (t0: T).(arity g c1
+t0 a0)) (let H5 \def (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0
+H4)) c1 c h H3 (CHead d (Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0:
+C).(\lambda (_: C).(drop i O c1 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop
+h (minus d0 i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d
+(Bind Abst) u)))) (arity g c1 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop i O c1 x0)).(\lambda (H7: (drop h (minus d0 i) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let H9 \def (eq_ind
+nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
+(minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
+H9 Abst d u H8) in (ex2_ind C (\lambda (c3: C).(clear x0 (CHead c3 (Bind
+Abst) (lift h (minus d0 (S i)) u)))) (\lambda (c3: C).(drop h (minus d0 (S
+i)) c3 d)) (arity g c1 (TLRef i) a0) (\lambda (x: C).(\lambda (H11: (clear x0
+(CHead x (Bind Abst) (lift h (minus d0 (S i)) u)))).(\lambda (H12: (drop h
+(minus d0 (S i)) x d)).(arity_abst g c1 x (lift h (minus d0 (S i)) u) i
+(getl_intro i c1 (CHead x (Bind Abst) (lift h (minus d0 (S i)) u)) x0 H6 H11)
+a0 (H2 x h (minus d0 (S i)) H12))))) H10)))))))) H5)) (lift h d0 (TLRef i))
+(lift_lref_lt i h d0 H4))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus
+i h)) (\lambda (t0: T).(arity g c1 t0 a0)) (arity_abst g c1 d u (plus i h)
+(drop_getl_trans_ge i c1 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) a0 H1)
+(lift h d0 (TLRef i)) (lift_lref_ge i h d0 H4)))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind b) u)) \to (arity g c1
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H5: (drop h d c1 c)).(eq_ind_r T (THead (Bind b) (lift h d u)
+(lift h (s (Bind b) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_bind
+g b H0 c1 (lift h d u) a1 (H2 c1 h d H5) (lift h (s (Bind b) d) t0) a2 (H4
+(CHead c1 (Bind b) (lift h d u)) h (s (Bind b) d) (drop_skip_bind h d c1 c H5
+b u))) (lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h
+d))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d u) (asucc g
+a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 (CHead c (Bind Abst) u)) \to (arity g c1
+(lift h d t0) a2))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H4: (drop h d c1 c)).(eq_ind_r T (THead (Bind Abst) (lift h d
+u) (lift h (s (Bind Abst) d) t0)) (\lambda (t1: T).(arity g c1 t1 (AHead a1
+a2))) (arity_head g c1 (lift h d u) a1 (H1 c1 h d H4) (lift h (s (Bind Abst)
+d) t0) a2 (H3 (CHead c1 (Bind Abst) (lift h d u)) h (s (Bind Abst) d)
+(drop_skip_bind h d c1 c H4 Abst u))) (lift h d (THead (Bind Abst) u t0))
+(lift_head (Bind Abst) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1
+(lift h d u) a1))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) (AHead
+a1 a2)))))))).(\lambda (c1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H4: (drop h d c1 c)).(eq_ind_r T (THead (Flat Appl) (lift h d u) (lift h (s
+(Flat Appl) d) t0)) (\lambda (t1: T).(arity g c1 t1 a2)) (arity_appl g c1
+(lift h d u) a1 (H1 c1 h d H4) (lift h (s (Flat Appl) d) t0) a2 (H3 c1 h (s
+(Flat Appl) d) H4)) (lift h d (THead (Flat Appl) u t0)) (lift_head (Flat
+Appl) u t0 h d))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c1:
+C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift
+h d u) (asucc g a0)))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
+a0)).(\lambda (H3: ((\forall (c1: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c1 c) \to (arity g c1 (lift h d t0) a0))))))).(\lambda (c1:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4: (drop h d c1
+c)).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d)
+t0)) (\lambda (t1: T).(arity g c1 t1 a0)) (arity_cast g c1 (lift h d u) a0
+(H1 c1 h d H4) (lift h (s (Flat Cast) d) t0) (H3 c1 h (s (Flat Cast) d) H4))
+(lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h
+d)))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c1: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c1 c) \to (arity g c1 (lift h d t0)
+a1))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c1:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H3: (drop h d c1
+c)).(arity_repl g c1 (lift h d t0) a1 (H1 c1 h d H3) a2 H2)))))))))))) c2 t a
+H))))).
+
+theorem arity_lift1:
+ \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds:
+PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t
+a) \to (arity g c1 (lift1 hds t) a))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t:
+T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))
+(\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda
+(H0: (arity g c2 t a)).(let H1 \def (match H return (\lambda (p:
+PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
+PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a))))))))
+with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda
+(H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0:
+C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C
+c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C
+c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda
+(H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda
+(H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
+PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False
+| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1)
+\to ((eq C c4 c2) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (arity g
+c1 t a))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal
+C c1) (refl_equal C c2))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(p: PList).(\lambda (H: ((\forall (c1: C).(\forall (t: T).((drop1 p c1 c2)
+\to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))).(\lambda (c1:
+C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c1 c2)).(\lambda (H1:
+(arity g c2 t a)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda
+(c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n
+n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 (lift n n0 (lift1 p
+t)) a)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil
+(PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let
+H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_:
+PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False]))
+I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (arity g
+c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4
+hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0
+p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 c2)).((let H7 \def
+(f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_:
+PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p]))
+(PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
+(\lambda (e: PList).(match e return (\lambda (_: PList).nat) with [PNil
+\Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p)
+H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return
+(\lambda (_: PList).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow
+n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1:
+nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2)
+\to ((drop n1 d c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0
+(lift1 p t)) a)))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda
+(n1: nat).((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n
+n1 c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 (lift1 p t))
+a))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0:
+PList).((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n n0 c0 c3) \to ((drop1 p0
+c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))) (\lambda (H12: (eq C
+c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c4 c2) \to ((drop n n0 c c3) \to
+((drop1 p c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a))))) (\lambda
+(H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c3) \to
+((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p t)) a)))) (\lambda (H14:
+(drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3 c2)).(arity_lift g c3 (lift1 p
+t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq C c4 c2 H13))) c0 (sym_eq C c0
+c1 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq
+nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n
+n0 p)) (refl_equal C c1) (refl_equal C c2))))))))))) hds)))).
+
+theorem arity_mono:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c
+t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H:
+(arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
+A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0:
+C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n)
+a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u
+a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a
+a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
+\def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
+C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
+(H5: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
+Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2))))).(ex2_2_ind C
+T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2))) (leq g a a2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abbr)
+x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def (eq_ind C (CHead d (Bind
+Abbr) u) (\lambda (c: C).(getl i c0 c)) H0 (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0
+(CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e:
+C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead
+_ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t:
+T).(getl i c0 (CHead x0 (Bind Abbr) t))) H8 u H10) in (let H13 \def (eq_ind_r
+T x1 (\lambda (t: T).(arity g x0 t a2)) H7 u H10) in (let H14 \def (eq_ind_r
+C x0 (\lambda (c: C).(getl i c0 (CHead c (Bind Abbr) u))) H12 d H11) in (let
+H15 \def (eq_ind_r C x0 (\lambda (c: C).(arity g c u a2)) H13 d H11) in (H2
+a2 H15))))))) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))).(ex2_2_ind C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abst)
+x1))).(\lambda (_: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind C
+(CHead d (Bind Abbr) u) (\lambda (c: C).(getl i c0 c)) H0 (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst)
+x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee:
+C).(match ee return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abbr) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind (leq g a a2)
+H9))))))) H5)) H4)))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
+u))).(\lambda (a: A).(\lambda (_: (arity g d u (asucc g a))).(\lambda (H2:
+((\forall (a2: A).((arity g d u a2) \to (leq g (asucc g a) a2))))).(\lambda
+(a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4 \def (arity_gen_lref
+g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
+c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
+u0 a2)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
+(Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g
+a2))))) (leq g a a2) (\lambda (H5: (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
+T).(arity g d u a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0:
+T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
+T).(arity g d0 u0 a2))) (leq g a a2) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g
+x0 x1 a2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c:
+C).(getl i c0 c)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind
+Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead
+d (Bind Abst) u) (\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void
+\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr)
+x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6))
+in (False_ind (leq g a a2) H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda
+(d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
+C).(\lambda (u: T).(arity g d u (asucc g a2)))))).(ex2_2_ind C T (\lambda
+(d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda
+(d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abst)
+x1))).(\lambda (H7: (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind C
+(CHead d (Bind Abst) u) (\lambda (c: C).(getl i c0 c)) H0 (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst)
+x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal
+C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead
+x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind
+Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1
+(\lambda (t: T).(getl i c0 (CHead x0 (Bind Abst) t))) H8 u H10) in (let H13
+\def (eq_ind_r T x1 (\lambda (t: T).(arity g x0 t (asucc g a2))) H7 u H10) in
+(let H14 \def (eq_ind_r C x0 (\lambda (c: C).(getl i c0 (CHead c (Bind Abst)
+u))) H12 d H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(arity g c u
+(asucc g a2))) H13 d H11) in (asucc_inj g a a2 (H2 (asucc g a2) H15))))))))
+H9))))))) H5)) H4)))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
+Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda (_: (arity
+g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g c0 u a3) \to (leq g a2
+a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_: (arity g (CHead c0
+(Bind b) u) t0 a3)).(\lambda (H4: ((\forall (a2: A).((arity g (CHead c0 (Bind
+b) u) t0 a2) \to (leq g a3 a2))))).(\lambda (a0: A).(\lambda (H5: (arity g c0
+(THead (Bind b) u t0) a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5)
+in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g
+(CHead c0 (Bind b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_:
+(arity g c0 u x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0
+H8)))) H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2:
+A).(\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3:
+A).((arity g c0 u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0:
+T).(\lambda (a3: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
+a3)).(\lambda (H3: ((\forall (a2: A).((arity g (CHead c0 (Bind Abst) u) t0
+a2) \to (leq g a3 a2))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead
+(Bind Abst) u t0) a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in
+(ex3_2_ind A A (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5))))
+(\lambda (a4: A).(\lambda (_: A).(arity g c0 u (asucc g a4)))) (\lambda (_:
+A).(\lambda (a5: A).(arity g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead
+a2 a3) a0) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0
+x1))).(\lambda (H7: (arity g c0 u (asucc g x0))).(\lambda (H8: (arity g
+(CHead c0 (Bind Abst) u) t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a:
+A).(leq g (AHead a2 a3) a)) (leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g
+x0) H7)) a3 x1 (H3 x1 H8)) a0 H6)))))) H5))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda
+(_: ((\forall (a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0:
+T).(\lambda (a3: A).(\lambda (_: (arity g c0 t0 (AHead a2 a3))).(\lambda (H3:
+((\forall (a4: A).((arity g c0 t0 a4) \to (leq g (AHead a2 a3)
+a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Flat Appl) u t0)
+a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0 H4) in (ex2_ind A (\lambda
+(a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity g c0 t0 (AHead a4 a0)))
+(leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u x)).(\lambda (H7:
+(arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x a0 (H3 (AHead x a0)
+H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a:
+A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: ((\forall (a2:
+A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda (t0:
+T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2: A).((arity g
+c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4: (arity g c0
+(THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u t0 a2 H4)
+in (and_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g a a2)
+(\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
+a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
+(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
+A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
+(leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
+g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
+
+theorem arity_cimp_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g
+c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d
+c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c
+c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i)
+a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let
+H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in
+(ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7:
+(cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def
+(eq_ind C (CHead x (Bind Abbr) u) (\lambda (c: C).(getl i c2 c)) H5 (CHead x0
+(Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind
+Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c _ _) \Rightarrow
+c])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono c2 (CHead x
+(Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11 \def (eq_ind_r
+C x0 (\lambda (c: C).(getl i c2 (CHead c (Bind Abbr) u))) H9 x H10) in (let
+H12 \def (eq_ind_r C x0 (\lambda (c: C).(cimp d c)) H7 x H10) in (arity_abbr
+g c2 x u i H11 a0 (H2 x H12))))))))) H6))))) H4))))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c2: C).((cimp d c2) \to (arity g c2 u (asucc g
+a0)))))).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(let H_x \def (H3 Abst d
+u i H0) in (let H4 \def H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5:
+(getl i c2 (CHead x (Bind Abst) u))).(let H_x0 \def (cimp_getl_conf c c2 H3
+Abst d u i H0) in (let H6 \def H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d
+d2)) (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2
+(TLRef i) a0) (\lambda (x0: C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl
+i c2 (CHead x0 (Bind Abst) u))).(let H9 \def (eq_ind C (CHead x (Bind Abst)
+u) (\lambda (c: C).(getl i c2 c)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2
+(CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow x | (CHead c _ _) \Rightarrow c])) (CHead x (Bind Abst) u)
+(CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0
+(Bind Abst) u) H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c: C).(getl i
+c2 (CHead c (Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0
+(\lambda (c: C).(cimp d c)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x
+H12))))))))) H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B
+b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_:
+(arity g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g
+c2 u a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b)
+u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c
+c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u)
+(cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
+((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
+a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to
+(arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c
+c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
+(cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
+(c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2:
+C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2:
+C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2
+H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
+(arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to
+(arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
+a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0
+a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1
+c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
+C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2:
+(leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2
+t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).
+
+theorem arity_aprem:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0:
+A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda
+(b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H1 \def (match H0 return
+(\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem n0 a
+a0)).((eq nat n0 i) \to ((eq A a (ASort O n)) \to ((eq A a0 b) \to (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g b))))))))))))) with [(aprem_zero a1 a2) \Rightarrow (\lambda (H0: (eq nat O
+i)).(\lambda (H1: (eq A (AHead a1 a2) (ASort O n))).(\lambda (H2: (eq A a1
+b)).(eq_ind nat O (\lambda (n0: nat).((eq A (AHead a1 a2) (ASort O n)) \to
+((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda
+(_: nat).(arity g d u (asucc g b))))))))) (\lambda (H3: (eq A (AHead a1 a2)
+(ASort O n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort O n) H3) in (False_ind ((eq A a1 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d
+u (asucc g b))))))) H4))) i H0 H1 H2)))) | (aprem_succ a2 a i0 H0 a1)
+\Rightarrow (\lambda (H1: (eq nat (S i0) i)).(\lambda (H2: (eq A (AHead a1
+a2) (ASort O n))).(\lambda (H3: (eq A a b)).(eq_ind nat (S i0) (\lambda (n0:
+nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a b) \to ((aprem i0 a2 a)
+\to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b)))))))))) (\lambda (H4: (eq A (AHead a1 a2)
+(ASort O n))).(let H5 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e
+return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _
+_) \Rightarrow True])) I (ASort O n) H4) in (False_ind ((eq A a b) \to
+((aprem i0 a2 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H5))) i H1 H2 H3
+H0))))]) in (H1 (refl_equal nat i) (refl_equal A (ASort O n)) (refl_equal A
+b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (i: nat).(\forall
+(b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i0:
+nat).(\lambda (b: A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b
+H3) in (let H4 \def H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1
+(asucc g b))).(let H_x0 \def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0
+u i H0) in (let H7 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0
+x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T
+nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x
+c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 (arity_lift g x0 x1
+(asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) H4))))))))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_:
+(arity g d u (asucc g a0))).(\lambda (H2: ((\forall (i: nat).(\forall (b:
+A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i0:
+nat).(\lambda (b: A).(\lambda (H3: (aprem i0 a0 b)).(let H4 \def (H2 i0 b
+(aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0:
+C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C
+T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O
+d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1
+(asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abst c0
+u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2)
+O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x
+c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 (arity_lift g x0 x1
+(asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) H4)))))))))))))
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda
+(u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_:
+((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead
+c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: nat).(\forall (b0:
+A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) u))))) (\lambda
+(d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: (aprem i a2
+b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead
+c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity
+g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0
+(CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9
+\def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S
+b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0
+(asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g
+a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1)
+b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3:
+((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead
+c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b:
+A).(\lambda (H4: (aprem i (AHead a1 a2) b)).((match i return (\lambda (n:
+nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) with [O
+\Rightarrow (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H6 \def (match H5
+return (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (aprem
+n a a0)).((eq nat n O) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d
+u (asucc g b))))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (_:
+(eq nat O O)).(\lambda (H5: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H6:
+(eq A a0 b)).((let H7 \def (f_equal A A (\lambda (e: A).(match e return
+(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow
+a])) (AHead a0 a3) (AHead a1 a2) H5) in ((let H8 \def (f_equal A A (\lambda
+(e: A).(match e return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 |
+(AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H5) in (eq_ind A a1
+(\lambda (a: A).((eq A a3 a2) \to ((eq A a b) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))
+(\lambda (H9: (eq A a3 a2)).(eq_ind A a2 (\lambda (_: A).((eq A a1 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d
+u (asucc g b)))))))) (\lambda (H10: (eq A a1 b)).(eq_ind A b (\lambda (_:
+A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b))))))) (eq_ind A a1 (\lambda (a: A).(ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g a))))))) (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g a1))))) c0 u O (drop_refl c0) H0) b H10) a1
+(sym_eq A a1 b H10))) a3 (sym_eq A a3 a2 H9))) a0 (sym_eq A a0 a1 H8))) H7))
+H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i)
+O)).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A a
+b)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H5) in (False_ind ((eq A (AHead a3 a0) (AHead a1 a2)) \to ((eq A a b) \to
+((aprem i a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda
+(_: nat).(arity g d u (asucc g b))))))))) H8)) H6 H7 H4))))]) in (H6
+(refl_equal nat O) (refl_equal A (AHead a1 a2)) (refl_equal A b)))) | (S n)
+\Rightarrow (\lambda (H5: (aprem (S n) (AHead a1 a2) b)).(let H6 \def (match
+H5 return (\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).(\lambda (_:
+(aprem n0 a a0)).((eq nat n0 (S n)) \to ((eq A a (AHead a1 a2)) \to ((eq A a0
+b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a0 a3)
+\Rightarrow (\lambda (H4: (eq nat O (S n))).(\lambda (H5: (eq A (AHead a0 a3)
+(AHead a1 a2))).(\lambda (H6: (eq A a0 b)).((let H7 \def (eq_ind nat O
+(\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O \Rightarrow
+True | (S _) \Rightarrow False])) I (S n) H4) in (False_ind ((eq A (AHead a0
+a3) (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda
+(d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H7))
+H5 H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i)
+(S n))).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A
+a b)).((let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return
+(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i)
+(S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a3 a0) (AHead a1
+a2)) \to ((eq A a b) \to ((aprem n0 a0 a) \to (ex2_3 C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda
+(d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))
+(\lambda (H9: (eq A (AHead a3 a0) (AHead a1 a2))).(let H10 \def (f_equal A A
+(\lambda (e: A).(match e return (\lambda (_: A).A) with [(ASort _ _)
+\Rightarrow a0 | (AHead _ a) \Rightarrow a])) (AHead a3 a0) (AHead a1 a2) H9)
+in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_:
+A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead
+a3 a0) (AHead a1 a2) H9) in (eq_ind A a1 (\lambda (_: A).((eq A a0 a2) \to
+((eq A a b) \to ((aprem n a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))
+(\lambda (H12: (eq A a0 a2)).(eq_ind A a2 (\lambda (a1: A).((eq A a b) \to
+((aprem n a1 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H13: (eq A a
+b)).(eq_ind A b (\lambda (a1: A).((aprem n a2 a1) \to (ex2_3 C T nat (\lambda
+(d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b)))))))) (\lambda (H14: (aprem n a2 b)).(let H_x \def (H3 n b H14) in (let
+H3 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus n j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O
+d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u
+(asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+nat).(\lambda (H15: (drop (plus n x2) O x0 (CHead c0 (Bind Abst)
+u))).(\lambda (H16: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus n x2) H15) H16)))))) H3)))) a
+(sym_eq A a b H13))) a0 (sym_eq A a0 a2 H12))) a3 (sym_eq A a3 a1 H11)))
+H10))) i (sym_eq nat i n H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n))
+(refl_equal A (AHead a1 a2)) (refl_equal A b))))]) H4))))))))))))) (\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to
+(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus
+i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d
+u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
+g c0 t0 (AHead a1 a2))).(\lambda (H3: ((\forall (i: nat).(\forall (b:
+A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i:
+nat).(\lambda (b: A).(\lambda (H4: (aprem i a2 b)).(let H5 \def (H3 (S i) b
+(aprem_succ a2 b i H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6:
+(drop (S (plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g
+b))).(C_ind (\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1
+x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0:
+T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n:
+nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity
+g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i
+x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda
+(j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0:
+T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0
+(CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O
+O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H11) in (False_ind (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) H13)))))
+(drop_gen_sort n (S (plus i x2)) O c0 H8))))) (\lambda (d: C).(\lambda (IHd:
+(((drop (S (plus i x2)) O d c0) \to ((arity g d x1 (asucc g b)) \to (ex2_3 C
+T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g b)))))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i
+x2)) O (CHead d k t1) c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g
+b))).((match k return (\lambda (k0: K).((arity g (CHead d k0 t1) x1 (asucc g
+b)) \to ((drop (r k0 (plus i x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))
+with [(Bind b0) \Rightarrow (\lambda (H10: (arity g (CHead d (Bind b0) t1) x1
+(asucc g b))).(\lambda (H11: (drop (r (Bind b0) (plus i x2)) O d
+c0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b))))) (CHead d (Bind b0) t1) x1 (S x2)
+(eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O (CHead d (Bind b0)
+t1) c0)) (drop_drop (Bind b0) (plus i x2) d c0 H11 t1) (plus i (S x2))
+(plus_n_Sm i x2)) H10))) | (Flat f) \Rightarrow (\lambda (H10: (arity g
+(CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Flat f) (plus
+i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g (CHead d (Flat f)
+t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in (ex2_3_ind C T nat
+(\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0
+c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0
+(asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda
+(_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x3: C).(\lambda (x4:
+T).(\lambda (x5: nat).(\lambda (H13: (drop (plus i x5) O x3 c0)).(\lambda
+(H14: (arity g x3 x4 (asucc g b))).(ex2_3_intro C T nat (\lambda (d0:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda
+(d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x3
+x4 x5 H13 H14)))))) H12))))]) H9 (drop_gen_drop k d c0 t1 (plus i x2)
+H8)))))))) x0 H6 H7)))))) H5)))))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_:
+((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a0) b) \to (ex2_3 C T
+nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d
+c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc
+g b))))))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda
+(H3: ((\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g
+b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i a0
+b)).(let H_x \def (H3 i b H4) in (let H5 \def H_x in (ex2_3_ind C T nat
+(\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0))))
+(\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g
+b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop
+(plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_:
+nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H6: (drop (plus i x2) O x0 c0)).(\lambda (H7:
+(arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda
+(_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda
+(u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7))))))
+H5)))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
+(_: (arity g c0 t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b:
+A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (a2:
+A).(\lambda (H2: (leq g a1 a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda
+(H3: (aprem i a2 b)).(let H_x \def (aprem_repl g a1 a2 H2 i b H3) in (let H4
+\def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem
+i a1 b1)) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x
+b)).(\lambda (H6: (aprem i a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def
+H_x0 in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_:
+T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda
+(H9: (arity g x0 x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d:
+C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8
+(arity_repl g x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5))))))))
+H7)))))) H4))))))))))))) c t a H))))).
+
+theorem arity_appls_cast:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs:
+TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to
+((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl)
+vs (THead (Flat Cast) u t)) a))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(\forall (a: A).((arity g c (THeads
+(Flat Appl) t0 u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) t0 t) a)
+\to (arity g c (THeads (Flat Appl) t0 (THead (Flat Cast) u t)) a)))))
+(\lambda (a: A).(\lambda (H: (arity g c u (asucc g a))).(\lambda (H0: (arity
+g c t a)).(arity_cast g c u a H t H0)))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (H: ((\forall (a: A).((arity g c (THeads (Flat Appl) t1 u)
+(asucc g a)) \to ((arity g c (THeads (Flat Appl) t1 t) a) \to (arity g c
+(THeads (Flat Appl) t1 (THead (Flat Cast) u t)) a)))))).(\lambda (a:
+A).(\lambda (H0: (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 u))
+(asucc g a))).(\lambda (H1: (arity g c (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 t)) a)).(let H2 \def (arity_gen_appl g c t0 (THeads (Flat Appl) t1
+t) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1)) (\lambda (a1:
+A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1 a))) (arity g c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))) a) (\lambda
+(x: A).(\lambda (H3: (arity g c t0 x)).(\lambda (H4: (arity g c (THeads (Flat
+Appl) t1 t) (AHead x a))).(let H5 \def (arity_gen_appl g c t0 (THeads (Flat
+Appl) t1 u) (asucc g a) H0) in (ex2_ind A (\lambda (a1: A).(arity g c t0 a1))
+(\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 u) (AHead a1 (asucc g
+a)))) (arity g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat
+Cast) u t))) a) (\lambda (x0: A).(\lambda (H6: (arity g c t0 x0)).(\lambda
+(H7: (arity g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g
+a)))).(arity_appl g c t0 x H3 (THeads (Flat Appl) t1 (THead (Flat Cast) u t))
+a (H (AHead x a) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x (asucc g
+a)) (arity_repl g c (THeads (Flat Appl) t1 u) (AHead x0 (asucc g a)) H7
+(AHead x (asucc g a)) (leq_head g x0 x (arity_mono g c t0 x0 H6 x H3) (asucc
+g a) (asucc g a) (leq_refl g (asucc g a)))) (asucc g (AHead x a)) (leq_refl g
+(asucc g (AHead x a)))) H4))))) H5))))) H2)))))))) vs))))).
+
+theorem arity_appls_abbr:
+ \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall
+(a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c
+(THeads (Flat Appl) vs (TLRef i)) a)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (vs:
+TList).(TList_ind (\lambda (t: TList).(\forall (a: A).((arity g c (THeads
+(Flat Appl) t (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) t
+(TLRef i)) a)))) (\lambda (a: A).(\lambda (H0: (arity g c (lift (S i) O v)
+a)).(arity_abbr g c d v i H a (arity_gen_lift g c v a (S i) O H0 d (getl_drop
+Abbr c d v i H))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0:
+((\forall (a: A).((arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) a) \to
+(arity g c (THeads (Flat Appl) t0 (TLRef i)) a))))).(\lambda (a: A).(\lambda
+(H1: (arity g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
+v))) a)).(let H2 \def (arity_gen_appl g c t (THeads (Flat Appl) t0 (lift (S
+i) O v)) a H1) in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1:
+A).(arity g c (THeads (Flat Appl) t0 (lift (S i) O v)) (AHead a1 a))) (arity
+g c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) a) (\lambda (x:
+A).(\lambda (H3: (arity g c t x)).(\lambda (H4: (arity g c (THeads (Flat
+Appl) t0 (lift (S i) O v)) (AHead x a))).(arity_appl g c t x H3 (THeads (Flat
+Appl) t0 (TLRef i)) a (H0 (AHead x a) H4))))) H2))))))) vs))))))).
+
+theorem arity_appls_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c:
+C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t:
+T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v)
+(THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat
+Appl) vs (THead (Bind b) v t)) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda
+(c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H0: (arity g c v
+a1)).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O t0) t) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Bind
+b) v t)) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g (CHead c (Bind b) v)
+t a2)).(arity_bind g b H c v a1 H0 t a2 H1))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (H1: ((\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads
+(Flat Appl) (lifts (S O) O t1) t) a2) \to (arity g c (THeads (Flat Appl) t1
+(THead (Bind b) v t)) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g (CHead
+c (Bind b) v) (THead (Flat Appl) (lift (S O) O t0) (THeads (Flat Appl) (lifts
+(S O) O t1) t)) a2)).(let H3 \def (arity_gen_appl g (CHead c (Bind b) v)
+(lift (S O) O t0) (THeads (Flat Appl) (lifts (S O) O t1) t) a2 H2) in
+(ex2_ind A (\lambda (a3: A).(arity g (CHead c (Bind b) v) (lift (S O) O t0)
+a3)) (\lambda (a3: A).(arity g (CHead c (Bind b) v) (THeads (Flat Appl)
+(lifts (S O) O t1) t) (AHead a3 a2))) (arity g c (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 (THead (Bind b) v t))) a2) (\lambda (x: A).(\lambda
+(H4: (arity g (CHead c (Bind b) v) (lift (S O) O t0) x)).(\lambda (H5: (arity
+g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O t1) t) (AHead x
+a2))).(arity_appl g c t0 x (arity_gen_lift g (CHead c (Bind b) v) t0 x (S O)
+O H4 c (drop_drop (Bind b) O c c (drop_refl c) v)) (THeads (Flat Appl) t1
+(THead (Bind b) v t)) a2 (H1 (AHead x a2) H5))))) H3))))))) vs))))))))).
+
+theorem arity_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
+c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u
+c1 t1 c2 t2) \to (arity g c2 t2 a))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(a0: A).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
+d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
+t2) \to (arity g c2 t2 a0))))))))))) (\lambda (c: C).(\lambda (n:
+nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i
+c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1:
+(fsubst0 i u c (TSort n) c2 t2)).(let H2 \def (fsubst0_gen_base c c2 (TSort
+n) t2 u i H1) in (or3_ind (land (eq C c c2) (subst0 i u (TSort n) t2)) (land
+(eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i u (TSort n) t2)
+(csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3: (land (eq C c
+c2) (subst0 i u (TSort n) t2))).(and_ind (eq C c c2) (subst0 i u (TSort n)
+t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c c2)).(\lambda (H5:
+(subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (ASort
+O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2 (ASort O n))) c2 H4))) H3))
+(\lambda (H3: (land (eq T (TSort n) t2) (csubst0 i u c c2))).(and_ind (eq T
+(TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 (ASort O n)) (\lambda (H4:
+(eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c c2)).(eq_ind T (TSort n)
+(\lambda (t: T).(arity g c2 t (ASort O n))) (arity_sort g c2 n) t2 H4))) H3))
+(\lambda (H3: (land (subst0 i u (TSort n) t2) (csubst0 i u c c2))).(and_ind
+(subst0 i u (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2 (ASort O n))
+(\lambda (H4: (subst0 i u (TSort n) t2)).(\lambda (_: (csubst0 i u c
+c2)).(subst0_gen_sort u t2 i n H4 (arity g c2 t2 (ASort O n))))) H3))
+H2))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
+A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: ((\forall (d1: C).(\forall
+(u0: T).(\forall (i: nat).((getl i d (CHead d1 (Bind Abbr) u0)) \to (\forall
+(c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 t2) \to (arity g c2 t2
+a0)))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda
+(H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2:
+T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H5 \def
+(fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (or3_ind (land (eq C c c2)
+(subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))
+(land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0)
+(\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))).(and_ind (eq C
+c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c
+c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind C c (\lambda (c0:
+C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 (lift (S i) O u0))
+(arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda (H10: (eq T t2 (lift
+(S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: T).(arity g c t a0))
+(let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind
+Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c
+(CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (let H13 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u)
+(CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1
+(Bind Abbr) u0) H11)) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono
+c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (\lambda
+(H15: (eq C d d1)).(let H16 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c
+(CHead d1 (Bind Abbr) t))) H12 u H14) in (eq_ind T u (\lambda (t: T).(arity g
+c (lift (S i) O t) a0)) (let H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i
+c (CHead c0 (Bind Abbr) u))) H16 d H15) in (arity_lift g d u a0 H1 c (S i) O
+(getl_drop Abbr c d u i H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref
+u0 t2 i0 i H8)) c2 H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2)
+(csubst0 i0 u0 c c2))).(and_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)
+(arity g c2 t2 a0) (\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0
+i0 u0 c c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e
+i i0 (arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def
+(csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abbr) u) H0) in (or4_ind
+(getl i c2 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c2
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (arity g c2 (TLRef i) a0)
+(\lambda (H11: (getl i c2 (CHead d (Bind Abbr) u))).(let H12 \def (eq_ind nat
+(minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u) (CHead d1
+(Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d
+(Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i)))
+(minus_x_Sy i0 i H9)) in (arity_abbr g c2 d u i H11 a0 H1))) (\lambda (H11:
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (S i)) u0 u w))))))).(ex3_4_ind B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef i) a0) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq C
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H13: (getl i c2
+(CHead x1 (Bind x0) x3))).(\lambda (H14: (subst0 (minus i0 (S i)) u0 x2
+x3)).(let H15 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B (\lambda (e: C).(match e
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2) H12) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
+t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H12) in (\lambda (H19:
+(eq B Abbr x0)).(\lambda (H20: (eq C d x1)).(let H21 \def (eq_ind_r T x2
+(\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H14 u H18) in (let H22
+\def (eq_ind_r C x1 (\lambda (c: C).(getl i c2 (CHead c (Bind x0) x3))) H13 d
+H20) in (let H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead d
+(Bind b) x3))) H22 Abbr H19) in (arity_abbr g c2 d x3 i H23 a0 (H2 d1 u0 (r
+(Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr)
+u0) u (minus i0 (S i)) H15) d x3 (fsubst0_snd (r (Bind Abbr) (minus i0 (S
+i))) u0 d u x3 H21))))))))) H17)) H16)))))))))) H11)) (\lambda (H11: (ex3_4 B
+C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl i c2 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind
+x0) x3))).(\lambda (H14: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H15 \def
+(eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) u)
+(CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3
+(CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0
+(S i))) (minus_x_Sy i0 i H9)) in (let H16 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12)
+in ((let H17 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in ((let H18
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H19: (eq B Abbr
+x0)).(\lambda (H20: (eq C d x1)).(let H21 \def (eq_ind_r T x3 (\lambda (t:
+T).(getl i c2 (CHead x2 (Bind x0) t))) H13 u H18) in (let H22 \def (eq_ind_r
+C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H14 d H20) in (let
+H23 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) u)))
+H21 Abbr H19) in (arity_abbr g c2 x2 u i H23 a0 (H2 d1 u0 (r (Bind Abbr)
+(minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr) u0) u
+(minus i0 (S i)) H15) x2 u (fsubst0_fst (r (Bind Abbr) (minus i0 (S i))) u0 d
+u x2 H22))))))))) H17)) H16)))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (arity g c2 (TLRef i) a0)
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H12: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind x0) x4))).(\lambda (H14:
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n
+(CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead
+d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abbr) u) i H0 (le_S_n i i0 (le_S (S i)
+i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal
+C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C B (\lambda (e: C).(match e
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
+t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20:
+(eq B Abbr x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3
+(\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23
+\def (eq_ind_r C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H15 d
+H21) in (let H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2
+(Bind b) x4))) H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r
+(Bind Abbr) (minus i0 (S i))) (getl_gen_S (Bind Abbr) d (CHead d1 (Bind Abbr)
+u0) u (minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abbr) (minus i0 (S
+i))) u0 d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10)))
+(\lambda (H9: (le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2
+u0 H8 (CHead d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land
+(subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (subst0 i0 u0
+(TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0
+i0 u0 (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(and_ind (eq nat i
+i0) (eq T t2 (lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i
+i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O
+u0) (\lambda (t: T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0
+(\lambda (n: nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r
+nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in
+(let H13 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: C).(getl i c
+c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H0
+(CHead d1 (Bind Abbr) u0) H12)) in (let H14 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead d1 (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abbr) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in
+((let H15 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
+i H0 (CHead d1 (Bind Abbr) u0) H12)) in (\lambda (H16: (eq C d d1)).(let H17
+\def (eq_ind_r T u0 (\lambda (t: T).(getl i c (CHead d1 (Bind Abbr) t))) H13
+u H15) in (let H18 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 i t c c2))
+H11 u H15) in (eq_ind T u (\lambda (t: T).(arity g c2 (lift (S i) O t) a0))
+(let H19 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr)
+u))) H17 d H16) in (arity_lift g d u a0 H1 c2 (S i) O (getl_drop Abbr c2 d u
+i (csubst0_getl_ge i i (le_n i) c c2 u H18 (CHead d (Bind Abbr) u) H19)))) u0
+H15))))) H14))))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H7)))) H6))
+H5))))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
+A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i: nat).((getl i d (CHead d1 (Bind Abbr) u0))
+\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 t2) \to (arity g
+c2 t2 (asucc g a0))))))))))).(\lambda (d1: C).(\lambda (u0: T).(\lambda (i0:
+nat).(\lambda (H3: (getl i0 c (CHead d1 (Bind Abbr) u0))).(\lambda (c2:
+C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H5
+\def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (or3_ind (land (eq C c
+c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i) t2) (csubst0 i0 u0 c
+c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2)) (arity g c2 t2
+a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))).(and_ind
+(eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0) (\lambda (H7: (eq
+C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind C c (\lambda (c0:
+C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2 (lift (S i) O u0))
+(arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda (H10: (eq T t2 (lift
+(S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t: T).(arity g c t a0))
+(let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind
+Abbr) u0))) H3 i H9) in (let H12 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (c0: C).(getl i c c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c
+(CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (let H13 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1
+(Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind
+Abbr) u0) H11)) in (False_ind (arity g c (lift (S i) O u0) a0) H13)))) t2
+H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7))) H6)) (\lambda (H6: (land (eq
+T (TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (eq T (TLRef i) t2) (csubst0
+i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq T (TLRef i) t2)).(\lambda
+(H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t
+a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let
+H10 \def (csubst0_getl_lt i0 i H9 c c2 u0 H8 (CHead d (Bind Abst) u) H0) in
+(or4_ind (getl i c2 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abst) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))
+(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b)
+u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c2 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))
+(arity g c2 (TLRef i) a0) (\lambda (H11: (getl i c2 (CHead d (Bind Abst)
+u))).(let H12 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n (CHead
+d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead d1 (Bind
+Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i) i0 H9)))
+(S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (arity_abst g c2 d u i H11 a0
+H1))) (\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b)
+u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c2
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))) (arity g c2 (TLRef
+i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H12: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H13: (getl i c2 (CHead x1 (Bind x0) x3))).(\lambda (H14:
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H15 \def (eq_ind nat (minus i0 i)
+(\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0)))
+(getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0
+(le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9))
+in (let H16 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d
+(Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H17 \def (f_equal C B
+(\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d
+(Bind Abst) u) (CHead x1 (Bind x0) x2) H12) in ((let H18 \def (f_equal C T
+(\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead
+x1 (Bind x0) x2) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C
+d x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i))
+u0 t x3)) H14 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c: C).(getl i
+c2 (CHead c (Bind x0) x3))) H13 d H20) in (let H23 \def (eq_ind_r B x0
+(\lambda (b: B).(getl i c2 (CHead d (Bind b) x3))) H22 Abst H19) in
+(arity_abst g c2 d x3 i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i)))
+(getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15) d
+x3 (fsubst0_snd (r (Bind Abst) (minus i0 (S i))) u0 d u x3 H21))))))))) H17))
+H16)))))))))) H11)) (\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead d (Bind Abst) u) (CHead
+e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u: T).(getl i c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u1: T).(getl i c2 (CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))
+(arity g c2 (TLRef i) a0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
+C).(\lambda (x3: T).(\lambda (H12: (eq C (CHead d (Bind Abst) u) (CHead x1
+(Bind x0) x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind x0) x3))).(\lambda
+(H14: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H15 \def (eq_ind nat (minus
+i0 i) (\lambda (n: nat).(getl n (CHead d (Bind Abst) u) (CHead d1 (Bind Abbr)
+u0))) (getl_conf_le i0 (CHead d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u)
+i H0 (le_S_n i i0 (le_S (S i) i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i
+H9)) in (let H16 \def (f_equal C C (\lambda (e: C).(match e return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c]))
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H17 \def
+(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort
+_) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d
+(Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C T
+(\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u) (CHead
+x1 (Bind x0) x3) H12) in (\lambda (H19: (eq B Abst x0)).(\lambda (H20: (eq C
+d x1)).(let H21 \def (eq_ind_r T x3 (\lambda (t: T).(getl i c2 (CHead x2
+(Bind x0) t))) H13 u H18) in (let H22 \def (eq_ind_r C x1 (\lambda (c:
+C).(csubst0 (minus i0 (S i)) u0 c x2)) H14 d H20) in (let H23 \def (eq_ind_r
+B x0 (\lambda (b: B).(getl i c2 (CHead x2 (Bind b) u))) H21 Abst H19) in
+(arity_abst g c2 x2 u i H23 a0 (H2 d1 u0 (r (Bind Abst) (minus i0 (S i)))
+(getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr) u0) u (minus i0 (S i)) H15)
+x2 u (fsubst0_fst (r (Bind Abst) (minus i0 (S i))) u0 d u x2 H22)))))))))
+H17)) H16)))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u0))))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl i c2 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i0 (S i)) u0 e1 e2)))))) (arity g c2 (TLRef i) a0) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H12: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H13: (getl i c2 (CHead x2 (Bind x0) x4))).(\lambda (H14:
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H15: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H16 \def (eq_ind nat (minus i0 i) (\lambda (n: nat).(getl n
+(CHead d (Bind Abst) u) (CHead d1 (Bind Abbr) u0))) (getl_conf_le i0 (CHead
+d1 (Bind Abbr) u0) c H3 (CHead d (Bind Abst) u) i H0 (le_S_n i i0 (le_S (S i)
+i0 H9))) (S (minus i0 (S i))) (minus_x_Sy i0 i H9)) in (let H17 \def (f_equal
+C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u) (CHead
+x1 (Bind x0) x3) H12) in ((let H18 \def (f_equal C B (\lambda (e: C).(match e
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3) H12) in ((let H19 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
+t])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H12) in (\lambda (H20:
+(eq B Abst x0)).(\lambda (H21: (eq C d x1)).(let H22 \def (eq_ind_r T x3
+(\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x4)) H14 u H19) in (let H23
+\def (eq_ind_r C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H15 d
+H21) in (let H24 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c2 (CHead x2
+(Bind b) x4))) H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r
+(Bind Abst) (minus i0 (S i))) (getl_gen_S (Bind Abst) d (CHead d1 (Bind Abbr)
+u0) u (minus i0 (S i)) H16) x2 x4 (fsubst0_both (r (Bind Abst) (minus i0 (S
+i))) u0 d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10)))
+(\lambda (H9: (le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2
+u0 H8 (CHead d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land
+(subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (subst0 i0 u0
+(TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0
+i0 u0 (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(and_ind (eq nat i
+i0) (eq T t2 (lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i
+i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O
+u0) (\lambda (t: T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0
+(\lambda (n: nat).(csubst0 n u0 c c2)) H8 i H9) in (let H12 \def (eq_ind_r
+nat i0 (\lambda (n: nat).(getl n c (CHead d1 (Bind Abbr) u0))) H3 i H9) in
+(let H13 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl i c
+c0)) H0 (CHead d1 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) i H0
+(CHead d1 (Bind Abbr) u0) H12)) in (let H14 \def (eq_ind C (CHead d (Bind
+Abst) u) (\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort
+_) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H12)) in
+(False_ind (arity g c2 (lift (S i) O u0) a0) H14))))) t2 H10)))
+(subst0_gen_lref u0 t2 i0 i H7)))) H6)) H5))))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda (H2: ((\forall
+(d1: C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to
+(arity g c2 t2 a1)))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (d1: C).(\forall
+(u0: T).(\forall (i: nat).((getl i (CHead c (Bind b) u) (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b)
+u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H6: (fsubst0 i u0 c (THead
+(Bind b) u t) c2 t2)).(let H7 \def (fsubst0_gen_base c c2 (THead (Bind b) u
+t) t2 u0 i H6) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead (Bind b) u
+t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (land
+(subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 a2)
+(\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind b) u t)
+t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2) (arity g c2
+t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0 (THead (Bind b)
+u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c t2 a2) (\lambda (H11: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c t0 a2))
+(arity_bind g b H0 c x a1 (H2 d1 u0 i H5 c x (fsubst0_snd i u0 c u x H13)) t
+a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b
+c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x) t (fsubst0_fst (S
+i) u0 (CHead c (Bind b) u) t (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u
+x H13 c)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3))) (\lambda (t2: T).(subst0 (s (Bind b) i) u0 t
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s
+(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity
+g c t0 a2)) (arity_bind g b H0 c u a1 H1 x a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c (Bind b) u) x (fsubst0_snd (S i) u0 (CHead c
+(Bind b) u) t x H13))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) u0 t t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
+(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t
+x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c t0 a2))
+(arity_bind g b H0 c x0 a1 (H2 d1 u0 i H5 c x0 (fsubst0_snd i u0 c u x0 H13))
+x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind
+b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both
+(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0)
+(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head
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+(Bind b) u t) t2) (csubst0 i u0 c c2))).(and_ind (eq T (THead (Bind b) u t)
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+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
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+(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s
+(Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
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+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t (fsubst0_fst (S i) u0 (CHead c
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+H10)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
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+(fsubst0_both (S i) u0 (CHead c (Bind b) u) t x H13 (CHead c2 (Bind b) u)
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+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind b) i) u0 t t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda
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+T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
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+x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c2 t0 a2))
+(arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 (fsubst0_both i u0 c u x0
+H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c
+u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x0)
+x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c2 (Bind b)
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+(subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) H7))))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
+(asucc g a1))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
+a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: C).(\forall (u0:
+T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead d1 (Bind Abbr)
+u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind
+Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda
+(u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Bind Abst) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Bind
+Abst) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead
+(Bind Abst) u t) t2)) (land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c
+c2)) (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))
+(arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land (eq C c c2) (subst0 i u0
+(THead (Bind Abst) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind
+Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (eq C c
+c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) t2)).(eq_ind C c
+(\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind (ex2 T (\lambda (u2:
+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3:
+T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u0 t t3)))) (arity g c t2 (AHead a1 a2)) (\lambda (H10: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2 (AHead a1 a2)) (\lambda
+(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12:
+(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0:
+T).(arity g c t0 (AHead a1 a2))) (arity_head g c x a1 (H1 d1 u0 i H4 c x
+(fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst
+(CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i
+H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t
+(CHead c (Bind Abst) x) (csubst0_snd_bind Abst i u0 u x H12 c)))) t2 H11))))
+H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u
+t3))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) u0 t t2)))).(ex2_ind T
+(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0
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+T).(arity g c t0 (AHead a1 a2))) (arity_head g c u a1 H0 x a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) u) x (fsubst0_snd (S i)
+u0 (CHead c (Bind Abst) u) t x H12))) t2 H11)))) H10)) (\lambda (H10: (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) u0 t t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity g c t2 (AHead
+a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Bind Abst) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Bind Abst) i) u0 t x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x0 a1 (H1 d1
+u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0)
+(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10))
+(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7:
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+c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0
+(AHead a1 a2))) (arity_head g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c
+u c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u)
+c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind
+Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind
+Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda
+(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c
+c2))).(and_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2)
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+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3:
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+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
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+Abst) i) u0 t t3)))) (arity g c2 t2 (AHead a1 a2)) (\lambda (H10: (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
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+(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x t))).(\lambda (H12:
+(subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t) (\lambda (t0:
+T).(arity g c2 t0 (AHead a1 a2))) (arity_head g c2 x a1 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 (S i) (getl_clear_bind
+Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr)
+u0) i H4) (CHead c2 (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind
+Abst) u) t (CHead c2 (Bind Abst) x) (csubst0_both_bind Abst i u0 u x H12 c c2
+H9)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u t3))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) u0 t
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
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+a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
+x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
+(Bind Abst) u x) (\lambda (t0: T).(arity g c2 t0 (AHead a1 a2))) (arity_head
+g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 (S
+i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) u) x (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x H12 (CHead c2 (Bind Abst) u)
+(csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H11)))) H10)) (\lambda (H10:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) u0 t t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
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+(AHead a1 a2)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2
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+u0 i H4 c2 x0 (fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind Abst) x0) x1 (fsubst0_both (S
+i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c2 (Bind Abst) x0)
+(csubst0_both_bind Abst i u0 u x0 H12 c c2 H9)))) t2 H11)))))) H10))
+(subst0_gen_head (Bind Abst) u0 u t t2 i H8)))) H7)) H6))))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
+a1)).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
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+((\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1
+(Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
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+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Flat Appl) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Flat
+Appl) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead
+(Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c
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+(arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2) (subst0 i u0 (THead (Flat
+Appl) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Flat Appl) u t)
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+(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda
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+Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c t2 a2)
+(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2
+(THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2
+a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x
+t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t)
+(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x a1 (H1 d1 u0 i H4 c x
+(fsubst0_snd i u0 c u x H12)) t a2 H2) t2 H11)))) H10)) (\lambda (H10: (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) u0 t t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))
+(arity g c t2 a2) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl)
+u x))).(\lambda (H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead
+(Flat Appl) u x) (\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c u a1 H0
+x a2 (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10))
+(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u0 t
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity
+g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Flat Appl) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Flat Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t0: T).(arity g c t0 a2)) (arity_appl g c x0 a1 (H1 d1 u0 i H4 c x0
+(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c
+t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9))
+c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0
+i u0 c c2))).(and_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst
+i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead
+(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H10:
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
+Appl) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x t))).(\lambda
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Appl) x t) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 x a1 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i
+u0 c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T
+t2 (THead (Flat Appl) u t3))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) u0 t
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Appl) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u0 t t3)) (arity g c2 t2 a2)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) u x))).(\lambda
+(H12: (subst0 (s (Flat Appl) i) u0 t x)).(eq_ind_r T (THead (Flat Appl) u x)
+(\lambda (t0: T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2
+u (fsubst0_fst i u0 c u c2 H9)) x a2 (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c
+t x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Appl) i) u0 t t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
+Appl) i) u0 t x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t0:
+T).(arity g c2 t0 a2)) (arity_appl g c2 x0 a1 (H1 d1 u0 i H4 c2 x0
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 a2 (H3 d1 u0 i H4 c2 x1
+(fsubst0_both i u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head
+(Flat Appl) u0 u t t2 i H8)))) H7)) H6)))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a0: A).(\lambda (H0: (arity g c u (asucc g
+a0))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
+nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
+a0))))))))))).(\lambda (t: T).(\lambda (H2: (arity g c t a0)).(\lambda (H3:
+((\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1
+(Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2
+t2) \to (arity g c2 t2 a0)))))))))).(\lambda (d1: C).(\lambda (u0:
+T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
+u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
+(Flat Cast) u t) c2 t2)).(let H6 \def (fsubst0_gen_base c c2 (THead (Flat
+Cast) u t) t2 u0 i H5) in (or3_ind (land (eq C c c2) (subst0 i u0 (THead
+(Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c
+c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2))
+(arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2) (subst0 i u0 (THead (Flat
+Cast) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Flat Cast) u t)
+t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0
+(THead (Flat Cast) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0))
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda
+(u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat
+Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c t2 a0)
+(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2
+(THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2
+a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x
+t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t)
+(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x a0 (H1 d1 u0 i H4 c x
+(fsubst0_snd i u0 c u x H12)) t H2) t2 H11)))) H10)) (\lambda (H10: (ex2 T
+(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3))) (\lambda (t2: T).(subst0
+(s (Flat Cast) i) u0 t t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))
+(arity g c t2 a0) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast)
+u x))).(\lambda (H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead
+(Flat Cast) u x) (\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c u a0 H0
+x (H3 d1 u0 i H4 c x (fsubst0_snd i u0 c t x H12))) t2 H11)))) H10)) (\lambda
+(H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) u0 t
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity
+g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead
+(Flat Cast) x0 x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13:
+(subst0 (s (Flat Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1)
+(\lambda (t0: T).(arity g c t0 a0)) (arity_cast g c x0 a0 (H1 d1 u0 i H4 c x0
+(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t
+x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2
+H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i
+u0 c c2))).(and_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
+(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda
+(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst
+i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
+H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead
+(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3)))) (arity g c2 t2 a0) (\lambda (H10:
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Flat
+Cast) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a0)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x t))).(\lambda
+(H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Flat Cast) x t) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 x a0 (H1 d1 u0 i H4 c2 x
+(fsubst0_both i u0 c u x H12 c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0
+c t c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2
+(THead (Flat Cast) u t3))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) u0 t
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Flat Cast) u t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u0 t t3)) (arity g c2 t2 a0)
+(\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) u x))).(\lambda
+(H12: (subst0 (s (Flat Cast) i) u0 t x)).(eq_ind_r T (THead (Flat Cast) u x)
+(\lambda (t0: T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2
+u (fsubst0_fst i u0 c u c2 H9)) x (H3 d1 u0 i H4 c2 x (fsubst0_both i u0 c t
+x H12 c2 H9))) t2 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Cast) i) u0 t t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u0 t t3))) (arity g c2 t2 a0) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Flat Cast) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Flat
+Cast) i) u0 t x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0:
+T).(arity g c2 t0 a0)) (arity_cast g c2 x0 a0 (H1 d1 u0 i H4 c2 x0
+(fsubst0_both i u0 c u x0 H12 c2 H9)) x1 (H3 d1 u0 i H4 c2 x1 (fsubst0_both i
+u0 c t x1 H13 c2 H9))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t
+t2 i H8)))) H7)) H6))))))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda
+(a1: A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (d1:
+C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u)) \to
+(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c t c2 t2) \to (arity g c2 t2
+a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (d1:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H3: (getl i c (CHead d1 (Bind
+Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i u c t
+c2 t2)).(let H5 \def (fsubst0_gen_base c c2 t t2 u i H4) in (or3_ind (land
+(eq C c c2) (subst0 i u t t2)) (land (eq T t t2) (csubst0 i u c c2)) (land
+(subst0 i u t t2) (csubst0 i u c c2)) (arity g c2 t2 a2) (\lambda (H6: (land
+(eq C c c2) (subst0 i u t t2))).(and_ind (eq C c c2) (subst0 i u t t2) (arity
+g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i u t
+t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2)) (arity_repl g c t2 a1
+(H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2) c2 H7))) H6)) (\lambda
+(H6: (land (eq T t t2) (csubst0 i u c c2))).(and_ind (eq T t t2) (csubst0 i u
+c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t t2)).(\lambda (H8: (csubst0 i
+u c c2)).(eq_ind T t (\lambda (t0: T).(arity g c2 t0 a2)) (arity_repl g c2 t
+a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2 H8)) a2 H2) t2 H7))) H6))
+(\lambda (H6: (land (subst0 i u t t2) (csubst0 i u c c2))).(and_ind (subst0 i
+u t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (subst0 i u t
+t2)).(\lambda (H8: (csubst0 i u c c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3
+c2 t2 (fsubst0_both i u c t t2 H7 c2 H8)) a2 H2))) H6)) H5)))))))))))))))) c1
+t1 a H))))).
+
+theorem arity_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c
+t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead
+d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2
+a)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (a: A).(\lambda (H:
+(arity g c t1 a)).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
+(subst0 i u t1 t2)).(arity_fsubst0 g c t1 a H d u i H0 c t2 (fsubst0_snd i u
+c t1 t2 H1)))))))))))).
-axiom csubst0_drop_gt_back: \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (drop n O c1 e))))))))) .
+inductive pr0: T \to (T \to Prop) \def
+| pr0_refl: \forall (t: T).(pr0 t t)
+| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1:
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1)
+(THead k u2 t2))))))))
+| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to
+(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1
+(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2))))))))
+| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1:
+T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0
+u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead
+(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t2)))))))))))))
+| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1:
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to
+(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
+| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall
+(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O
+t1)) t2))))))
+| pr0_epsilon: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u:
+T).(pr0 (THead (Flat Cast) u t1) t2)))).
+
+theorem pr0_gen_sort:
+ \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
+\def
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(let H0
+\def (match H return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t
+t0)).((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))) with
+[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TSort n))).(\lambda (H1: (eq
+T t x)).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TSort
+n)))) (\lambda (H2: (eq T (TSort n) x)).(eq_ind T (TSort n) (\lambda (t0:
+T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H2)) t (sym_eq T t (TSort
+n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T
+(THead k u1 t1) (TSort n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4
+\def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead
+k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4))
+H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TSort n))).(\lambda (H3: (eq
+T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead
+(Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TSort
+n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3)
+\Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
+(TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H4) in (False_ind ((eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to
+((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))))
+H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow
+(\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TSort n))).(\lambda (H4: (eq T
+(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1
+t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to
+((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TSort
+n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda
+(H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TSort n))).(\lambda (H3: (eq
+T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1))
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst))
+\to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1
+t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TSort
+n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u
+t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T
+x (TSort n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TSort n)) (refl_equal T
+x))))).
+
+theorem pr0_gen_lref:
+ \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
+\def
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(let H0
+\def (match H return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t
+t0)).((eq T t (TLRef n)) \to ((eq T t0 x) \to (eq T x (TLRef n))))))) with
+[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TLRef n))).(\lambda (H1: (eq
+T t x)).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TLRef
+n)))) (\lambda (H2: (eq T (TLRef n) x)).(eq_ind T (TLRef n) (\lambda (t0:
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H2)) t (sym_eq T t (TLRef
+n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T
+(THead k u1 t1) (TLRef n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4
+\def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead
+k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4))
+H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TLRef n))).(\lambda (H3: (eq
+T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead
+(Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TLRef
+n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3)
+\Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
+(TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H4) in (False_ind ((eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to
+((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n)))))))
+H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow
+(\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TLRef n))).(\lambda (H4: (eq T
+(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1
+t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to
+((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TLRef
+n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda
+(H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TLRef n))).(\lambda (H3: (eq
+T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1))
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst))
+\to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1
+t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TLRef
+n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u
+t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T
+x (TLRef n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TLRef n)) (refl_equal T
+x))))).
+
+theorem pr0_gen_abst:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1
+t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Bind Abst) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abst) u1 t1)) \to ((eq
+T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2))))))))) with [(pr0_refl t) \Rightarrow
+(\lambda (H0: (eq T t (THead (Bind Abst) u1 t1))).(\lambda (H1: (eq T t
+x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).((eq T t0 x) \to
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H2: (eq T (THead (Bind Abst)
+u1 t1) x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T
+(THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2))) u1 t1 (refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl
+t1)) x H2)) t (sym_eq T t (THead (Bind Abst) u1 t1) H0) H1))) | (pr0_comp u0
+u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead
+(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H5 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in (eq_ind K (Bind Abst)
+(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2)
+x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))))
+(\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to
+((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T
+t1 (\lambda (t: T).((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 u1 u2) \to
+((pr0 t t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) (\lambda (H9: (eq T (THead (Bind
+Abst) u2 t2) x)).(eq_ind T (THead (Bind Abst) u2 t2) (\lambda (t: T).((pr0 u1
+u2) \to ((pr0 t1 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) (\lambda (H10: (pr0 u1
+u2)).(\lambda (H11: (pr0 t1 t2)).(ex3_2_intro T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
+t1 t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H10 H11))) x H9)) t0
+(sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abst)
+H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow
+(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead
+(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let
+H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H2) in (False_ind ((eq T
+(THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2
+t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind
+b) u0 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Abst) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1
+v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
+\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst)
+u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
+(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind ((eq T
+(THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O
+u2 t2 w) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b
+H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O)
+O t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t:
+T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
+t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
+(Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in (eq_ind B Abst (\lambda
+(b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to
+((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T
+(lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0
+t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T (lift (S O) O t0)
+t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B
+Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))))) (\lambda (H9: (eq
+T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t0 t)
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))))) (\lambda (H10: (not (eq
+B Abst Abst))).(\lambda (_: (pr0 t0 x)).(False_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
+(S O) O t0) t3)))) (H10 (refl_equal B Abst))))) t2 (sym_eq T t2 x H9))) t1
+H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abst H6))) H5)) H4)) H3 H0 H1))) |
+(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u
+t0) (THead (Bind Abst) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def
+(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
+t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abst) u1 t1))
+(refl_equal T x)))))).
+
+theorem pr0_gen_appl:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1
+t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2))))))))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Flat Appl) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Appl) u1 t1)) \to ((eq
+T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0:
+(eq T t (THead (Flat Appl) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T
+(THead (Flat Appl) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2))))))))))) (\lambda (H2: (eq T (THead (Flat Appl) u1 t1)
+x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0
+t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl)
+u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat
+Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
+t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl)
+u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Appl)
+u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2:
+(eq T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead
+k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1)
+H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in
+((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in (eq_ind K
+(Flat Appl) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T
+(THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1
+(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Appl) u2 t2) x) \to
+((pr0 t u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H8:
+(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Appl) u2 t2)
+x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))))) (\lambda (H9: (eq T (THead (Flat Appl) u2 t2)
+x)).(eq_ind T (THead (Flat Appl) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to
+((pr0 t1 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t
+(THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11:
+(pr0 t1 t2)).(or3_intro0 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl)
+u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Flat
+Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3)
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex3_2_intro T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Appl)
+u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7)))
+k (sym_eq K k (Flat Appl) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0
+t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind
+Abbr) v2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u t0) |
+(TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) \Rightarrow
+t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1
+t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t0)) (THead (Flat Appl) u1 t1) H2) in (eq_ind T u1 (\lambda (t: T).((eq T
+(THead (Bind Abst) u t0) t1) \to ((eq T (THead (Bind Abbr) v2 t2) x) \to
+((pr0 t v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x
+(THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1
+v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H6:
+(eq T (THead (Bind Abst) u t0) t1)).(eq_ind T (THead (Bind Abst) u t0)
+(\lambda (t: T).((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 u1 v2) \to
+((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b)
+v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))))) (\lambda (H7: (eq T (THead (Bind Abbr) v2 t2)
+x)).(eq_ind T (THead (Bind Abbr) v2 t2) (\lambda (t: T).((pr0 u1 v2) \to
+((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t
+(THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1
+v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H8: (pr0
+u1 v2)).(\lambda (H9: (pr0 t0 t2)).(or3_intro1 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda
+(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl)
+(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))) u t0 v2 t2 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T
+(THead (Bind Abbr) v2 t2)) H8 H9)))) x H7)) t1 H6)) v1 (sym_eq T v1 u1 H5)))
+H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow
+(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
+(Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead
+(Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
+(Flat Appl) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (eq_ind T u1 (\lambda (t:
+T).((eq T (THead (Bind b) u0 t0) t1) \to ((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 t
+v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))))))))))) (\lambda (H8: (eq T (THead (Bind b) u0 t0)
+t1)).(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).((eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to
+((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H9: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) x)).(eq_ind T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (\lambda (t: T).((not (eq B b
+Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0
+t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda
+(b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1
+z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b0) v3 (THead (Flat
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))))))))) (\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 u1
+v2)).(\lambda (H12: (pr0 u0 u2)).(\lambda (H13: (pr0 t0 t2)).(or3_intro2
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead
+(Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))
+(THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b)
+u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0)
+v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
+(eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0
+t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))) b u0 t0 v2 u2 t2 H10 (refl_equal T (THead (Bind b)
+u0 t0)) (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t2))) H11 H12 H13)))))) x H9)) t1 H8)) v1 (sym_eq T v1 u1 H7))) H6)) H5 H0 H1
+H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T
+(THead (Bind Abbr) u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T
+(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0
+t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind ((eq
+T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0
+O u2 t2 w) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x
+(THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1
+u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0))
+(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind
+T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Appl) u1 t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b
+Abst)) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda
+(b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind b0) y1 z1)))))))) (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda
+(t3: T).(eq T x (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O u2)
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) H4)) H3 H0 H1))) |
+(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u
+t0) (THead (Flat Appl) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def
+(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
+(Flat Appl) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead
+(Flat Appl) u1 t1)) (refl_equal T x)))))).
+
+theorem pr0_gen_cast:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Flat Cast) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Cast) u1 t1)) \to ((eq
+T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))))) with [(pr0_refl t)
+\Rightarrow (\lambda (H0: (eq T t (THead (Flat Cast) u1 t1))).(\lambda (H1:
+(eq T t x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0: T).((eq T t0 x)
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
+Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))) (\lambda (H2: (eq T (THead
+(Flat Cast) u1 t1) x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0:
+T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat
+Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
+u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
+(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t
+(THead (Flat Cast) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k)
+\Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Cast) u1
+t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H5 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H6 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (eq_ind K (Flat Cast)
+(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2)
+x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 x)))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda
+(t: T).((eq T t0 t1) \to ((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 t u2)
+\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x
+(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) (\lambda
+(H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Cast) u2
+t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 x)))))) (\lambda (H9: (eq T (THead (Flat Cast) u2 t2)
+x)).(eq_ind T (THead (Flat Cast) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to
+((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t))))) (\lambda (H10:
+(pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2))
+(ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast)
+u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T
+(THead (Flat Cast) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0
+(sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Cast) H6))) H5)) H4)) H3 H0 H1))) |
+(pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3:
+(eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
+(Flat Cast) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to
+((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+x))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3)
+\Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0))
+(THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
+(Flat Cast) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to
+((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+x))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
+\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast)
+u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
+(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1
+t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to
+((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 x)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u)
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead
+(Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T
+(THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1
+t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t0 t2)
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) |
+(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u
+t0) (THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1) H1) in ((let H4 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t]))
+(THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1) H1) in (eq_ind T u1
+(\lambda (_: T).((eq T t0 t1) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) (\lambda (H5: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 x))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x
+(\lambda (t: T).((pr0 t1 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))
+(\lambda (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)
+H7)) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 t1 H5))) u (sym_eq T u u1 H4)))
+H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T
+x)))))).
+
+theorem pr0_lift:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall
+(d: nat).(pr0 (lift h d t1) (lift h d t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t)
+(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0
+(lift h d u1) (lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda
+(_: (pr0 t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0
+(lift h d t0) (lift h d t3)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda
+(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t:
+T).(pr0 t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2)
+(lift h (s k d) t3)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k
+d) t0)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d)
+t0) (lift h (s k d) t3) (H3 h (s k d)) k) (lift h d (THead k u2 t3))
+(lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) (lift_head k u1 t0 h
+d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
+(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
+d v1) (lift h d v2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0
+t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0)
+(lift h d t3)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead
+(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u
+t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r
+T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s
+(Flat Appl) d)) t0)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t)
+(lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r T (THead (Bind Abbr) (lift h
+d v2) (lift h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl)
+(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s
+(Bind Abst) (s (Flat Appl) d)) t0))) t)) (pr0_beta (lift h (s (Flat Appl) d)
+u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl)
+d)) t0) (lift h (s (Bind Abbr) d) t3) (H3 h (s (Bind Abbr) d))) (lift h d
+(THead (Bind Abbr) v2 t3)) (lift_head (Bind Abbr) v2 t3 h d)) (lift h (s
+(Flat Appl) d) (THead (Bind Abst) u t0)) (lift_head (Bind Abst) u t0 h (s
+(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t0)))
+(lift_head (Flat Appl) v1 (THead (Bind Abst) u t0) h d))))))))))))) (\lambda
+(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (pr0 t0 t3)).(\lambda (H6: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (h: nat).(\lambda (d:
+nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d)
+(THead (Bind b) u1 t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead (Bind b)
+(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t0))
+(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead
+(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O)
+O v2) t3))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead
+(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d))
+t0))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O
+v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t3)) (\lambda (t: T).(pr0 (THead
+(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift
+h (s (Bind b) (s (Flat Appl) d)) t0))) (THead (Bind b) (lift h d u2) t)))
+(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h
+d v1) (THead (Bind b) (lift h d u1) (lift h n t0))) (THead (Bind b) (lift h d
+u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t3)))))
+(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat
+Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d)
+t0))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O)
+d) t3))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d
+u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t0) (lift h (plus (S O) d)
+t3) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d
+v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b)
+d) (THead (Flat Appl) (lift (S O) O v2) t3)) (lift_head (Flat Appl) (lift (S
+O) O v2) t3 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t3) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t0))
+(lift_head (Bind b) u1 t0 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
+v1 (THead (Bind b) u1 t0))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t0)
+h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
+u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1)
+(lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0
+t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0)
+(lift h d t3)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t3 w)).(\lambda
+(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift
+h (s (Bind Abbr) d) t0)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr)
+u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr)
+d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind
+Abbr) d) t0)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S
+d) t0) (lift h (S d) t3) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in
+(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2)
+(lift h d' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (lt_le_S O
+(S d) (le_lt_n_Sm O d (le_O_n d))) h) d (eq_ind nat d (\lambda (n: nat).(eq
+nat n d)) (refl_equal nat d) (minus d O) (minus_n_O d))))) (lift h d (THead
+(Bind Abbr) u2 w)) (lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind
+Abbr) u1 t0)) (lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (pr0 t0 t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d:
+nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (u: T).(\lambda (h:
+nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s
+(Bind b) d) (lift (S O) O t0))) (\lambda (t: T).(pr0 t (lift h d t3)))
+(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Bind b) (lift h d
+u) (lift h n (lift (S O) O t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O
+(lift h d t0)) (\lambda (t: T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d
+t3))) (pr0_zeta b H0 (lift h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift
+h (plus (S O) d) (lift (S O) O t0)) (lift_d t0 h (S O) d O (le_O_n d))) (S d)
+(refl_equal nat (S d))) (lift h d (THead (Bind b) u (lift (S O) O t0)))
+(lift_head (Bind b) u (lift (S O) O t0) h d))))))))))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).(\lambda (H1: ((\forall (h:
+nat).(\forall (d: nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (u:
+T).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Flat Cast) (lift h
+d u) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(pr0 t (lift h d t3)))
+(pr0_epsilon (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 h d) (lift h d
+u)) (lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h
+d))))))))) t1 t2 H))).
+
+theorem pr0_gen_abbr:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Bind Abbr) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq
+T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))))) with
+[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abbr) u1
+t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda
+(t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y:
+T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O
+x))))) (\lambda (H2: (eq T (THead (Bind Abbr) u1 t1) x)).(eq_ind T (THead
+(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
+(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
+t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
+t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u2 y t2)))))) u1 t1 (refl_equal T (THead (Bind
+Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y:
+T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t
+(sym_eq T t (THead (Bind Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1
+k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1
+t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr)
+(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2)
+x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
+t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0
+u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr)
+u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
+t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind
+T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 u1 u2) \to
+((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x
+(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
+t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O
+x))))))) (\lambda (H9: (eq T (THead (Bind Abbr) u2 t2) x)).(eq_ind T (THead
+(Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))
+(\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead (Bind
+Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T
+(THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
+t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0
+t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
+t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k
+(sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0
+t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind
+Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H2) in
+(False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0
+t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))) H4))
+H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow
+(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
+(Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl)
+v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
+t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2)
+\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x
+(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
+t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O
+x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
+\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr)
+u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in ((let H6 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in (eq_ind T u1
+(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) u2 w) x) \to
+((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
+O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) x) \to
+((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
+O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (THead
+(Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t:
+T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
+O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))) (\lambda (H9: (pr0 u1
+u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0 O u2 t2
+w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
+(Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1
+(lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr)
+u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda
+(y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda
+(y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7)))
+u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u)
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead
+(Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
+t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
+(Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr (\lambda
+(b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to
+((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
+t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind
+T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not
+(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
+t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O
+t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not
+(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t
+t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y
+t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T
+x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3:
+T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O
+t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift
+(S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0
+x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y:
+T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0
+(lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq
+T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5))
+H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T
+(THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2
+x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
+(\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))) H3)) H2
+H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T x)))))).
+
+theorem pr0_gen_void:
+ \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1
+t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
+\def
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
+(Bind Void) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Void) u1 t1)) \to ((eq
+T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))))) with
+[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Void) u1
+t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda
+(t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1) x)).(eq_ind T (THead
+(Bind Void) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void)
+u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Bind Void)
+u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2:
+(eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead
+k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1)
+H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in
+((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K
+(Bind Void) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T
+(THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0
+u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void)
+u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T
+t1 (\lambda (t: T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to
+((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x
+(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+x))))))) (\lambda (H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead
+(Bind Void) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda
+(H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void)
+u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
+(Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2
+(refl_equal T (THead (Bind Void) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0
+t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Void) H6))) H5)) H4))
+H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1
+t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Void) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2)
+x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (lift (S O) O x)))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1
+v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl)
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(\lambda (H5: (eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst))
+\to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) H6)) H5 H0 H1 H2 H3))) |
+(pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead
+(Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H4: (eq T (THead
+(Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead
+(Bind Void) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to
+((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) |
+(pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2
+x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d:
+nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f
+d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
+(d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f
+d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O
+t0)) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S
+O) O t0)) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B
+(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in
+(eq_ind B Void (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1)
+\to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u
+u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2
+x) \to ((not (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0)
+t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B
+Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift
+(S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not
+(eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
+(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_:
+(not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x))
+(pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u
+u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2
+H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind
+Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead
+(Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H1) in
+(False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 (refl_equal T
+(THead (Bind Void) u1 t1)) (refl_equal T x)))))).
+
+theorem pr0_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0
+(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
+(t2: T).(pr0 t1 t2)))))))
+\def
+ \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t
+x)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr0 t1
+t2))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat d (\lambda (n:
+nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h n
+t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: T).(\forall
+(x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h
+x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: T).(\lambda
+(t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (ex2
+T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0
+t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1:
+(eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h x1
+t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x:
+T).(\forall (x0: nat).((eq T u1 (lift h x0 x)) \to (ex2 T (\lambda (t2:
+T).(eq T u2 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall (x:
+T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2:
+T).(eq T t3 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (k:
+K).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead k u1 t2)
+(lift h x1 x0))).(K_ind (\lambda (k0: K).((eq T (THead k0 u1 t2) (lift h x1
+x0)) \to (ex2 T (\lambda (t4: T).(eq T (THead k0 u2 t3) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x0 t4))))) (\lambda (b: B).(\lambda (H6: (eq T (THead
+(Bind b) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z:
+T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T
+u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1)
+z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1
+x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b)
+x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3)
+(lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4:
+T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T
+(\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4:
+T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3
+(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1)
+x4) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift
+h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5:
+T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2
+x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+(THead (Bind b) t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Bind b) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead
+(Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4:
+T).(pr0 (THead (Bind b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h
+x1 (THead (Bind b) x5 x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4))
+(lift_bind b x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2
+H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 (S x1) H9)) x0 H7))))))
+(lift_gen_bind b u1 t2 x0 h x1 H6)))) (\lambda (f: F).(\lambda (H6: (eq T
+(THead (Flat f) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0:
+T).(\lambda (z: T).(eq T x0 (THead (Flat f) y0 z)))) (\lambda (y0:
+T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2
+t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H7: (eq T x0 (THead (Flat f) x2 x3))).(\lambda (H8: (eq T
+u1 (lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T
+(THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead
+(Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq
+T t3 (lift h x1 x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 t) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex2_ind T
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Flat f) u2 (lift h x1 x4)) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x5: T).(\lambda
+(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T
+(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f)
+t (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2
+x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Flat f) (lift h x1 x5)
+(lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3)
+t4)) (THead (Flat f) x5 x4) (sym_eq T (lift h x1 (THead (Flat f) x5 x4))
+(THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift_flat f x5 x4 h x1))
+(pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 H_x0)))) (H2 x2 x1 H8)) t3
+H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat f u1 t2 x0 h x1 H6)))) k
+H5))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x: T).(\forall (x0: nat).((eq T v1
+(lift h x0 x)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x0 t2))) (\lambda
+(t2: T).(pr0 x t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0
+t2 t3)).(\lambda (H4: ((\forall (x: T).(\forall (x0: nat).((eq T t2 (lift h
+x0 x)) \to (ex2 T (\lambda (t2: T).(eq T t3 (lift h x0 t2))) (\lambda (t2:
+T).(pr0 x t2)))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t2)) (lift h x1 x0))).(ex3_2_ind T
+T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
+(\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1 y0)))) (\lambda (_:
+T).(\lambda (z: T).(eq T (THead (Bind Abst) u t2) (lift h x1 z)))) (ex2 T
+(\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda
+(t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H6: (eq T
+x0 (THead (Flat Appl) x2 x3))).(\lambda (H7: (eq T v1 (lift h x1
+x2))).(\lambda (H8: (eq T (THead (Bind Abst) u t2) (lift h x1 x3))).(eq_ind_r
+T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+(THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
+(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind Abst)
+y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda
+(H9: (eq T x3 (THead (Bind Abst) x4 x5))).(\lambda (_: (eq T u (lift h x1
+x4))).(\lambda (H11: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T (THead (Bind
+Abst) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr)
+v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4))))
+(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4:
+T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift
+h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4
+x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1)
+x6))).(\lambda (H12: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))))
+(ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0
+x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 (lift h (S x1)
+x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead
+(Bind Abst) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T v2 (lift h
+x1 x7))).(\lambda (H13: (pr0 x2 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t (lift h (S x1) x6))
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind
+Abst) x4 x5)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind Abbr)
+(lift h x1 x7) (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)) (THead (Bind Abbr) x7
+x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x7 x6)) (THead (Bind Abbr) (lift
+h x1 x7) (lift h (S x1) x6)) (lift_bind Abbr x7 x6 h x1)) (pr0_beta x4 x2 x7
+H13 x5 x6 H12)) v2 H_x0)))) (H2 x2 x1 H7)) t3 H_x)))) (H4 x5 (S x1) H11)) x3
+H9)))))) (lift_gen_bind Abst u t2 x3 h x1 H8)) x0 H6)))))) (lift_gen_flat
+Appl v1 (THead (Bind Abst) u t2) x0 h x1 H5)))))))))))))) (\lambda (b:
+B).(\lambda (H1: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (H3: ((\forall (x: T).(\forall (x0:
+nat).((eq T v1 (lift h x0 x)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x0
+t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u1 u2)).(\lambda (H5: ((\forall (x: T).(\forall (x0:
+nat).((eq T u1 (lift h x0 x)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x0
+t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (_: (pr0 t2 t3)).(\lambda (H7: ((\forall (x: T).(\forall (x0:
+nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2: T).(eq T t3 (lift h x0
+t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t2)) (lift
+h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead
+(Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h x1
+y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead (Bind b) u1 t2) (lift h
+x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead (Flat Appl)
+x2 x3))).(\lambda (H10: (eq T v1 (lift h x1 x2))).(\lambda (H11: (eq T (THead
+(Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl) x2 x3)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
+(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 (THead (Bind b) y0
+z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H12: (eq T x3 (THead (Bind b) x4 x5))).(\lambda
+(H13: (eq T u1 (lift h x1 x4))).(\lambda (H14: (eq T t2 (lift h (S x1)
+x5))).(eq_ind_r T (THead (Bind b) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 t) t4)))) (ex2_ind T
+(\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x5 t4))
+(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2
+(THead (Bind b) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift
+h (S x1) x6))).(\lambda (H15: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq
+T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift h (S
+x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead
+(Bind b) x4 x5)) t4))) (\lambda (x7: T).(\lambda (H_x0: (eq T u2 (lift h x1
+x7))).(\lambda (H16: (pr0 x4 x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) t (THead (Flat Appl) (lift
+(S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex2_ind T (\lambda (t4:
+T).(eq T v2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O
+v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T
+v2 (lift h x1 x8))).(\lambda (H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7)
+(THead (Flat Appl) (lift (S O) O t) (lift h (S x1) x6))) (lift h x1 t4)))
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))))
+(eq_ind T (lift h (plus (S O) x1) (lift (S O) O x8)) (\lambda (t: T).(ex2 T
+(\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) t
+(lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (S x1) (THead (Flat
+Appl) (lift (S O) O x8) x6)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+(THead (Bind b) (lift h x1 x7) t) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) (ex_intro2 T (\lambda
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat
+Appl) (lift (S O) O x8) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead
+(Flat Appl) x2 (THead (Bind b) x4 x5)) t4)) (THead (Bind b) x7 (THead (Flat
+Appl) (lift (S O) O x8) x6)) (sym_eq T (lift h x1 (THead (Bind b) x7 (THead
+(Flat Appl) (lift (S O) O x8) x6))) (THead (Bind b) (lift h x1 x7) (lift h (S
+x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift_bind b x7 (THead (Flat
+Appl) (lift (S O) O x8) x6) h x1)) (pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5
+x6 H15)) (THead (Flat Appl) (lift h (S x1) (lift (S O) O x8)) (lift h (S x1)
+x6)) (lift_flat Appl (lift (S O) O x8) x6 h (S x1))) (lift (S O) O (lift h x1
+x8)) (lift_d x8 h (S O) x1 O (le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2
+H_x0)))) (H5 x4 x1 H13)) t3 H_x)))) (H7 x5 (S x1) H14)) x3 H12))))))
+(lift_gen_bind b u1 t2 x3 h x1 H11)) x0 H9)))))) (lift_gen_flat Appl v1
+(THead (Bind b) u1 t2) x0 h x1 H8))))))))))))))))))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x:
+T).(\forall (x0: nat).((eq T u1 (lift h x0 x)) \to (ex2 T (\lambda (t2:
+T).(eq T u2 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall (x:
+T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2:
+T).(eq T t3 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (w:
+T).(\lambda (H5: (subst0 O u2 t3 w)).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t2) (lift h x1
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind
+Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u1 (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda
+(t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead
+(Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 x2))).(\lambda (H9:
+(eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda
+(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1
+t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3
+(lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T (\lambda (t4:
+T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0
+(THead (Bind Abbr) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3
+(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(let H11 \def (eq_ind T t3
+(\lambda (t: T).(subst0 O u2 t w)) H5 (lift h (S x1) x4) H_x) in (ex2_ind T
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x5: T).(\lambda (H_x0:
+(eq T u2 (lift h x1 x5))).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1
+x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) t w)
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (let
+H13 \def (eq_ind T u2 (\lambda (t: T).(subst0 O t (lift h (S x1) x4) w)) H11
+(lift h x1 x5) H_x0) in (let H14 \def (refl_equal nat (S (plus O x1))) in
+(let H15 \def (eq_ind nat (S x1) (\lambda (n: nat).(subst0 O (lift h x1 x5)
+(lift h n x4) w)) H13 (S (plus O x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq
+T w (lift h (S (plus O x1)) t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) w) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda (x6:
+T).(\lambda (H16: (eq T w (lift h (S (plus O x1)) x6))).(\lambda (H17:
+(subst0 O x5 x4 x6)).(eq_ind_r T (lift h (S (plus O x1)) x6) (\lambda (t:
+T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) t) (lift h
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)))) (ex_intro2 T
+(\lambda (t4: T).(eq T (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O
+x1)) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3)
+t4)) (THead (Bind Abbr) x5 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x5
+x6)) (THead (Bind Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6))
+(lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta x2 x5 H12 x3 x4 H10 x6 H17))
+w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 H15))))) u2 H_x0)))) (H2 x2 x1
+H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) (lift_gen_bind Abbr u1 t2 x0 h x1
+H6))))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H3: ((\forall
+(x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2:
+T).(eq T t3 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (u:
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T (THead (Bind b) u
+(lift (S O) O t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda
+(z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
+T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2)
+(lift h (S x1) z)))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4)))
+(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H5: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (_: (eq T u (lift h x1
+x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r T
+(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift
+h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S
+O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1))
+(plus x1 (S O)) (plus_comm x1 (S O))) in (let H9 \def (eq_ind nat (S x1)
+(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O))
+H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4:
+T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift
+h x1 x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t)
+t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
+T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5:
+T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4
+x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T
+t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O
+x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5
+(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4
+x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0
+H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall
+(x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to (ex2 T (\lambda (t2:
+T).(eq T t3 (lift h x0 t2))) (\lambda (t2: T).(pr0 x t2)))))))).(\lambda (u:
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast)
+u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
+x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift
+h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast)
+x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h
+x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4))))
+(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0
+x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4:
+T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T
+t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4)
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda
+(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4:
+T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat
+Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_epsilon x3 x4 H7 x2))
+t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1
+H3)))))))))) y x H0))))) H))))).
+
+theorem pr0_subst0_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3)))))))))
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1
+v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t:
+T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0)
+(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u0: T).(\lambda
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1:
+((\forall (u2: T).((pr0 u2 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t))
+(\lambda (t: T).(pr0 t u0))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
+(u3: T).(\lambda (H2: (pr0 u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3
+u1 t0)) (\lambda (t0: T).(pr0 t0 u0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u0 t)))) (\lambda (x:
+T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 x u0)).(ex_intro2 T
+(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0
+(THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp x u0
+H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda (v:
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_: (subst0
+(s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T
+(\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t
+t0))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind
+T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t t0)) (ex2
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 t
+(THead k u t0)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3
+x)).(\lambda (H4: (pr0 x t0)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
+(THead k u t3) t)) (\lambda (t: T).(pr0 t (THead k u t0))) (THead k u x)
+(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) x t0 H4 k))))) (H1
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u2:
+T).((pr0 u2 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t)) (\lambda (t:
+T).(pr0 t u0))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u1:
+T).((pr0 u1 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t0 t)) (\lambda
+(t: T).(pr0 t t3))))))).(\lambda (u3: T).(\lambda (H4: (pr0 u3 v)).(ex2_ind T
+(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t t3)) (ex2 T
+(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t
+(THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0
+x)).(\lambda (H6: (pr0 x t3)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t))
+(\lambda (t: T).(pr0 t u0)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1
+t0) t)) (\lambda (t: T).(pr0 t (THead k u0 t3)))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 x0 u0)).(ex_intro2 T (\lambda
+(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t (THead k u0
+t3))) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp x0 u0
+H8 x t3 H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
+T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T
+(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4)))))))))
+(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v
+u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t:
+T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0)
+(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u0: T).(\lambda
+(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1:
+((\forall (u2: T).((pr0 v u2) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t))
+(\lambda (t: T).(pr0 u0 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
+(u3: T).(\lambda (H2: (pr0 v u3)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3
+u1 t0)) (\lambda (t0: T).(pr0 u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3
+(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u0 t) t0))) (\lambda (x:
+T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 u0 x)).(ex_intro2 T
+(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0
+(THead k u0 t) t0)) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp u0
+x H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda
+(v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_:
+(subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to
+(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0
+t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind
+T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 t)) (ex2
+T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0
+(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3
+x)).(\lambda (H4: (pr0 t0 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
+(THead k u t3) t)) (\lambda (t: T).(pr0 (THead k u t0) t)) (THead k u x)
+(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) t0 x H4 k))))) (H1
+u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u2:
+T).((pr0 v u2) \to (ex2 T (\lambda (t: T).(subst0 i0 u2 u1 t)) (\lambda (t:
+T).(pr0 u0 t))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u1:
+T).((pr0 v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t0 t)) (\lambda
+(t: T).(pr0 t3 t))))))).(\lambda (u3: T).(\lambda (H4: (pr0 v u3)).(ex2_ind T
+(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T
+(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead
+k u0 t3) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0
+x)).(\lambda (H6: (pr0 t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t))
+(\lambda (t: T).(pr0 u0 t)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1
+t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) t))) (\lambda (x0: T).(\lambda
+(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 u0 x0)).(ex_intro2 T (\lambda
+(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3)
+t)) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp u0 x0 H8
+t3 x H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))).
+
+theorem pr0_subst0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall
+(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1
+v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t2 w2))))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0
+w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
+nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1
+v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd
+v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0:
+(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2
+w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
+t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
+nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2:
+T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1
+(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5:
+T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4))
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k
+u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T w1 (THead k u2
+t3))) (\lambda (u2: T).(subst0 i v1 u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq
+T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1
+(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1
+(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3)
+(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2)
+(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))
+(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead
+k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda
+(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2
+(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0
+(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3)
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0:
+T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror
+(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x
+t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2
+(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k)
+(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7))))
+H6)) (\lambda (H6: (ex2 T (\lambda (t2: T).(eq T w1 (THead k u1 t2)))
+(\lambda (t2: T).(subst0 (s k i) v1 t3 t2)))).(ex2_ind T (\lambda (t5: T).(eq
+T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0
+w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1
+(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k
+u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind
+(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k
+i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4)
+w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k)))
+(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
+k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2:
+T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2
+(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda
+(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0
+(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead
+k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2))))))
+H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T w1 (THead k u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i) v1 t3 t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda
+(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0
+i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))
+(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda
+(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1
+x0)).(\lambda (H9: (subst0 (s k i) v1 t3 x1)).(eq_ind_r T (THead k x0 x1)
+(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t
+w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x1
+t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2
+t4 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4)
+w2)))) (\lambda (H10: (pr0 x1 t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2:
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0
+x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (H11: (pr0 x0
+u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2:
+T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4)
+w2))) (pr0_comp x0 u2 H11 x1 t4 H10 k))) (\lambda (H11: (ex2 T (\lambda (w2:
+T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda
+(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k
+x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda
+(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k
+x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda
+(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2
+t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i
+H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda
+(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T
+(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or
+(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x:
+T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4
+x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2:
+T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2
+(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k
+x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2))
+(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda
+(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2
+t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4
+i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda
+(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2))
+(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2
+t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2:
+T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0
+x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1)
+(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda
+(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0
+(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))
+(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k
+t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2
+H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4)))))))))))))))))
+(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1
+v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1
+v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2
+w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
+t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda
+(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3))
+w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda
+(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda
+(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat
+Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind
+Abst) u t3) t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1
+u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead
+(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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+x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind
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+H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7))
+(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b:
+B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1:
+T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2)
+\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda
+(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift
+(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda
+(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b)
+u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or
+(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b)
+u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda
+(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
+(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u
+x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda
+(H5: (ex2 T (\lambda (t2: T).(eq T w1 (THead (Bind b) u t2))) (\lambda (t2:
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t2)))).(ex2_ind T (\lambda
+(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b)
+i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6:
+(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift
+(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5)))
+(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda
+(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(eq_ind_r T (THead (Bind
+b) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2))
+(\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S O) O x0)
+(\lambda (t: T).(or (pr0 (THead (Bind b) u t) t4) (ex2 T (\lambda (w2:
+T).(pr0 (THead (Bind b) u t) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))
+(let H10 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n v1 t3
+x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0
+x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind b) u
+(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift
+(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H11: (pr0
+x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H11 u))) (\lambda (H11: (ex2 T
+(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2
+t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda
+(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H12: (pr0 x0
+x1)).(\lambda (H13: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u
+(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift
+(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H12 u) H13))))) H11)) (H2 v1
+x0 i H10 v2 H4))) x H8) w1 H6)))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i)
+(S O) O H7 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i)
+(le_lt_n_Sm O i (le_O_n i)))))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T w1 (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T
+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i)
+v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O
+t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or
+(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O
+x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(eq_ind_r
+T (THead (Bind b) x0 x1) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2:
+T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S
+O) O x) (\lambda (t: T).(or (pr0 (THead (Bind b) x0 t) t4) (ex2 T (\lambda
+(w2: T).(pr0 (THead (Bind b) x0 t) w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n
+v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 T (\lambda (w2:
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind
+b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0
+(lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H12:
+(pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H12 x0))) (\lambda (H12: (ex2 T
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))).(ex2_ind
+T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) (or (pr0
+(THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead
+(Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (x2: T).(\lambda (H13: (pr0 x x2)).(\lambda (H14: (subst0 i v2 t4
+x2)).(or_intror (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda
+(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind b)
+x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) x2 (pr0_zeta
+b H0 x x2 H13 x0) H14))))) H12)) (H2 v1 x i H11 v2 H4))) x1 H9) w1 H6))))
+(subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S O) O H8 (le_S_n (S O) (S i)
+(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n
+i)))))))))))) H5)) (subst0_gen_head (Bind b) v1 u (lift (S O) O t3) w1 i
+H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3
+t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i:
+nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1
+t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i:
+nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) w1)).(\lambda (v2:
+T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u2: T).(eq T w1
+(THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2))) (ex2 T
+(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0
+(s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat
+Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2))
+(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: (ex2 T (\lambda (u2:
+T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3)))
+(\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2:
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x:
+T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda (_: (subst0 i
+v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: T).(or (pr0 t t4)
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))
+(or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 T (\lambda (w2: T).(pr0
+(THead (Flat Cast) x t3) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
+(pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t2:
+T).(eq T w1 (THead (Flat Cast) u t2))) (\lambda (t2: T).(subst0 (s (Flat
+Cast) i) v1 t3 t2)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Flat Cast)
+u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5)) (or (pr0 w1 t4)
+(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
+(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) u x))).(\lambda
+(H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T (THead (Flat Cast) u x)
+(\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda
+(w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2:
+T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or
+(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat
+Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H7: (pr0 x
+t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
+(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
+(pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 x w2))
+(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda
+(w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or
+(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat
+Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
+T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4
+x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
+(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
+(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7)) (H1 v1 x (s
+(Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T w1 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Flat Cast) i) v1 t3 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
+T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2:
+T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0
+x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast)
+i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0
+t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
+(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0
+x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda
+(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0
+(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast)
+x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_epsilon x1 t4 H8
+x0))) (\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2:
+T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1
+w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead
+(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1)
+w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9:
+(pr0 x1 x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror
+(pr0 (THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat
+Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T
+(\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0
+i v2 t4 w2)) x (pr0_epsilon x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat
+Cast) i) H7 v2 H3)) w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1
+i H2))))))))))))) t1 t2 H))).
+
+theorem pr0_confluence__pr0_cong_upsilon_refl:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
+T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to
+(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x)
+\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4))
+t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t)))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda
+(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda
+(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda
+(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead
+(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4
+t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5)
+(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S
+O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind
+b))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_upsilon_cong:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2:
+T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall
+(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5:
+T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t)))))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2
+x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2
+x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda
+(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0))
+(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat
+Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp
+(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat
+Appl)) (Bind b))))))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_upsilon_delta:
+ (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w:
+T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x:
+T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2
+x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to
+((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))))))))
+\def
+ \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2:
+T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2:
+T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2:
+(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2
+x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda
+(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T
+(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead
+(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H
+u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O
+v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2)
+(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind
+Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2:
+T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda
+(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl)
+u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3
+(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8:
+(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))
+(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon
+Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift
+(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O)
+O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl))
+(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9
+(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1
+H5))))))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_upsilon_zeta:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
+T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0
+u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1:
+T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat
+Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) (lift (S O) O x))) t)))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda
+(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda
+(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x:
+T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4:
+(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O
+(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1
+t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat
+Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl)
+(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O)
+O)))))))))))))))).
+
+theorem pr0_confluence__pr0_cong_delta:
+ \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to
+(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall
+(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind
+Abbr) u3 w) t))))))))))))))
+\def
+ \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3
+t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda
+(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3
+x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2:
+T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))
+(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3
+x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2))
+(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w
+w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)))
+(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0
+x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta
+u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4))
+(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))).
+
+theorem pr0_confluence__pr0_upsilon_upsilon:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
+T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1:
+T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to
+(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2
+x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl)
+(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) t)))))))))))))))))))
+\def
+ \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
+(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2
+x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1
+x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1)
+t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0)
+x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat
+Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0)
+(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1
+H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O
+x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S
+O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))).
+
+theorem pr0_confluence__pr0_delta_delta:
+ \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
+(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to
+(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0)
+\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))))))))))))))))
+\def
+ \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2
+t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0:
+(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2:
+(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5
+x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2:
+T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0
+x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2:
+T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w
+x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp
+u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr))))
+(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O
+x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0
+O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7:
+(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr))
+(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x
+H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2:
+T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda
+(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2
+w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1:
+T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0
+w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0
+w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1
+H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda
+(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T
+(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda
+(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t:
+T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1)
+(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2
+x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in
+(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x
+H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda
+(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x
+x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t:
+T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3:
+T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1
+x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta
+u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11))
+(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))
+(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2
+w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9
+(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0
+x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5))
+(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))).
+
+theorem pr0_confluence__pr0_delta_epsilon:
+ \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to
+(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2
+t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2
+t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda
+(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda
+(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))
+(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S
+O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t:
+T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w
+(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
+(le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H3 (ex2 T (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))))))))
+(pr0_gen_lift t4 t3 (S O) O H0)))))))).
+
+theorem pr0_confluence:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0
+t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
+\def
+ \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to
+(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3))
+(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall
+(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0
+v t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2
+t))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2:
+T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 return (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).((eq T t0 t) \to ((eq T t3 t1)
+\to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with
+[(pr0_refl t0) \Rightarrow (\lambda (H2: (eq T t0 t)).(\lambda (H3: (eq T t0
+t1)).(eq_ind T t (\lambda (t: T).((eq T t t1) \to (ex2 T (\lambda (t2:
+T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda (H4: (eq T t
+t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t2: T).(pr0 t1 t2))
+(\lambda (t1: T).(pr0 t2 t1)))) (let H5 \def (match H1 return (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).((eq T t0 t) \to ((eq T t3 t2)
+\to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with
+[(pr0_refl t3) \Rightarrow (\lambda (H5: (eq T t3 t)).(\lambda (H6: (eq T t3
+t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H7: (eq T t
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind T t (\lambda (t: T).(eq
+T t3 t)) H5 t2 H7) in (let H1 \def (eq_ind T t (\lambda (t: T).(eq T t t1))
+H4 t2 H7) in (let H2 \def (eq_ind T t (\lambda (t: T).(eq T t0 t)) H2 t2 H7)
+in (let H3 \def (eq_ind T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H t2 H7)
+in (let H4 \def (eq_ind T t2 (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H3 t1 H1)
+in (eq_ind_r T t1 (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t t0)))) (let H8 \def (eq_ind T t2 (\lambda (t: T).(eq
+T t0 t)) H2 t1 H1) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t:
+T).(pr0 t1 t)) t1 (pr0_refl t1) (pr0_refl t1))) t2 H1)))))) t (sym_eq T t t2
+H7))) t3 (sym_eq T t3 t H5) H6))) | (pr0_comp u1 u2 H4 t3 t4 H5 k)
+\Rightarrow (\lambda (H6: (eq T (THead k u1 t3) t)).(\lambda (H7: (eq T
+(THead k u2 t4) t2)).(eq_ind T (THead k u1 t3) (\lambda (_: T).((eq T (THead
+k u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead
+k u2 t4) t2)).(eq_ind T (THead k u2 t4) (\lambda (t: T).((pr0 u1 u2) \to
+((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t
+t0)))))) (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t3 t4)).(let H0 \def
+(eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead k u1 t3) H6) in (eq_ind
+T (THead k u1 t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0))
+(\lambda (t0: T).(pr0 (THead k u2 t4) t0)))) (let H1 \def (eq_ind_r T t
+(\lambda (t: T).(eq T t0 t)) H2 (THead k u1 t3) H6) in (let H2 \def (eq_ind_r
+T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v
+t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k u1 t3) H6) in (ex_intro2 T
+(\lambda (t: T).(pr0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t4)
+t)) (THead k u2 t4) (pr0_comp u1 u2 H9 t3 t4 H10 k) (pr0_refl (THead k u2
+t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_beta u v1 v2 H4 t3 t4 H5)
+\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) t)).(\lambda (H7: (eq T (THead (Bind Abbr) v2 t4) t2)).(eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v2 t4) t2) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead
+(Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t:
+T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t t0)))))) (\lambda (H9: (pr0 v1 v2)).(\lambda (H10:
+(pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t4) t0)))) (let H1
+\def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6)
+in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t4) t)) (THead (Bind
+Abbr) v2 t4) (pr0_beta u v1 v2 H9 t3 t4 H10) (pr0_refl (THead (Bind Abbr) v2
+t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_upsilon b H4 v1 v2 H5 u1 u2
+H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead
+(Bind b) u1 t3)) t)).(\lambda (H9: (eq T (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to
+((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H10: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t: T).((not (eq B b
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda
+(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H11: (not
+(eq B b Abst))).(\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 u1
+u2)).(\lambda (H14: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T t t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (t: T).(ex2
+T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) t0)))) (let H1 \def (eq_ind_r T t
+(\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+H8) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t)
+\to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in
+(pr0_confluence__pr0_cong_upsilon_refl b H11 u1 u2 H13 t3 t4 H14 v1 v2 v2 H12
+(pr0_refl v2)))) t1 H0)))))) t2 H10)) t H8 H9 H4 H5 H6 H7))) | (pr0_delta u1
+u2 H4 t3 t4 H5 w H6) \Rightarrow (\lambda (H7: (eq T (THead (Bind Abbr) u1
+t3) t)).(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead
+(Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to
+((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda
+(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H9: (eq T
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t:
+T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda
+(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H10: (pr0 u1
+u2)).(\lambda (H11: (pr0 t3 t4)).(\lambda (H12: (subst0 O u2 t4 w)).(let H0
+\def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Bind Abbr) u1 t3)
+H7) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t: T).(ex2 T (\lambda
+(t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))))
+(let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Bind Abbr)
+u1 t3) H7) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v:
+T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v
+t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2
+t0)))))))))) H (THead (Bind Abbr) u1 t3) H7) in (ex_intro2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u1 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+u2 w) t)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H10 t3 t4 H11 w H12)
+(pr0_refl (THead (Bind Abbr) u2 w))))) t1 H0))))) t2 H9)) t H7 H8 H4 H5 H6)))
+| (pr0_zeta b H4 t3 t4 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Bind b)
+u (lift (S O) O t3)) t)).(\lambda (H7: (eq T t4 t2)).(eq_ind T (THead (Bind
+b) u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t2) \to ((not (eq B b
+Abst)) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda
+(t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T t4 t2)).(eq_ind T t2 (\lambda
+(t: T).((not (eq B b Abst)) \to ((pr0 t3 t) \to (ex2 T (\lambda (t0: T).(pr0
+t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (not (eq B b
+Abst))).(\lambda (H10: (pr0 t3 t2)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T t t1)) H4 (THead (Bind b) u (lift (S O) O t3)) H6) in (eq_ind T
+(THead (Bind b) u (lift (S O) O t3)) (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def (eq_ind_r T t
+(\lambda (t: T).(eq T t0 t)) H2 (THead (Bind b) u (lift (S O) O t3)) H6) in
+(let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Bind b) u (lift (S O) O t3)) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Bind b) u (lift (S O) O t3)) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_zeta b
+H9 t3 t2 H10 u) (pr0_refl t2)))) t1 H0)))) t4 (sym_eq T t4 t2 H8))) t H6 H7
+H4 H5))) | (pr0_epsilon t3 t4 H4 u) \Rightarrow (\lambda (H5: (eq T (THead
+(Flat Cast) u t3) t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T (THead (Flat
+Cast) u t3) (\lambda (_: T).((eq T t4 t2) \to ((pr0 t3 t4) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H7:
+(eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t3 t) \to (ex2 T (\lambda
+(t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (pr0 t3
+t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat
+Cast) u t3) H5) in (eq_ind T (THead (Flat Cast) u t3) (\lambda (t: T).(ex2 T
+(\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def
+(eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Cast) u t3) H5) in
+(let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Flat Cast) u t3) H5) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast)
+u t3) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_epsilon t3 t2 H8 u) (pr0_refl
+t2)))) t1 H0))) t4 (sym_eq T t4 t2 H7))) t H5 H6 H4)))]) in (H5 (refl_equal T
+t) (refl_equal T t2))) t (sym_eq T t t1 H4))) t0 (sym_eq T t0 t H2) H3))) |
+(pr0_comp u1 u2 H2 t0 t3 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1
+t0) t)).(\lambda (H5: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u1 t0)
+(\lambda (_: T).((eq T (THead k u2 t3) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3)
+\to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))))
+(\lambda (H6: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u2 t3) (\lambda
+(t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1))
+(\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda (H8:
+(pr0 t0 t3)).(let H9 \def (match H1 return (\lambda (t0: T).(\lambda (t1:
+T).(\lambda (_: (pr0 t0 t1)).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T
+(\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t))))))))
+with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq
+T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10:
+(eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead
+k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t
+(\lambda (t: T).(eq T t t2)) H10 (THead k u1 t0) H4) in (eq_ind T (THead k u1
+t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0))
+(\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq
+T t4 t)) H6 (THead k u1 t0) H4) in (let H2 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead k u1 t0) H4) in (ex_intro2 T (\lambda (t:
+T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k u1 t0) t)) (THead k
+u2 t3) (pr0_refl (THead k u2 t3)) (pr0_comp u1 u2 H7 t0 t3 H8 k)))) t2 H0)) t
+(sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u0 u3 H6 t4 t5
+H7 k0) \Rightarrow (\lambda (H9: (eq T (THead k0 u0 t4) t)).(\lambda (H10:
+(eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u0 t4) (\lambda (_: T).((eq T
+(THead k0 u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda
+(H11: (eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u3 t5) (\lambda (t:
+T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2
+t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 u0
+u3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T (THead k u1 t0) t)) H4 (THead k0 u0 t4) H9) in (let H1 \def (match
+H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k0 u0 t4))
+\to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0
+(THead k0 u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T
+(THead k u1 t0) (THead k0 u0 t4))).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead k0
+u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
+| (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead k0 u0 t4) H0) in
+((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u1 t0) (THead k0 u0 t4) H0) in (eq_ind K k0
+(\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t:
+T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t))))))
+(\lambda (H10: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: T).(pr0
+(THead k0 u3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda
+(_: T).(ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead k0 u3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead k0 u0 t4) H9) in (let H5 \def (eq_ind T t0
+(\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda
+(t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t))
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t))
+(\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0
+u2 x)).(\lambda (H8: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t))
+(\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t))
+(\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x0: T).(\lambda (H9:
+(pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0
+(THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)) (THead k0 x
+x0) (pr0_comp u2 x H7 t3 x0 H9 k0) (pr0_comp u3 x H8 t5 x0 H12 k0))))) (H4 t4
+(tlt_head_dx k0 u0 t4) t3 H5 t5 H13))))) (H4 u0 (tlt_head_sx k0 u0 t4) u2 H6
+u3 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k
+k0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead k0 u0 t4))))))) t2 H11)) t
+H9 H10 H6 H7))) | (pr0_beta u v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H9:
+(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) t)).(\lambda (H10: (eq
+T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0
+v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0))
+(\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2
+t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to
+((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda
+(t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 t4
+t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4
+(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl)
+v1 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2
+t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead
+(Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t]))
+(THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let
+H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t4)) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T u1 v1) \to ((eq T t0
+(THead (Bind Abst) u t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) (\lambda (H10: (eq T
+u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind Abst) u t4)) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) (\lambda (H11: (eq T t0 (THead
+(Bind Abst) u t4))).(eq_ind T (THead (Bind Abst) u t4) (\lambda (_: T).(ex2 T
+(\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0
+(THead (Bind Abbr) v2 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9)
+in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind
+Abst) u t4) H11) in (let H6 \def (match H5 return (\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abst) u t4)) \to ((eq
+T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1))
+(\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1)))))))) with [(pr0_refl t)
+\Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u t4))).(\lambda (H5:
+(eq T t t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: T).((eq T t0
+t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda
+(t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1))))) (\lambda (H6: (eq T (THead
+(Bind Abst) u t4) t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0:
+T).(ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t0) t1)) (\lambda (t1:
+T).(pr0 (THead (Bind Abbr) v2 t5) t1)))) (let H1 \def (eq_ind T u1 (\lambda
+(t: T).(pr0 t u2)) H7 v1 H10) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0))
+(\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl)
+u2 (THead (Bind Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2
+t5) t0))) (\lambda (x: T).(\lambda (H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2
+x)).(ex_intro2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind
+Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)) (THead
+(Bind Abbr) x t5) (pr0_beta u u2 x H2 t4 t5 H13) (pr0_comp v2 x H3 t5 t5
+(pr0_refl t5) (Bind Abbr)))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead
+(Bind Abst) u t4)) u2 H1 v2 H12))) t3 H6)) t (sym_eq T t (THead (Bind Abst) u
+t4) H0) H5))) | (pr0_comp u0 u3 H0 t1 t2 H4 k0) \Rightarrow (\lambda (H5: (eq
+T (THead k0 u0 t1) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead k0
+u3 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1
+| (THead _ _ t) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4)
+H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t _) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in
+((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _)
+\Rightarrow k])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in (eq_ind K
+(Bind Abst) (\lambda (k: K).((eq T u0 u) \to ((eq T t1 t4) \to ((eq T (THead
+k u3 t2) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq
+T t1 t4) \to ((eq T (THead (Bind Abst) u3 t2) t3) \to ((pr0 t u3) \to ((pr0
+t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0))
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) (\lambda (H9: (eq
+T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t2) t3)
+\to ((pr0 u u3) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))
+(\lambda (H11: (eq T (THead (Bind Abst) u3 t2) t3)).(eq_ind T (THead (Bind
+Abst) u3 t2) (\lambda (t: T).((pr0 u u3) \to ((pr0 t4 t2) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) v2 t5) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H15: (pr0 t4
+t2)).(let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H10) in
+(ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) (\lambda (x: T).(\lambda
+(H10: (pr0 u2 x)).(\lambda (H12: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0
+t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat
+Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind
+Abbr) v2 t5) t))) (\lambda (x0: T).(\lambda (H13: (pr0 t2 x0)).(\lambda (H16:
+(pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))
+(THead (Bind Abbr) x x0) (pr0_beta u3 u2 x H10 t2 x0 H13) (pr0_comp v2 x H12
+t5 x0 H16 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind Abst) u t4) t4
+(THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (tlt_head_dx (Bind Abst) u
+t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t4))) t2 H15 t5 H13)))))
+(H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t4)) u2 H7 v2
+H12))))) t3 H11)) t1 (sym_eq T t1 t4 H9))) u0 (sym_eq T u0 u H6))) k0 (sym_eq
+K k0 (Bind Abst) H3))) H2)) H1)) H8 H0 H4))) | (pr0_beta u0 v0 v3 H0 t1 t2
+H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat Appl) v0 (THead (Bind Abst)
+u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead (Bind Abbr) v3
+t2) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0
+t1)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H5) in (False_ind
+((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) \to ((pr0 t1 t2) \to
+(ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) H1)) H8 H0 H4))) | (pr0_upsilon b
+H0 v0 v3 H4 u0 u3 H5 t1 t2 H8) \Rightarrow (\lambda (H11: (eq T (THead (Flat
+Appl) v0 (THead (Bind b) u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H12:
+(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let
+H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t1)) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u t4) H11) in (False_ind ((eq T (THead (Bind
+b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b Abst))
+\to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+v2 t5) t)))))))) H1)) H12 H0 H4 H5 H8))) | (pr0_delta u0 u3 H0 t1 t2 H4 w H5)
+\Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u0 t1) (THead (Bind Abst)
+u t4))).(\lambda (H11: (eq T (THead (Bind Abbr) u3 w) t3)).((let H1 \def
+(eq_ind T (THead (Bind Abbr) u0 t1) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (THead (Bind Abst) u t4) H8) in (False_ind ((eq T
+(THead (Bind Abbr) u3 w) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to ((subst0 O
+u3 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))) H1)) H11 H0 H4 H5)))
+| (pr0_zeta b H0 t1 t2 H4 u0) \Rightarrow (\lambda (H5: (eq T (THead (Bind b)
+u0 (lift (S O) O t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T t2
+t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d:
+nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f
+d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t1) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
+(d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) |
+(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f
+d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t1) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O
+t1)) (THead (Bind Abst) u t4) H5) in ((let H2 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef
+_) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift
+(S O) O t1)) (THead (Bind Abst) u t4) H5) in ((let H3 \def (f_equal T B
+(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in
+(eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t1) t4)
+\to ((eq T t2 t3) \to ((not (eq B b0 Abst)) \to ((pr0 t1 t2) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u
+(\lambda (_: T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq
+B Abst Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))))
+(\lambda (H9: (eq T (lift (S O) O t1) t4)).(eq_ind T (lift (S O) O t1)
+(\lambda (_: T).((eq T t2 t3) \to ((not (eq B Abst Abst)) \to ((pr0 t1 t2)
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) (\lambda (H7: (eq T t2
+t3)).(eq_ind T t3 (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t1 t) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) (\lambda (H11: (not (eq B Abst
+Abst))).(\lambda (_: (pr0 t1 t3)).(let H10 \def (match (H11 (refl_equal B
+Abst)) return (\lambda (_: False).(ex2 T (\lambda (t: T).(pr0 (THead (Flat
+Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))) with
+[]) in H10))) t2 (sym_eq T t2 t3 H7))) t4 H9)) u0 (sym_eq T u0 u H6))) b
+(sym_eq B b Abst H3))) H2)) H1)) H8 H0 H4))) | (pr0_epsilon t1 t2 H0 u0)
+\Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u0 t1) (THead (Bind Abst)
+u t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat
+Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H4) in
+(False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5)
+t))))) H1)) H5 H0)))]) in (H6 (refl_equal T (THead (Bind Abst) u t4))
+(refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind Abst) u t4) H11))) u1
+(sym_eq T u1 v1 H10))) k (sym_eq K k (Flat Appl) H3))) H2)) H1)))]) in (H1
+(refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))))))) t2 H11))
+t H9 H10 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u0 u3 H8 t4 t5 H9) \Rightarrow
+(\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t4))
+t)).(\lambda (H11: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t4))
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
+((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda
+(t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to
+((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda
+(H13: (not (eq B b Abst))).(\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 u0
+u3)).(\lambda (H16: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 t4))
+H10) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal
+\Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Appl) v1 (THead
+(Bind b) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead
+(Bind b) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat
+Appl) v1 (THead (Bind b) u0 t4)) H0) in ((let H3 \def (f_equal T K (\lambda
+(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
+(TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0)
+(THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (eq_ind K (Flat Appl)
+(\lambda (k: K).((eq T u1 v1) \to ((eq T t0 (THead (Bind b) u0 t4)) \to (ex2
+T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))) (\lambda (H11: (eq T
+u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind b) u0 t4)) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))
+(\lambda (H12: (eq T t0 (THead (Bind b) u0 t4))).(eq_ind T (THead (Bind b) u0
+t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3)
+t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
+O v2) t5)) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v:
+T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v
+t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2
+t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H10) in (let H5
+\def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind b) u0 t4) H12)
+in (let H6 \def (match H5 return (\lambda (t: T).(\lambda (t0: T).(\lambda
+(_: (pr0 t t0)).((eq T t (THead (Bind b) u0 t4)) \to ((eq T t0 t3) \to (ex2 T
+(\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1)))))))) with
+[(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind b) u0
+t4))).(\lambda (H5: (eq T t t3)).(eq_ind T (THead (Bind b) u0 t4) (\lambda
+(t0: T).((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2
+t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v2) t5)) t1))))) (\lambda (H6: (eq T (THead (Bind b) u0 t4) t3)).(eq_ind
+T (THead (Bind b) u0 t4) (\lambda (t0: T).(ex2 T (\lambda (t1: T).(pr0 (THead
+(Flat Appl) u2 t0) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat
+Appl) (lift (S O) O v2) t5)) t1)))) (let H1 \def (eq_ind T u1 (\lambda (t:
+T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0))
+(\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl)
+u2 (THead (Bind b) u0 t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t5)) t0))) (\lambda (x: T).(\lambda (H2:
+(pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl
+b H13 u0 u3 H15 t4 t5 H16 u2 v2 x H2 H3)))) (H4 v1 (tlt_head_sx (Flat Appl)
+v1 (THead (Bind b) u0 t4)) u2 H1 v2 H14))) t3 H6)) t (sym_eq T t (THead (Bind
+b) u0 t4) H0) H5))) | (pr0_comp u4 u5 H0 t1 t2 H4 k0) \Rightarrow (\lambda
+(H5: (eq T (THead k0 u4 t1) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T
+(THead k0 u5 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _)
+\Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind
+b) u0 t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4
+| (THead _ t _) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5)
+in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_:
+T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _
+_) \Rightarrow k])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in (eq_ind K
+(Bind b) (\lambda (k: K).((eq T u4 u0) \to ((eq T t1 t4) \to ((eq T (THead k
+u5 t2) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t))))))))) (\lambda (H6: (eq T u4
+u0)).(eq_ind T u0 (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind b) u5
+t2) t3) \to ((pr0 t u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H12: (eq T t1
+t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind b) u5 t2) t3) \to ((pr0
+u0 u5) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2
+t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v2) t5)) t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t2)
+t3)).(eq_ind T (THead (Bind b) u5 t2) (\lambda (t: T).((pr0 u0 u5) \to ((pr0
+t4 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0))
+(\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t0)))))) (\lambda (H17: (pr0 u0 u5)).(\lambda (H18: (pr0 t4 t2)).(let H7
+\def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T
+(\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))
+(\lambda (x: T).(\lambda (H9: (pr0 u2 x)).(\lambda (H11: (pr0 v2 x)).(ex2_ind
+T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))
+(\lambda (x0: T).(\lambda (H14: (pr0 t2 x0)).(\lambda (H16: (pr0 t5
+x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 t)) (ex2
+T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t))
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda (H19: (pr0 u3
+x1)).(pr0_confluence__pr0_cong_upsilon_cong b H13 u2 v2 x H9 H11 t2 t5 x0 H14
+H16 u5 u3 x1 H15 H19)))) (H4 u0 (tlt_trans (THead (Bind b) u0 t4) u0 (THead
+(Flat Appl) v1 (THead (Bind b) u0 t4)) (tlt_head_sx (Bind b) u0 t4)
+(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t4))) u5 H17 u3 H15))))) (H4
+t4 (tlt_trans (THead (Bind b) u0 t4) t4 (THead (Flat Appl) v1 (THead (Bind b)
+u0 t4)) (tlt_head_dx (Bind b) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind
+b) u0 t4))) t2 H18 t5 H16))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead
+(Bind b) u0 t4)) u2 H7 v2 H14))))) t3 H8)) t1 (sym_eq T t1 t4 H12))) u4
+(sym_eq T u4 u0 H6))) k0 (sym_eq K k0 (Bind b) H3))) H2)) H1)) H10 H0 H4))) |
+(pr0_beta u v0 v3 H0 t1 t2 H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat
+Appl) v0 (THead (Bind Abst) u t1)) (THead (Bind b) u0 t4))).(\lambda (H10:
+(eq T (THead (Bind Abbr) v3 t2) t3)).((let H1 \def (eq_ind T (THead (Flat
+Appl) v0 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0
+t4) H5) in (False_ind ((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3)
+\to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3)
+t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t)))))) H1)) H10 H0 H4))) | (pr0_upsilon b0 H0 v0 v3 H4 u4 u5 H5 t1
+t2 H10) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind
+b0) u4 t1)) (THead (Bind b) u0 t4))).(\lambda (H14: (eq T (THead (Bind b0) u5
+(THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let H1 \def (eq_ind T (THead
+(Flat Appl) v0 (THead (Bind b0) u4 t1)) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind b) u0 t4) H13) in (False_ind ((eq T (THead (Bind b0) u5 (THead
+(Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b0 Abst)) \to ((pr0 v0
+v3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead
+(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat
+Appl) (lift (S O) O v2) t5)) t)))))))) H1)) H14 H0 H4 H5 H10))) | (pr0_delta
+u4 u5 H0 t1 t2 H4 w H5) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr)
+u4 t1) (THead (Bind b) u0 t4))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w)
+t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _
+t) \Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in
+((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _)
+\Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in
+((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: T).B)
+with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0
+t4) H10) in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u0) \to ((eq T t1 t4)
+\to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to
+((subst0 O u5 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3)
+t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t)))))))))) (\lambda (H6: (eq T u4 u0)).(eq_ind T u0 (\lambda (t:
+T).((eq T t1 t4) \to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 t u5) \to
+((pr0 t1 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t0))))))))) (\lambda (H8: (eq T t1
+t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t3) \to
+((pr0 u0 u5) \to ((pr0 t t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H18:
+(eq T (THead (Bind Abbr) u5 w) t3)).(eq_ind T (THead (Bind Abbr) u5 w)
+(\lambda (t: T).((pr0 u0 u5) \to ((pr0 t4 t2) \to ((subst0 O u5 t2 w) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5))
+t0))))))) (\lambda (H19: (pr0 u0 u5)).(\lambda (H20: (pr0 t4 t2)).(\lambda
+(H21: (subst0 O u5 t2 w)).(let H9 \def (eq_ind_r B b (\lambda (b: B).(\forall
+(v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (\forall
+(t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t:
+T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))) H4 Abbr H3) in (let H12
+\def (eq_ind_r B b (\lambda (b: B).(eq T t0 (THead (Bind b) u0 t4))) H12 Abbr
+H3) in (let H13 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13
+Abbr H3) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11)
+in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t))) (\lambda (x: T).(\lambda (H11: (pr0 u2 x)).(\lambda (H14: (pr0
+v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t))
+(ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w))
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O)
+O v2) t5)) t))) (\lambda (x0: T).(\lambda (H16: (pr0 t2 x0)).(\lambda (H22:
+(pr0 t5 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3
+t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5
+w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift
+(S O) O v2) t5)) t))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda
+(H23: (pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_delta H13 u5 t2 w H21 u2
+v2 x H11 H14 t5 x0 H16 H22 u3 x1 H15 H23)))) (H9 u0 (tlt_trans (THead (Bind
+Abbr) u0 t4) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_sx
+(Bind Abbr) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) u5
+H19 u3 H15))))) (H9 t4 (tlt_trans (THead (Bind Abbr) u0 t4) t4 (THead (Flat
+Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_dx (Bind Abbr) u0 t4)
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) t2 H20 t5 H16)))))
+(H9 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) u2 H7 v2
+H14))))))))) t3 H18)) t1 (sym_eq T t1 t4 H8))) u4 (sym_eq T u4 u0 H6))) b
+H3)) H2)) H1)) H17 H0 H4 H5))) | (pr0_zeta b0 H0 t1 t2 H4 u) \Rightarrow
+(\lambda (H5: (eq T (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0
+t4))).(\lambda (H10: (eq T t2 t3)).((let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x (S O))) O t1) | (TLRef _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x (S O))) O t1) | (THead _ _ t) \Rightarrow t]))
+(THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in ((let H2
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4)
+H5) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead
+k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O
+t1)) (THead (Bind b) u0 t4) H5) in (eq_ind B b (\lambda (b1: B).((eq T u u0)
+\to ((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b1 Abst))
+\to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3)
+t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t))))))))) (\lambda (H6: (eq T u u0)).(eq_ind T u0 (\lambda (_:
+T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b Abst))
+\to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3)
+t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
+O v2) t5)) t0)))))))) (\lambda (H13: (eq T (lift (S O) O t1) t4)).(eq_ind T
+(lift (S O) O t1) (\lambda (_: T).((eq T t2 t3) \to ((not (eq B b Abst)) \to
+((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0))
+(\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t0))))))) (\lambda (H8: (eq T t2 t3)).(eq_ind T t3 (\lambda (t: T).((not
+(eq B b Abst)) \to ((pr0 t1 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) t5)) t0)))))) (\lambda (H17: (not (eq B b Abst))).(\lambda
+(H18: (pr0 t1 t3)).(let H9 \def (eq_ind_r T t4 (\lambda (t: T).(\forall (v:
+T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t))) \to (\forall (t1:
+T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H4 (lift (S O) O t1)
+H13) in (let H12 \def (eq_ind_r T t4 (\lambda (t: T).(eq T t0 (THead (Bind b)
+u0 t))) H12 (lift (S O) O t1) H13) in (let H16 \def (eq_ind_r T t4 (\lambda
+(t: T).(pr0 t t5)) H16 (lift (S O) O t1) H13) in (ex2_ind T (\lambda (t3:
+T).(eq T t5 (lift (S O) O t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda
+(t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda
+(H19: (eq T t5 (lift (S O) O x))).(\lambda (H20: (pr0 t1 x)).(eq_ind_r T
+(lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) t)) t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t
+u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t:
+T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t))
+(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+(lift (S O) O x))) t))) (\lambda (x0: T).(\lambda (H11: (pr0 u2 x0)).(\lambda
+(H14: (pr0 v2 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0
+t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t:
+T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
+x))) t))) (\lambda (x1: T).(\lambda (H21: (pr0 x x1)).(\lambda (H22: (pr0 t3
+x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H17 u0 u3 H15 u2 v2 x0 H11 H14
+x t3 x1 H21 H22)))) (H9 t1 (tlt_trans (THead (Bind b) u0 (lift (S O) O t1))
+t1 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1))) (lift_tlt_dx
+(Bind b) u0 t1 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift
+(S O) O t1)))) x H20 t3 H18))))) (H9 v1 (tlt_head_sx (Flat Appl) v1 (THead
+(Bind b) u0 (lift (S O) O t1))) u2 H7 v2 H14))) t5 H19)))) (pr0_gen_lift t1
+t5 (S O) O H16))))))) t2 (sym_eq T t2 t3 H8))) t4 H13)) u (sym_eq T u u0
+H6))) b0 (sym_eq B b0 b H3))) H2)) H1)) H10 H0 H4))) | (pr0_epsilon t1 t2 H0
+u) \Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u t1) (THead (Bind b)
+u0 t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat
+Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H4) in
+(False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0
+(THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t))))) H1)) H5 H0)))]) in (H6 (refl_equal
+T (THead (Bind b) u0 t4)) (refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind
+b) u0 t4) H12))) u1 (sym_eq T u1 v1 H11))) k (sym_eq K k (Flat Appl) H3)))
+H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0
+t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u0 u3 H6 t4 t5 H7
+w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda
+(H10: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t4)
+(\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to
+((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead
+(Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t:
+T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda
+(H12: (pr0 u0 u3)).(\lambda (H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u3 t5
+w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4
+(THead (Bind Abbr) u0 t4) H9) in (let H1 \def (match H0 return (\lambda (t:
+T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u3 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead
+k u1 t0) (THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead
+(Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Bind
+Abbr) u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Bind
+Abbr) u0 t4) H0) in (eq_ind K (Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to
+((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u3 w) t)))))) (\lambda (H10: (eq T u1
+u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u3 w) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) u3 w) t0)))) (let H4 \def (eq_ind_r T t (\lambda
+(t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to
+(\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H9) in (let
+H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def
+(eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda
+(t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0
+(THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w)
+t))) (\lambda (x: T).(\lambda (H7: (pr0 u2 x)).(\lambda (H8: (pr0 u3
+x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) u3 w) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3
+x0)).(\lambda (H12: (pr0 t5 x0)).(pr0_confluence__pr0_cong_delta u3 t5 w H14
+u2 x H7 H8 t3 x0 H9 H12)))) (H4 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H5 t5
+H13))))) (H4 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H6 u3 H12))))) t0 (sym_eq
+T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k (Bind Abbr) H3))) H2))
+H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H11)) t H9
+H10 H6 H7 H8))) | (pr0_zeta b H6 t4 t5 H7 u) \Rightarrow (\lambda (H9: (eq T
+(THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind
+T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to
+((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T t5
+t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2
+t0)))))) (\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 t4 t2)).(let
+H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead
+(Bind b) u (lift (S O) O t4)) H9) in (let H1 \def (match H0 return (\lambda
+(t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b) u (lift (S O) O
+t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k
+u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H2 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t]))
+(THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H3 \def
+(f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort
+_) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind K (Bind
+b) (\lambda (k: K).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T
+(\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t))))))
+(\lambda (H10: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O)
+O t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t3) t0)) (\lambda
+(t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T t0 (lift (S O) O t4))).(eq_ind T
+(lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+b) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t
+(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t4))
+H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (lift (S O) O
+t4) H11) in (ex2_ind T (\lambda (t2: T).(eq T t3 (lift (S O) O t2))) (\lambda
+(t2: T).(pr0 t4 t2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 t3) t))
+(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H6: (eq T t3 (lift (S
+O) O x))).(\lambda (H8: (pr0 t4 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u
+H10) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 t))
+(ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) (\lambda
+(t: T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (H9: (pr0 x x0)).(\lambda (H13:
+(pr0 t2 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O)
+O x)) t)) (\lambda (t: T).(pr0 t2 t)) x0 (pr0_zeta b H12 x x0 H9 u2) H13))))
+(H4 t4 (lift_tlt_dx (Bind b) u t4 (S O) O) x H8 t2 H13))) t3 H6))))
+(pr0_gen_lift t4 t3 (S O) O H5)))) t0 (sym_eq T t0 (lift (S O) O t4) H11)))
+u1 (sym_eq T u1 u H10))) k (sym_eq K k (Bind b) H3))) H2)) H1)))]) in (H1
+(refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2
+H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u) \Rightarrow (\lambda (H9:
+(eq T (THead (Flat Cast) u t4) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T
+(THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2
+t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4
+t) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0:
+T).(pr0 t2 t0))))) (\lambda (H12: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t
+(\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Cast) u t4) H9) in
+(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T
+t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3)
+t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda
+(H0: (eq T (THead k u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (f_equal T
+T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H2 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t]))
+(THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H3 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u1 t0) (THead (Flat Cast) u t4) H0) in (eq_ind K (Flat Cast)
+(\lambda (k: K).((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t:
+T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10:
+(eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))
+(\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda
+(t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))
+(let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Flat Cast) u t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t
+t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u
+H10) in (ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t2 t))
+(ex2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) (\lambda (t:
+T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t3 x)).(\lambda (H8: (pr0
+t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t))
+(\lambda (t: T).(pr0 t2 t)) x (pr0_epsilon t3 x H7 u2) H8)))) (H4 t4
+(tlt_head_dx (Flat Cast) u t4) t3 H5 t2 H12))))) t0 (sym_eq T t0 t4 H11))) u1
+(sym_eq T u1 u H10))) k (sym_eq K k (Flat Cast) H3))) H2)) H1)))]) in (H1
+(refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9
+H10 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2
+H3))) | (pr0_beta u v1 v2 H2 t0 t3 H3) \Rightarrow (\lambda (H4: (eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t0)) t)).(\lambda (H5: (eq T (THead (Bind
+Abbr) v2 t3) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0))
+(\lambda (_: T).((eq T (THead (Bind Abbr) v2 t3) t1) \to ((pr0 v1 v2) \to
+((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0
+t2 t1))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t3) t1)).(eq_ind T
+(THead (Bind Abbr) v2 t3) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t0 t3) \to
+(ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1))))))
+(\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t0 t3)).(let H9 \def (match H1
+return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (pr0 t0 t1)).((eq T t0
+t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2
+t3) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow
+(\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda
+(t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2
+t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind
+T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3)
+t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T t t2)) H10 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (t:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0:
+T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6
+(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (let H2 \def (eq_ind_r
+T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v
+t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) H4) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2
+t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t0))
+t)) (THead (Bind Abbr) v2 t3) (pr0_refl (THead (Bind Abbr) v2 t3)) (pr0_beta
+u v1 v2 H7 t0 t3 H8)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6)
+H9))) | (pr0_comp u1 u2 H6 t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead
+k u1 t4) t)).(\lambda (H10: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1
+t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4
+t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda
+(t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k u2 t5) t2)).(eq_ind T
+(THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t
+t0)))))) (\lambda (H12: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def
+(eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst)
+u t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 return (\lambda
+(t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead k u2
+t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4))).(let H1 \def (f_equal T
+T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind
+Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind Abst) u t0)) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda
+(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
+(TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl)
+v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in ((let H3 \def (f_equal T
+K (\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _)
+\Rightarrow k])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1
+t4) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T v1 u1) \to ((eq T
+(THead (Bind Abst) u t0) t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind
+Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H10:
+(eq T v1 u1)).(eq_ind T u1 (\lambda (_: T).((eq T (THead (Bind Abst) u t0)
+t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda
+(t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))) (\lambda (H11: (eq T (THead
+(Bind Abst) u t0) t4)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Flat Appl) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda
+(k: K).(eq T (THead k u1 t4) t)) H9 (Flat Appl) H3) in (let H5 \def (eq_ind_r
+T t4 (\lambda (t: T).(pr0 t t5)) H13 (THead (Bind Abst) u t0) H11) in (let H6
+\def (match H5 return (\lambda (t: T).(\lambda (t1: T).(\lambda (_: (pr0 t
+t1)).((eq T t (THead (Bind Abst) u t0)) \to ((eq T t1 t5) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u2 t5) t0)))))))) with [(pr0_refl t2) \Rightarrow (\lambda (H0:
+(eq T t2 (THead (Bind Abst) u t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T
+(THead (Bind Abst) u t0) (\lambda (t: T).((eq T t t5) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u2 t5) t0))))) (\lambda (H2: (eq T (THead (Bind Abst) u t0)
+t5)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u2 t) t0)))) (let H3 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead
+(Flat Appl) u1 t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H4 \def
+(eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1:
+T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1
+(THead (Bind Abst) u t0)) H3) in (let H5 \def (eq_ind T v1 (\lambda (t:
+T).(pr0 t v2)) H7 u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda
+(t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t))
+(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t)))
+(\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u2
+x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda
+(t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t)) (THead (Bind
+Abbr) x t3) (pr0_comp v2 x H6 t3 t3 (pr0_refl t3) (Bind Abbr)) (pr0_beta u u2
+x H7 t0 t3 H8))))) (H4 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u
+t0)) v2 H5 u2 H12))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind Abst) u t0) H0)
+H1))) | (pr0_comp u0 u3 H0 t2 t6 H1 k) \Rightarrow (\lambda (H5: (eq T (THead
+k u0 t2) (THead (Bind Abst) u t0))).(\lambda (H13: (eq T (THead k u3 t6)
+t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _
+t) \Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H3
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _)
+\Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H6
+\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in (eq_ind K
+(Bind Abst) (\lambda (k0: K).((eq T u0 u) \to ((eq T t2 t0) \to ((eq T (THead
+k0 u3 t6) t5) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl)
+u2 t5) t))))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq
+T t2 t0) \to ((eq T (THead (Bind Abst) u3 t6) t5) \to ((pr0 t u3) \to ((pr0
+t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0))
+(\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H14: (eq
+T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t6) t5)
+\to ((pr0 u u3) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))
+(\lambda (H15: (eq T (THead (Bind Abst) u3 t6) t5)).(eq_ind T (THead (Bind
+Abst) u3 t6) (\lambda (t: T).((pr0 u u3) \to ((pr0 t0 t6) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u2 t) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H17: (pr0 t0
+t6)).(let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u1
+t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H11 \def (eq_ind_r T t
+(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 (THead (Bind
+Abst) u t0)) H4) in (let H7 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7
+u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u2 t))
+(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t))) (\lambda (x:
+T).(\lambda (H10: (pr0 v2 x)).(\lambda (H12: (pr0 u2 x)).(ex2_ind T (\lambda
+(t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0
+(THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2
+(THead (Bind Abst) u3 t6)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t6
+x0)).(\lambda (H18: (pr0 t3 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind Abst) u3 t6)) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H10 t3 x0 H18
+(Bind Abbr)) (pr0_beta u3 u2 x H12 t6 x0 H8))))) (H11 t0 (tlt_trans (THead
+(Bind Abst) u t0) t0 (THead (Flat Appl) u1 (THead (Bind Abst) u t0))
+(tlt_head_dx (Bind Abst) u t0) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst)
+u t0))) t6 H17 t3 H8))))) (H11 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind
+Abst) u t0)) v2 H7 u2 H12))))))) t5 H15)) t2 (sym_eq T t2 t0 H14))) u0
+(sym_eq T u0 u H9))) k (sym_eq K k (Bind Abst) H6))) H3)) H2)) H13 H0 H1))) |
+(pr0_beta u0 v0 v3 H0 t2 t6 H1) \Rightarrow (\lambda (H4: (eq T (THead (Flat
+Appl) v0 (THead (Bind Abst) u0 t2)) (THead (Bind Abst) u t0))).(\lambda (H11:
+(eq T (THead (Bind Abbr) v3 t6) t5)).((let H2 \def (eq_ind T (THead (Flat
+Appl) v0 (THead (Bind Abst) u0 t2)) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u
+t0) H4) in (False_ind ((eq T (THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3)
+\to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3)
+t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))))) H2)) H11 H0 H1)))
+| (pr0_upsilon b H0 v0 v3 H1 u0 u3 H4 t2 t6 H11) \Rightarrow (\lambda (H12:
+(eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t2)) (THead (Bind Abst) u
+t0))).(\lambda (H13: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
+O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b)
+u0 t2)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H12) in (False_ind
+((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t5) \to
+((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to
+(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 t5) t)))))))) H2)) H13 H0 H1 H4 H11))) |
+(pr0_delta u0 u3 H0 t2 t6 H1 w H4) \Rightarrow (\lambda (H11: (eq T (THead
+(Bind Abbr) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H12: (eq T (THead
+(Bind Abbr) u3 w) t5)).((let H2 \def (eq_ind T (THead (Bind Abbr) u0 t2)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead
+(Bind Abst) u t0) H11) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t5) \to
+((pr0 u0 u3) \to ((pr0 t2 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl)
+u2 t5) t))))))) H2)) H12 H0 H1 H4))) | (pr0_zeta b H0 t2 t6 H1 u0)
+\Rightarrow (\lambda (H4: (eq T (THead (Bind b) u0 (lift (S O) O t2)) (THead
+(Bind Abst) u t0))).(\lambda (H11: (eq T t6 t5)).((let H2 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u
+t0) H4) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
+| (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead
+(Bind Abst) u t0) H4) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u0
+(lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in (eq_ind B Abst (\lambda
+(b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to
+((not (eq B b0 Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead
+(Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5)
+t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T
+(lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0
+t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0))
+(\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H12: (eq
+T (lift (S O) O t2) t0)).(eq_ind T (lift (S O) O t2) (\lambda (_: T).((eq T
+t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u2 t5) t0))))))) (\lambda (H7: (eq T t6 t5)).(eq_ind T t5 (\lambda (t:
+T).((not (eq B Abst Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2
+t5) t0)))))) (\lambda (H13: (not (eq B Abst Abst))).(\lambda (_: (pr0 t2
+t5)).(let H8 \def (match (H13 (refl_equal B Abst)) return (\lambda (_:
+False).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 t5) t)))) with []) in H8))) t6 (sym_eq T t6 t5
+H7))) t0 H12)) u0 (sym_eq T u0 u H6))) b (sym_eq B b Abst H5))) H3)) H2)) H11
+H0 H1))) | (pr0_epsilon t2 t6 H0 u0) \Rightarrow (\lambda (H1: (eq T (THead
+(Flat Cast) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H4: (eq T t6
+t5)).((let H2 \def (eq_ind T (THead (Flat Cast) u0 t2) (\lambda (e: T).(match
+e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Abst) u t0) H1) in (False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to
+(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t:
+T).(pr0 (THead (Flat Appl) u2 t5) t))))) H2)) H4 H0)))]) in (H6 (refl_equal T
+(THead (Bind Abst) u t0)) (refl_equal T t5))))) t4 H11)) v1 (sym_eq T v1 u1
+H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2
+H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v0 v3 H6 t4 t5 H7) \Rightarrow
+(\lambda (H9: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4))
+t)).(\lambda (H10: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead
+(Flat Appl) v0 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))
+(\lambda (H11: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Bind
+Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t
+t0)))))) (\lambda (H12: (pr0 v0 v3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def
+(eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst)
+u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H9) in (let H1
+\def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
+(THead (Flat Appl) v0 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) v3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow (match t return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow
+t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0
+(THead (Bind Abst) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef
+_) \Rightarrow u | (THead _ _ t) \Rightarrow (match t return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0
+_) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead
+(Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in ((let H3 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t]))
+(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead
+(Bind Abst) u0 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq T u u0) \to ((eq
+T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0))
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) (\lambda (H10: (eq T
+u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) v3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda
+(t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to
+(\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+Abst) u0 t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8
+t4 H11) in (let H6 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 v0 H3) in
+(ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 t)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) v3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 v2
+x)).(\lambda (H8: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda
+(t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t))
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t))) (\lambda (x0: T).(\lambda
+(H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr)
+v3 t5) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H7 t3 x0 H9 (Bind Abbr))
+(pr0_comp v3 x H8 t5 x0 H12 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind
+Abst) u0 t4) t4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) (tlt_head_dx
+(Bind Abst) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t4))) t3
+H5 t5 H13))))) (H4 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t4))
+v2 H6 v3 H12))))) t0 (sym_eq T t0 t4 H11))) u (sym_eq T u u0 H10))) v1
+(sym_eq T v1 v0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0
+(THead (Bind Abst) u0 t4)))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_upsilon b
+H6 v0 v3 H7 u1 u2 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat
+Appl) v0 (THead (Bind b) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl)
+v0 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0
+v3) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12:
+(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5))
+t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5))
+(\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to
+((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0))
+(\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H13: (not (eq B b
+Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
+t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1
+t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ?
+? t)).((eq T t (THead (Flat Appl) v0 (THead (Bind b) u1 t4))) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead
+(Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)))).(let H1
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow (match t return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0
+| (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1
+t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ _ t) \Rightarrow (match t return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])]))
+(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead
+(Bind b) u1 t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _)
+\Rightarrow Abst | (THead _ _ t) \Rightarrow (match t return (\lambda (_:
+T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead
+k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead
+(Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let
+H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
+Appl) v0 (THead (Bind b) u1 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B
+Abst b) \to ((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v3) t5)) t0))))))) (\lambda (H11: (eq B Abst
+b)).(eq_ind B Abst (\lambda (b: B).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) (\lambda
+(H12: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0
+(THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))))
+(\lambda (H14: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda
+(t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead
+(Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def
+(eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 Abst H11) in (let H6
+\def (match (H5 (refl_equal B Abst)) return (\lambda (_: False).(ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0
+(THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))) with
+[]) in H6)) t0 (sym_eq T t0 t4 H14))) u (sym_eq T u u1 H12))) b H11)) v1
+(sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat
+Appl) v0 (THead (Bind b) u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) |
+(pr0_delta u1 u2 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead
+(Bind Abbr) u1 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w)
+t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind
+Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u2 w)
+t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to
+((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0
+u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u2 t5 w)).(let H0
+\def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) t)) H4 (THead (Bind Abbr) u1 t4) H9) in (let H1 \def (match H0
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind Abbr)
+u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0))
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))))) with [refl_equal
+\Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
+t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3)
+t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) H1)))]) in (H1
+(refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8)))
+| (pr0_zeta b H6 t4 t5 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b)
+u0 (lift (S O) O t4)) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T (THead (Bind
+b) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b
+Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr)
+v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T t5
+t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4
+t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1
+(THead (Bind Abst) u t0)) t)) H4 (THead (Bind b) u0 (lift (S O) O t4)) H8) in
+(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T
+t (THead (Bind b) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead
+(Bind Abst) u t0)) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u0 (lift (S O) O t4)) H0) in (False_ind (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2
+t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 (lift (S O) O t4))))))))
+t5 (sym_eq T t5 t2 H10))) t H8 H9 H6 H7))) | (pr0_epsilon t4 t5 H6 u0)
+\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8:
+(eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5
+t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2
+t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5
+t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))
+(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Cast) u0
+t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead
+(Bind Abst) u t0)) (THead (Flat Cast) u0 t4))).(let H1 \def (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow
+False])])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T (\lambda
+(t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t)))
+H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5
+t2 H9))) t H7 H8 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1
+H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t0 t3 H5)
+\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0))
+t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t3)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0))
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t3)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to
+((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0
+t2 t1))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2)
+\to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1))
+(\lambda (t1: T).(pr0 t2 t1)))))))) (\lambda (H9: (not (eq B b
+Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda
+(H12: (pr0 t0 t3)).(let H13 \def (match H1 return (\lambda (t0: T).(\lambda
+(t1: T).(\lambda (_: (pr0 t0 t1)).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t3)) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow
+(\lambda (H8: (eq T t4 t)).(\lambda (H13: (eq T t4 t2)).(eq_ind T t (\lambda
+(t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2
+t0))))) (\lambda (H14: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda
+(t: T).(eq T t t2)) H14 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (t: T).(ex2
+T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t
+(\lambda (t: T).(eq T t4 t)) H8 (THead (Flat Appl) v1 (THead (Bind b) u1 t0))
+H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t)
+\to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (ex2_sym T (pr0 (THead (Flat
+Appl) v1 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11
+t0 t3 H12 v1 v2 v2 H10 (pr0_refl v2))))) t2 H0)) t (sym_eq T t t2 H14))) t4
+(sym_eq T t4 t H8) H13))) | (pr0_comp u0 u3 H8 t4 t5 H9 k) \Rightarrow
+(\lambda (H13: (eq T (THead k u0 t4) t)).(\lambda (H14: (eq T (THead k u3 t5)
+t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to
+((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2
+t0))))))) (\lambda (H15: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u3 t5)
+(\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 t t0)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17:
+(pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat
+Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead k u0 t4) H13) in (let H1 \def
+(match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k
+u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead k u3 t5)
+t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl)
+v1 (THead (Bind b) u1 t0)) (THead k u0 t4))).(let H1 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow (THead (Bind b) u1 t0) | (TLRef _) \Rightarrow (THead (Bind b) u1
+t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t0)) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match
+e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind b) u1 t0)) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda
+(e: T).(match e return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat
+Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) \Rightarrow k]))
+(THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in (eq_ind
+K (Flat Appl) (\lambda (k: K).((eq T v1 u0) \to ((eq T (THead (Bind b) u1 t0)
+t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead k u3 t5) t))))))
+(\lambda (H14: (eq T v1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T (THead (Bind
+b) u1 t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u3 t5) t0))))) (\lambda (H15: (eq T (THead (Bind b) u1 t0) t4)).(eq_ind
+T (THead (Bind b) u1 t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0:
+T).(pr0 (THead (Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda
+(k: K).(eq T (THead k u0 t4) t)) H13 (Flat Appl) H3) in (let H5 \def
+(eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H17 (THead (Bind b) u1 t0) H15) in
+(let H6 \def (match H5 return (\lambda (t: T).(\lambda (t1: T).(\lambda (_:
+(pr0 t t1)).((eq T t (THead (Bind b) u1 t0)) \to ((eq T t1 t5) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) with
+[(pr0_refl t2) \Rightarrow (\lambda (H0: (eq T t2 (THead (Bind b) u1
+t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda
+(t: T).((eq T t t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u3 t5) t0))))) (\lambda (H2: (eq T (THead (Bind b) u1 t0)
+t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))) (let H3 \def (eq_ind_r
+T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b)
+u1 t0) H15) in (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v:
+T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v
+t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2
+t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H3) in (let H5
+\def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T
+(\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t))
+(\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0)) t)))
+(\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u3 x)).(ex2_sym
+T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)))
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t0 t3 H12 u3 v2 x H7
+H6))))) (H4 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H5 u3
+H16))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind b) u1 t0) H0) H1))) | (pr0_comp
+u4 u5 H0 t2 t6 H1 k) \Rightarrow (\lambda (H6: (eq T (THead k u4 t2) (THead
+(Bind b) u1 t0))).(\lambda (H13: (eq T (THead k u5 t6) t5)).((let H2 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t]))
+(THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H3 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t]))
+(THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H5 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u4 t2) (THead (Bind b) u1 t0) H6) in (eq_ind K (Bind b) (\lambda
+(k0: K).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T (THead k0 u5 t6) t5) \to
+((pr0 u4 u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead
+(Flat Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1
+(\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind b) u5 t6) t5) \to ((pr0
+t u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T t2 t0)).(eq_ind T t0
+(\lambda (t: T).((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 u1 u5) \to ((pr0
+t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5)
+t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t6) t5)).(eq_ind T (THead
+(Bind b) u5 t6) (\lambda (t: T).((pr0 u1 u5) \to ((pr0 t0 t6) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))))) (\lambda
+(H18: (pr0 u1 u5)).(\lambda (H19: (pr0 t0 t6)).(let H15 \def (eq_ind_r T t4
+(\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1
+t0) H15) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt
+v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to
+(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H
+(THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H15) in (let H4 \def (eq_ind T
+v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0
+v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0
+(THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x: T).(\lambda
+(H10: (pr0 v2 x)).(\lambda (H14: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0
+t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0
+(THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x0: T).(\lambda
+(H12: (pr0 t6 x0)).(\lambda (H16: (pr0 t3 x0)).(ex2_ind T (\lambda (t:
+T).(pr0 u5 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t:
+T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x1:
+T).(\lambda (H11: (pr0 u5 x1)).(\lambda (H21: (pr0 u2 x1)).(ex2_sym T (pr0
+(THead (Flat Appl) u3 (THead (Bind b) u5 t6))) (pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_cong b
+H9 u3 v2 x H14 H10 t6 t3 x0 H12 H16 u5 u2 x1 H11 H21))))) (H20 u1 (tlt_trans
+(THead (Bind b) u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t0))
+(tlt_head_sx (Bind b) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1
+t0))) u5 H18 u2 H11))))) (H20 t0 (tlt_trans (THead (Bind b) u1 t0) t0 (THead
+(Flat Appl) u0 (THead (Bind b) u1 t0)) (tlt_head_dx (Bind b) u1 t0)
+(tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t0))) t6 H19 t3 H12))))) (H20
+u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H4 u3 H16))))))) t5
+H8)) t2 (sym_eq T t2 t0 H17))) u4 (sym_eq T u4 u1 H7))) k (sym_eq K k (Bind
+b) H5))) H3)) H2)) H13 H0 H1))) | (pr0_beta u v0 v3 H0 t2 t6 H1) \Rightarrow
+(\lambda (H6: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (THead
+(Bind b) u1 t0))).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t5)).((let
+H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u1 t0) H6) in (False_ind ((eq T
+(THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) \to ((pr0 t2 t6) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))) H2)) H13 H0
+H1))) | (pr0_upsilon b0 H0 v0 v3 H1 u4 u5 H6 t2 t6 H13) \Rightarrow (\lambda
+(H14: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t2)) (THead (Bind b) u1
+t0))).(\lambda (H15: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O)
+O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind
+b0) u4 t2)) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H14) in
+(False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3)
+t6)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to
+((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3
+t5) t)))))))) H2)) H15 H0 H1 H6 H13))) | (pr0_delta u4 u5 H0 t2 t6 H1 w H6)
+\Rightarrow (\lambda (H13: (eq T (THead (Bind Abbr) u4 t2) (THead (Bind b) u1
+t0))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) t5)).((let H2 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H5 \def
+(f_equal T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort
+_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abbr])])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13)
+in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T
+(THead (Bind Abbr) u5 w) t5) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to ((subst0 O
+u5 t6 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3
+t5) t)))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 (\lambda (t: T).((eq
+T t2 t0) \to ((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 t u5) \to ((pr0 t2
+t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0
+(THead (Flat Appl) u3 t5) t0))))))))) (\lambda (H18: (eq T t2 t0)).(eq_ind T
+t0 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 u1 u5) \to
+((pr0 t t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0:
+T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H19: (eq T (THead
+(Bind Abbr) u5 w) t5)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t:
+T).((pr0 u1 u5) \to ((pr0 t0 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda
+(t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3))
+t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0))))))) (\lambda (H20:
+(pr0 u1 u5)).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 O u5 t6
+w)).(let H15 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u1 t0)
+t4)) H15 Abbr H5) in (let H9 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b
+Abst))) H9 Abbr H5) in (let H23 \def (eq_ind_r B b (\lambda (b: B).(eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)) H8 Abbr H5)
+in (let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0
+t0) t)) H4 (THead (Bind Abbr) u1 t0) H15) in (let H8 \def (eq_ind_r T t
+(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind
+Abbr) u1 t0)) H4) in (let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2))
+H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3
+t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead
+(Bind Abbr) u5 w)) t))) (\lambda (x: T).(\lambda (H14: (pr0 v2 x)).(\lambda
+(H16: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0
+t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead
+(Bind Abbr) u5 w)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 t6 x0)).(\lambda
+(H24: (pr0 t3 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t:
+T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3
+(THead (Bind Abbr) u5 w)) t))) (\lambda (x1: T).(\lambda (H11: (pr0 u5
+x1)).(\lambda (H25: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead
+(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O)
+O v2) t3))) (pr0_confluence__pr0_cong_upsilon_delta H9 u5 t6 w H22 u3 v2 x
+H16 H14 t3 x0 H12 H24 u2 x1 H11 H25))))) (H8 u1 (tlt_trans (THead (Bind Abbr)
+u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) (tlt_head_sx (Bind
+Abbr) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t0))) u5 H20
+u2 H11))))) (H8 t0 (tlt_trans (THead (Bind Abbr) u1 t0) t0 (THead (Flat Appl)
+u0 (THead (Bind Abbr) u1 t0)) (tlt_head_dx (Bind Abbr) u1 t0) (tlt_head_dx
+(Flat Appl) u0 (THead (Bind Abbr) u1 t0))) t6 H21 t3 H12))))) (H8 u0
+(tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) v2 H10 u3
+H16))))))))))) t5 H19)) t2 (sym_eq T t2 t0 H18))) u4 (sym_eq T u4 u1 H7))) b
+H5)) H3)) H2)) H17 H0 H1 H6))) | (pr0_zeta b0 H0 t2 t6 H1 u) \Rightarrow
+(\lambda (H6: (eq T (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1
+t0))).(\lambda (H13: (eq T t6 t5)).((let H2 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x (S O))) O t2) | (TLRef _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) \Rightarrow t]))
+(THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in ((let H3
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0)
+H6) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead
+k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O
+t2)) (THead (Bind b) u1 t0) H6) in (eq_ind B b (\lambda (b1: B).((eq T u u1)
+\to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b1 Abst))
+\to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat
+Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_:
+T).((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b Abst))
+\to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat
+Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T (lift (S O) O t2) t0)).(eq_ind T
+(lift (S O) O t2) (\lambda (_: T).((eq T t6 t5) \to ((not (eq B b Abst)) \to
+((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3
+t5) t0))))))) (\lambda (H8: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: T).((not
+(eq B b Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0
+(THead (Flat Appl) u3 t5) t0)))))) (\lambda (H18: (not (eq B b
+Abst))).(\lambda (H19: (pr0 t2 t5)).(let H9 \def (eq_ind_r T t0 (\lambda (t:
+T).(eq T (THead (Bind b) u1 t) t4)) H15 (lift (S O) O t2) H17) in (let H15
+\def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4
+(THead (Bind b) u1 (lift (S O) O t2)) H9) in (let H20 \def (eq_ind_r T t
+(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b)
+u1 (lift (S O) O t2))) H15) in (let H12 \def (eq_ind_r T t0 (\lambda (t:
+T).(pr0 t t3)) H12 (lift (S O) O t2) H17) in (ex2_ind T (\lambda (t4: T).(eq
+T t3 (lift (S O) O t4))) (\lambda (t3: T).(pr0 t2 t3)) (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t))
+(\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x: T).(\lambda
+(H21: (eq T t3 (lift (S O) O x))).(\lambda (H22: (pr0 t2 x)).(eq_ind_r T
+(lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t)) t0)) (\lambda (t0: T).(pr0 (THead
+(Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind T v1 (\lambda (t: T).(pr0 t
+v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t:
+T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead
+(Flat Appl) u3 t5) t))) (\lambda (x0: T).(\lambda (H10: (pr0 v2 x0)).(\lambda
+(H14: (pr0 u3 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0
+t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead (Flat
+Appl) u3 t5) t))) (\lambda (x1: T).(\lambda (H16: (pr0 x x1)).(\lambda (H23:
+(pr0 t5 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t5)) (pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))))
+(pr0_confluence__pr0_cong_upsilon_zeta b H18 u1 u2 H11 u3 v2 x0 H14 H10 x t5
+x1 H16 H23))))) (H20 t2 (tlt_trans (THead (Bind b) u1 (lift (S O) O t2)) t2
+(THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) (lift_tlt_dx
+(Bind b) u1 t2 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift
+(S O) O t2)))) x H22 t5 H19))))) (H20 u0 (tlt_head_sx (Flat Appl) u0 (THead
+(Bind b) u1 (lift (S O) O t2))) v2 H4 u3 H16))) t3 H21)))) (pr0_gen_lift t2
+t3 (S O) O H12)))))))) t6 (sym_eq T t6 t5 H8))) t0 H17)) u (sym_eq T u u1
+H7))) b0 (sym_eq B b0 b H5))) H3)) H2)) H13 H0 H1))) | (pr0_epsilon t2 t6 H0
+u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t2) (THead (Bind b)
+u1 t0))).(\lambda (H6: (eq T t6 t5)).((let H2 \def (eq_ind T (THead (Flat
+Cast) u t2) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H1) in
+(False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t:
+T).(pr0 (THead (Flat Appl) u3 t5) t))))) H2)) H6 H0)))]) in (H6 (refl_equal T
+(THead (Bind b) u1 t0)) (refl_equal T t5))))) t4 H15)) v1 (sym_eq T v1 u0
+H14))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) t2
+H15)) t H13 H14 H8 H9))) | (pr0_beta u v0 v3 H8 t4 t5 H9) \Rightarrow
+(\lambda (H10: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t4))
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead
+(Flat Appl) v0 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H14: (eq T (THead (Bind Abbr) v3
+t5) t2)).(eq_ind T (THead (Bind Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to
+((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda
+(_: (pr0 v0 v3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t
+(\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6
+(THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H1 \def (match
+H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat
+Appl) v0 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) with [refl_equal \Rightarrow
+(\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead
+(Flat Appl) v0 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda
+(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead
+_ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0))
+(THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal
+T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match
+t return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0)
+in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_:
+T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t)
+\Rightarrow (match t return (\lambda (_: T).B) with [(TSort _) \Rightarrow b
+| (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0
+(THead (Bind Abst) u t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0)
+in (eq_ind T v0 (\lambda (_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t0
+t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5)
+t0))))))) (\lambda (H13: (eq B b Abst)).(eq_ind B Abst (\lambda (b: B).((eq T
+u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) v3 t5) t)))))) (\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda
+(_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead
+(Bind Abbr) v3 t5) t0))))) (\lambda (H15: (eq T t0 t4)).(eq_ind T t4 (\lambda
+(_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5)
+t0)))) (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t)
+\to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H6 \def (eq_ind T t0
+(\lambda (t: T).(pr0 t t3)) H12 t4 H15) in (let H7 \def (eq_ind T u1 (\lambda
+(t: T).(pr0 t u2)) H11 u H14) in (let H8 \def (eq_ind B b (\lambda (b:
+B).(not (eq B b Abst))) H9 Abst H13) in (let H9 \def (match (H8 (refl_equal B
+Abst)) return (\lambda (_: False).(ex2 T (\lambda (t: T).(pr0 (THead (Bind
+Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) v3 t5) t)))) with []) in H9))))) t0 (sym_eq T t0 t4 H15)))
+u1 (sym_eq T u1 u H14))) b (sym_eq B b Abst H13))) v1 (sym_eq T v1 v0 H4)))
+H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind
+Abst) u t4)))))))) t2 H14)) t H10 H13 H8 H9))) | (pr0_upsilon b0 H8 v0 v3 H9
+u0 u3 H10 t4 t5 H11) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0
+(THead (Bind b0) u0 t4)) t)).(\lambda (H14: (eq T (THead (Bind b0) u3 (THead
+(Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) v0
+(THead (Bind b0) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u3 (THead
+(Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0
+v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0:
+T).(pr0 t2 t0))))))))) (\lambda (H15: (eq T (THead (Bind b0) u3 (THead (Flat
+Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat
+Appl) (lift (S O) O v3) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0
+v0 v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda
+(t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H17:
+(pr0 v0 v3)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t4 t5)).(let H0
+\def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind
+b) u1 t0)) t)) H6 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H13) in (let
+H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
+(THead (Flat Appl) v0 (THead (Bind b0) u0 t4))) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O
+v3) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead
+(Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0)
+u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead
+_ _ t) \Rightarrow (match t return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow
+t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0
+(THead (Bind b0) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef
+_) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match t return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0
+_) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead
+(Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H3 \def (f_equal T B
+(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) \Rightarrow (match t
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])])) (THead (Flat
+Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0
+t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in (eq_ind T v0
+(\lambda (_: T).((eq B b b0) \to ((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift
+(S O) O v3) t5)) t0))))))) (\lambda (H14: (eq B b b0)).(eq_ind B b0 (\lambda
+(b: B).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t:
+T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))))))
+(\lambda (H15: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S
+O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat
+Appl) (lift (S O) O v3) t5)) t0))))) (\lambda (H16: (eq T t0 t4)).(eq_ind T
+t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead
+(Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def
+(eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1:
+T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0
+(THead (Bind b0) u0 t4)) H13) in (let H6 \def (eq_ind T t0 (\lambda (t:
+T).(pr0 t t3)) H12 t4 H16) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0
+t u2)) H11 u0 H15) in (let H8 \def (eq_ind B b (\lambda (b: B).(not (eq B b
+Abst))) H9 b0 H14) in (let H9 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2))
+H10 v0 H4) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3
+t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift
+(S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat
+Appl) (lift (S O) O v3) t5)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2
+x)).(\lambda (H11: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 u2 t))
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead
+(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x0:
+T).(\lambda (H12: (pr0 u2 x0)).(\lambda (H13: (pr0 u3 x0)).(ex2_ind T
+(\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t))
+(\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3)
+t5)) t))) (\lambda (x1: T).(\lambda (H17: (pr0 t3 x1)).(\lambda (H18: (pr0 t5
+x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H8 v2 v3 x H10 H11 u2 u3 x0 H12
+H13 t3 t5 x1 H17 H18)))) (H5 t4 (tlt_trans (THead (Bind b0) u0 t4) t4 (THead
+(Flat Appl) v0 (THead (Bind b0) u0 t4)) (tlt_head_dx (Bind b0) u0 t4)
+(tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t4))) t3 H6 t5 H19))))) (H5
+u0 (tlt_trans (THead (Bind b0) u0 t4) u0 (THead (Flat Appl) v0 (THead (Bind
+b0) u0 t4)) (tlt_head_sx (Bind b0) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead
+(Bind b0) u0 t4))) u2 H7 u3 H18))))) (H5 v0 (tlt_head_sx (Flat Appl) v0
+(THead (Bind b0) u0 t4)) v2 H9 v3 H17))))))) t0 (sym_eq T t0 t4 H16))) u1
+(sym_eq T u1 u0 H15))) b (sym_eq B b b0 H14))) v1 (sym_eq T v1 v0 H4))) H3))
+H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind b0) u0
+t4)))))))))) t2 H15)) t H13 H14 H8 H9 H10 H11))) | (pr0_delta u0 u3 H8 t4 t5
+H9 w H10) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4)
+t)).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind
+Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0
+u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda
+(t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq T (THead (Bind Abbr) u3 w)
+t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: T).((pr0 u0 u3) \to
+((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0:
+T).(pr0 t t0))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4
+t5)).(\lambda (_: (subst0 O u3 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Bind
+Abbr) u0 t4) H11) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda
+(_: (eq ? ? t)).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0)))))) with [refl_equal
+\Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0))
+(THead (Bind Abbr) u0 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u0
+t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind
+Abbr) u3 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0
+t4)))))))) t2 H13)) t H11 H12 H8 H9 H10))) | (pr0_zeta b0 H8 t4 t5 H9 u)
+\Rightarrow (\lambda (H10: (eq T (THead (Bind b0) u (lift (S O) O t4))
+t)).(\lambda (H11: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O
+t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t4 t5)
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H12: (eq T
+t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 t4 t)
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq
+B b0 Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda
+(t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead
+(Bind b0) u (lift (S O) O t4)) H10) in (let H1 \def (match H0 return (\lambda
+(t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b0) u (lift (S O) O
+t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead
+(Bind b) u1 t0)) (THead (Bind b0) u (lift (S O) O t4)))).(let H1 \def (eq_ind
+T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind b0) u (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t))
+(\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b0) u
+(lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H12))) t H10 H11 H8 H9))) |
+(pr0_epsilon t4 t5 H8 u) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u
+t4) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4)
+(\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2
+(\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2
+t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat
+Cast) u t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_:
+(eq ? ? t)).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0))
+(\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0:
+(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Cast) u
+t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0))
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t4)
+H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1
+(refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9
+H10 H8)))]) in (H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7
+H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 t0 t3 H3 w H4) \Rightarrow (\lambda (H5:
+(eq T (THead (Bind Abbr) u1 t0) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2
+w) t1)).(eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (_: T).((eq T (THead
+(Bind Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3
+w) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2
+t1)))))))) (\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead
+(Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0
+O u2 t3 w) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2
+t1))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t0 t3)).(\lambda
+(H10: (subst0 O u2 t3 w)).(let H11 \def (match H1 return (\lambda (t0:
+T).(\lambda (t1: T).(\lambda (_: (pr0 t0 t1)).((eq T t0 t) \to ((eq T t1 t2)
+\to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq T t4
+t)).(\lambda (H11: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 t2 t0))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H12
+(THead (Bind Abbr) u1 t0) H5) in (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda
+(t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda
+(t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t))
+H7 (THead (Bind Abbr) u1 t0) H5) in (let H2 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 t0) H5) in (ex_intro2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u1 t0) t)) (THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr)
+u2 w)) (pr0_delta u1 u2 H8 t0 t3 H9 w H10)))) t2 H0)) t (sym_eq T t t2 H12)))
+t4 (sym_eq T t4 t H7) H11))) | (pr0_comp u0 u3 H7 t4 t5 H8 k) \Rightarrow
+(\lambda (H11: (eq T (THead k u0 t4) t)).(\lambda (H12: (eq T (THead k u3 t5)
+t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to
+((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T
+(THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) (\lambda (t: T).((pr0 u0 u3)
+\to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w)
+t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda
+(H15: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead
+(Bind Abbr) u1 t0) t)) H5 (THead k u0 t4) H11) in (let H1 \def (match H0
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k u0 t4)) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 (THead k u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0:
+(eq T (THead (Bind Abbr) u1 t0) (THead k u0 t4))).(let H1 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k _ _)
+\Rightarrow k])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in (eq_ind K
+(Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+k u3 t5) t)))))) (\lambda (H12: (eq T u1 u0)).(eq_ind T u0 (\lambda (_:
+T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w)
+t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) t0))))) (\lambda (H13:
+(eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0
+(THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5)
+t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u0 t4) t))
+H11 (Bind Abbr) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall
+(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0
+v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2
+t0)))))))))) H (THead (Bind Abbr) u0 t4) H4) in (let H6 \def (eq_ind T t0
+(\lambda (t: T).(pr0 t t3)) H9 t4 H13) in (let H7 \def (eq_ind T u1 (\lambda
+(t: T).(pr0 t u2)) H8 u0 H12) in (ex2_ind T (\lambda (t: T).(pr0 u2 t))
+(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2
+w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) t))) (\lambda (x:
+T).(\lambda (H8: (pr0 u2 x)).(\lambda (H9: (pr0 u3 x)).(ex2_ind T (\lambda
+(t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0
+(THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5)
+t))) (\lambda (x0: T).(\lambda (H11: (pr0 t3 x0)).(\lambda (H14: (pr0 t5
+x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 t5)) (pr0 (THead (Bind Abbr) u2
+w)) (pr0_confluence__pr0_cong_delta u2 t3 w H10 u3 x H9 H8 t5 x0 H14 H11)))))
+(H5 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H6 t5 H15))))) (H5 u0 (tlt_head_sx
+(Bind Abbr) u0 t4) u2 H7 u3 H14)))))) t0 (sym_eq T t0 t4 H13))) u1 (sym_eq T
+u1 u0 H12))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4)))))))
+t2 H13)) t H11 H12 H7 H8))) | (pr0_beta u v1 v2 H7 t4 t5 H8) \Rightarrow
+(\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4))
+t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))
+(\lambda (H11: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind
+Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t
+t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def
+(eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead
+(Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 return
+(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl) v1
+(THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))
+with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0)
+(THead (Flat Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (eq_ind T
+(THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1
+(THead (Bind Abst) u t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0
+(THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5)
+t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t4)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u0 u3 H9
+t4 t5 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead
+(Bind b) u0 t4)) t)).(\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat
+Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl)
+(lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to
+((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H13: (eq T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t:
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5)
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def
+(eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead
+(Flat Appl) v1 (THead (Bind b) u0 t4)) H11) in (let H1 \def (match H0 return
+(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl) v1
+(THead (Bind b) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr)
+u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
+(S O) O v2) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T
+(THead (Bind Abbr) u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0
+t4)))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (False_ind
+(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0
+(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in
+(H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))))))))) t2
+H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u0 u3 H7 t4 t5 H8 w0 H9)
+\Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H12:
+(eq T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t4)
+(\lambda (_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to
+((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T (\lambda (t0: T).(pr0 (THead
+(Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq
+T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda
+(t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t
+t0))))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t4 t5)).(\lambda
+(H16: (subst0 O u3 t5 w0)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T
+(THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind Abbr) u0 t4) H11) in (let H1
+\def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t
+(THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0))))))
+with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0)
+(THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match
+e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0)
+(THead (Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef
+_) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0)
+(THead (Bind Abbr) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4)
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 (THead (Bind Abbr) u3 w0) t0))))) (\lambda (H12: (eq T t0
+t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0)))) (let
+H3 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall
+(t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0
+t4) H11) in (let H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 t4 H12)
+in (let H5 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u0 H2) in
+(ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u3 w0) t))) (\lambda (x: T).(\lambda (H6: (pr0 u2 x)).(\lambda
+(H7: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0
+t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t3
+x0)).(\lambda (H9: (pr0 t5 x0)).(pr0_confluence__pr0_delta_delta u2 t3 w H10
+u3 t5 w0 H16 x H6 H7 x0 H8 H9)))) (H3 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3
+H4 t5 H15))))) (H3 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H5 u3 H14))))) t0
+(sym_eq T t0 t4 H12))) u1 (sym_eq T u1 u0 H2))) H1)))]) in (H1 (refl_equal T
+(THead (Bind Abbr) u0 t4)))))))) t2 H13)) t H11 H12 H7 H8 H9))) | (pr0_zeta b
+H7 t4 t5 H8 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O)
+O t4)) t)).(\lambda (H12: (eq T t5 t2)).(eq_ind T (THead (Bind b) u (lift (S
+O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4
+t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda
+(t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2 (\lambda
+(t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0
+(THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda
+(H14: (not (eq B b Abst))).(\lambda (H15: (pr0 t4 t2)).(let H0 \def (eq_ind_r
+T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind b) u
+(lift (S O) O t4)) H11) in (let H1 \def (match H0 return (\lambda (t:
+T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b) u (lift (S O) O t4)))
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead
+(Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let
+H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _)
+\Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O
+t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow
+Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) with
+[(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr)
+u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind B Abbr (\lambda
+(_: B).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t:
+T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda
+(H12: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) O t4))
+\to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 t2 t0))))) (\lambda (H13: (eq T t0 (lift (S O) O t4))).(eq_ind T
+(lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r B b
+(\lambda (b: B).(not (eq B b Abst))) H14 Abbr H3) in (let H5 \def (eq_ind_r B
+b (\lambda (b: B).(eq T (THead (Bind b) u (lift (S O) O t4)) t)) H11 Abbr H3)
+in (let H6 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Bind Abbr) u (lift (S O) O t4)) H5) in (let H7 \def (eq_ind T t0 (\lambda
+(t: T).(pr0 t t3)) H9 (lift (S O) O t4) H13) in (ex2_ind T (\lambda (t2:
+T).(eq T t3 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t4 t2)) (ex2 T (\lambda
+(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))
+(\lambda (x: T).(\lambda (H9: (eq T t3 (lift (S O) O x))).(\lambda (H11: (pr0
+t4 x)).(let H10 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H10
+(lift (S O) O x) H9) in (let H8 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2))
+H8 u H12) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2
+t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t:
+T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0
+t2 x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H10 x
+(pr0_refl (lift (S O) O x)) t2)))) (H6 t4 (lift_tlt_dx (Bind Abbr) u t4 (S O)
+O) x H11 t2 H15))))))) (pr0_gen_lift t4 t3 (S O) O H7)))))) t0 (sym_eq T t0
+(lift (S O) O t4) H13))) u1 (sym_eq T u1 u H12))) b H3)) H2)) H1)))]) in (H1
+(refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2
+H13))) t H11 H12 H7 H8))) | (pr0_epsilon t4 t5 H7 u) \Rightarrow (\lambda
+(H8: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T
+(THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to
+(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) (\lambda (H10: (eq T t5 t2)).(eq_ind T t2 (\lambda (t:
+T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))
+(\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def
+(eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead
+(Flat Cast) u t4) H8) in (let H1 \def (match H0 return (\lambda (t:
+T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2
+t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr)
+u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (eq_ind T (THead (Bind Abbr)
+u1 t0) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (THead (Flat Cast) u t4) H0) in (False_ind (ex2 T
+(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2
+t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T
+t5 t2 H10))) t H8 H9 H7)))]) in (H11 (refl_equal T t) (refl_equal T t2))))))
+t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t0 t3 H3 u) \Rightarrow (\lambda
+(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) t)).(\lambda (H5: (eq T t3
+t1)).(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (_: T).((eq T t3
+t1) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2:
+T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T t3
+t1)).(eq_ind T t1 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t0 t) \to
+(ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))
+(\lambda (H7: (not (eq B b Abst))).(\lambda (H8: (pr0 t0 t1)).(let H9 \def
+(match H1 return (\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0
+t3)).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t))
+(\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda
+(H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t:
+T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_:
+T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let
+H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead (Bind b) u
+(lift (S O) O t0)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t0))
+(\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t
+t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead
+(Bind b) u (lift (S O) O t0)) H4) in (let H2 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t0)) H4) in
+(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b)
+u (lift (S O) O t0)) t)) t1 (pr0_refl t1) (pr0_zeta b H7 t0 t1 H8 u)))) t2
+H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u1 u2 H6
+t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) t)).(\lambda
+(H10: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_:
+T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda
+(H11: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t:
+T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13:
+(pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b)
+u (lift (S O) O t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k u1 t4)) \to
+(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead k u2 t5)
+t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u
+(lift (S O) O t0)) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec
+lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0)
+\Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map)
+(\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t]))
+(THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H2 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t]))
+(THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H3 \def
+(f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort
+_) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _)
+\Rightarrow k])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in
+(eq_ind K (Bind b) (\lambda (k: K).((eq T u u1) \to ((eq T (lift (S O) O t0)
+t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2
+t5) t)))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T
+(lift (S O) O t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 (THead (Bind b) u2 t5) t0))))) (\lambda (H11: (eq T (lift (S O) O t0)
+t4)).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0
+t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t5) t0)))) (let H4 \def
+(eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H9 (Bind b) H3) in
+(let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0)
+H11) in (ex2_ind T (\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda
+(t2: T).(pr0 t0 t2)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0
+(THead (Bind b) u2 t5) t))) (\lambda (x: T).(\lambda (H6: (eq T t5 (lift (S
+O) O x))).(\lambda (H9: (pr0 t0 x)).(let H12 \def (eq_ind_r T t4 (\lambda
+(t0: T).(eq T (THead (Bind b) u1 t0) t)) H4 (lift (S O) O t0) H11) in (let
+H13 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to
+(\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead
+(Bind b) u1 (lift (S O) O t0)) H12) in (eq_ind_r T (lift (S O) O x) (\lambda
+(t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead
+(Bind b) u2 t) t0)))) (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t:
+T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead
+(Bind b) u2 (lift (S O) O x)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 x
+x0)).(\lambda (H14: (pr0 t1 x0)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t))
+(\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) x0 H14
+(pr0_zeta b H7 x x0 H8 u2))))) (H13 t0 (lift_tlt_dx (Bind b) u1 t0 (S O) O) x
+H9 t1 H8)) t5 H6)))))) (pr0_gen_lift t0 t5 (S O) O H5)))) t4 H11)) u (sym_eq
+T u u1 H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4)))))))
+t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v1 v2 H6 t4 t5 H7) \Rightarrow
+(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))
+t)).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat
+Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr)
+v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0
+t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T (THead (Bind
+Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0
+v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda
+(t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4
+t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift
+(S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H8) in
+(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T
+t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O
+t0)) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind
+T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T
+(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5)
+t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t4)))))))) t2 H10)) t H8 H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8
+t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind
+b0) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl)
+(lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0)
+u1 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift
+(S O) O v2) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1
+u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b0) u2 (THead (Flat
+Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat
+Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0
+v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1
+t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0
+Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
+t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u
+(lift (S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) H10)
+in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (THead (Flat Appl) v1 (THead (Bind b0) u1 t4))) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal
+\Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead
+(Flat Appl) v1 (THead (Bind b0) u1 t4)))).(let H1 \def (eq_ind T (THead (Bind
+b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1
+(THead (Bind b0) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t))
+(\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2)
+t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b0)
+u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t4 t5
+H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t4)
+t)).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
+Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
+u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0
+t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind
+Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1
+u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0
+t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda
+(H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r
+T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead
+(Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 return (\lambda (t:
+T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w)
+t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
+t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
+(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
+(Bind Abbr) u1 t4) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in (eq_ind B Abbr (\lambda
+(_: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t:
+T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)))))) (\lambda
+(H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t4)
+\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind
+Abbr) u2 w) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) t4)).(eq_ind T
+(lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) (let H4 \def (eq_ind_r
+T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) H11) in (ex2_ind T
+(\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t0 t2))
+(ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2
+w) t))) (\lambda (x: T).(\lambda (H5: (eq T t5 (lift (S O) O x))).(\lambda
+(H6: (pr0 t0 x)).(let H9 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead
+(Bind Abbr) u1 t0) t)) H9 (lift (S O) O t0) H11) in (let H12 \def (eq_ind_r T
+t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O
+t0)) H9) in (let H13 \def (eq_ind T t5 (\lambda (t: T).(subst0 O u2 t w)) H14
+(lift (S O) O x) H5) in (let H7 \def (eq_ind B b (\lambda (b: B).(not (eq B b
+Abst))) H7 Abbr H3) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t:
+T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead
+(Bind Abbr) u2 w) t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_:
+(pr0 t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1)
+(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H13 x (pr0_refl
+(lift (S O) O x)) t1))))) (H12 t0 (lift_tlt_dx (Bind Abbr) u1 t0 (S O) O) x
+H6 t1 H8))))))))) (pr0_gen_lift t0 t5 (S O) O H4))) t4 H11)) u (sym_eq T u u1
+H10))) b (sym_eq B b Abbr H3))) H2)) H1)))]) in (H1 (refl_equal T (THead
+(Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b0 H6 t4
+t5 H7 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) O
+t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S
+O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0
+t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2
+t0))))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq
+B b0 Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda
+(t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H13:
+(pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b)
+u (lift (S O) O t0)) t)) H4 (THead (Bind b0) u0 (lift (S O) O t4)) H9) in
+(let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T
+t (THead (Bind b0) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1
+t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda
+(H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S
+O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind
+b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in ((let
+H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0
+(lift (S O) O t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e
+return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B)
+with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in (eq_ind B b0
+(\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t0) (lift (S O) O t4))
+\to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))
+(\lambda (H10: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O
+t0) (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0))))) (\lambda (H11: (eq T (lift (S O) O t0) (lift (S O) O
+t4))).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t
+(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b0) u0 (lift (S O) O
+t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H8 t4
+(lift_inj t0 t4 (S O) O H11)) in (let H6 \def (eq_ind B b (\lambda (b:
+B).(not (eq B b Abst))) H7 b0 H3) in (ex2_ind T (\lambda (t: T).(pr0 t1 t))
+(\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t:
+T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t1 x)).(\lambda (H8: (pr0
+t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) x
+H7 H8)))) (H4 t4 (lift_tlt_dx (Bind b0) u0 t4 (S O) O) t1 H5 t2 H13)))))
+(lift (S O) O t4) H11)) u (sym_eq T u u0 H10))) b (sym_eq B b b0 H3))) H2))
+H1)))]) in (H1 (refl_equal T (THead (Bind b0) u0 (lift (S O) O t4)))))))) t5
+(sym_eq T t5 t2 H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u0)
+\Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8:
+(eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5
+t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t:
+T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0
+t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Cast) u0
+t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1
+t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda
+(H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Cast) u0
+t4))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T
+(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1
+(refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8
+H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t3 (sym_eq T t3 t1
+H6))) t H4 H5 H2 H3))) | (pr0_epsilon t0 t3 H2 u) \Rightarrow (\lambda (H3:
+(eq T (THead (Flat Cast) u t0) t)).(\lambda (H4: (eq T t3 t1)).(eq_ind T
+(THead (Flat Cast) u t0) (\lambda (_: T).((eq T t3 t1) \to ((pr0 t0 t3) \to
+(ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))
+(\lambda (H5: (eq T t3 t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2
+T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda
+(H6: (pr0 t0 t1)).(let H7 \def (match H1 return (\lambda (t0: T).(\lambda
+(t3: T).(\lambda (_: (pr0 t0 t3)).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T
+(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl
+t4) \Rightarrow (\lambda (H5: (eq T t4 t)).(\lambda (H7: (eq T t4
+t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (eq T t
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t:
+T).(eq T t t2)) H8 (THead (Flat Cast) u t0) H3) in (eq_ind T (THead (Flat
+Cast) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda
+(t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t))
+H5 (THead (Flat Cast) u t0) H3) in (let H2 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall
+(t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t0) H3) in (ex_intro2 T
+(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u t0) t))
+t1 (pr0_refl t1) (pr0_epsilon t0 t1 H6 u)))) t2 H0)) t (sym_eq T t t2 H8)))
+t4 (sym_eq T t4 t H5) H7))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow
+(\lambda (H7: (eq T (THead k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5)
+t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to
+((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T (THead k u2 t5)
+t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5)
+\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))
+(\lambda (_: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H0 \def (eq_ind_r
+T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead k u1 t4) H7)
+in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ? ?
+t)).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 (THead k u2 t5) t0)))))) with [refl_equal \Rightarrow
+(\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead k u1 t4))).(let H1 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t]))
+(THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k _ _)
+\Rightarrow k])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in (eq_ind K
+(Flat Cast) (\lambda (k: K).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T
+(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 t5) t))))))
+(\lambda (H8: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to
+(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast)
+u2 t5) t0))))) (\lambda (H9: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2
+T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) u2
+t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4)
+t)) H7 (Flat Cast) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall
+(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0
+v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2
+t0)))))))))) H (THead (Flat Cast) u1 t4) H4) in (let H6 \def (eq_ind T t0
+(\lambda (t: T).(pr0 t t1)) H6 t4 H9) in (ex2_ind T (\lambda (t: T).(pr0 t1
+t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda
+(t: T).(pr0 (THead (Flat Cast) u2 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0
+t1 x)).(\lambda (H10: (pr0 t5 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t))
+(\lambda (t: T).(pr0 (THead (Flat Cast) u2 t5) t)) x H7 (pr0_epsilon t5 x H10
+u2))))) (H5 t4 (tlt_head_dx (Flat Cast) u1 t4) t1 H6 t5 H11))))) t0 (sym_eq T
+t0 t4 H9))) u (sym_eq T u u1 H8))) k H3)) H2)) H1)))]) in (H1 (refl_equal T
+(THead k u1 t4))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u0 v1 v2 H5 t4 t5
+H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst)
+u0 t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T
+(THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda
+(H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5)
+(\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1
+v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq
+T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t4)) H7) in (let H1 \def (match H0 return (\lambda (t: T).(\lambda (_: (eq ?
+? t)).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5)
+t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast)
+u t0) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind
+T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
+(Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T (\lambda
+(t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))
+H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t4)))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4
+t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b)
+u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4))
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to
+((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0
+t2 t0))))))))) (\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2)
+\to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda
+(_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0
+\def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3
+(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in (let H1 \def (match H0
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Appl)
+v1 (THead (Bind b) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda
+(t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5))
+t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast)
+u t0) (THead (Flat Appl) v1 (THead (Bind b) u1 t4)))).(let H1 \def (eq_ind T
+(THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
+(Flat Appl) v1 (THead (Bind b) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t:
+T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl)
+v1 (THead (Bind b) u1 t4)))))))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) |
+(pr0_delta u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead
+(Bind Abbr) u1 t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w)
+t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind
+Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to
+(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))
+(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
+Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2
+t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t
+t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_:
+(subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead
+(Flat Cast) u t0) t)) H3 (THead (Bind Abbr) u1 t4) H8) in (let H1 \def (match
+H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind
+Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0
+(THead (Bind Abbr) u2 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0:
+(eq T (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t4))).(let H1 \def
+(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
+t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0
+(THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr)
+u1 t4)))))))) t2 H10)) t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u0)
+\Rightarrow (\lambda (H7: (eq T (THead (Bind b) u0 (lift (S O) O t4))
+t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O
+t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5)
+\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))
+(\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst))
+\to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4
+t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u
+t0) t)) H3 (THead (Bind b) u0 (lift (S O) O t4)) H7) in (let H1 \def (match
+H0 return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Bind b)
+u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead
+(Flat Cast) u t0) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def
+(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0
+(lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t))
+(\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0
+(lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) |
+(pr0_epsilon t4 t5 H5 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast)
+u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4)
+(\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5
+t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (pr0 t4
+t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u
+t0) t)) H3 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return
+(\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead (Flat Cast) u0 t4))
+\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))
+with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0)
+(THead (Flat Cast) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match
+e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0)
+(THead (Flat Cast) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0)
+(THead (Flat Cast) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4)
+\to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))
+(\lambda (H8: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H3 \def (eq_ind_r T t
+(\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1)
+\to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0))
+(\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u0 t4) H7) in (let
+H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H6 t4 H8) in (ex2_ind T
+(\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t:
+T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H5:
+(pr0 t1 x)).(\lambda (H6: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1
+t)) (\lambda (t: T).(pr0 t2 t)) x H5 H6)))) (H3 t4 (tlt_head_dx (Flat Cast)
+u0 t4) t1 H4 t2 H10)))) t0 (sym_eq T t0 t4 H8))) u (sym_eq T u u0 H2)))
+H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5
+t2 H9))) t H7 H8 H5)))]) in (H7 (refl_equal T t) (refl_equal T t2)))) t3
+(sym_eq T t3 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) (refl_equal T
+t1))))))))) t0).
+
+theorem pr0_delta1:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall
+(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead
+(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1:
+(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind
+Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind
+Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H
+t1 t2 H0 t0 H2))) w H1)))))))).
+
+theorem pr0_subst1_back:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1:
+T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda
+(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x
+H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
+
+theorem pr0_subst1_fwd:
+ \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1
+i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
+\def
+ \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1:
+T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda
+(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t:
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x
+H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
+
+theorem pr0_subst1:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall
+(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1
+v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2
+w2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1:
+T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1
+w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))))))
+(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0
+t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2))))
+(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda
+(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2))
+(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2))
+(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2
+T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3
+(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2))
+(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0
+w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0
+w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4:
+(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2:
+T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i
+v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).
+
+theorem nf0_dec:
+ \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t1 t2))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to
+(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl
+(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T
+(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda
+(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n)
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
+(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl
+(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T
+(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda
+(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n)
+t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T
+(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t:
+T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0
+t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(match k return (\lambda (k0:
+K).(or (\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) with [(Bind b)
+\Rightarrow (match b return (\lambda (b0: B).(or (\forall (t2: T).((pr0
+(THead (Bind b0) t t0) t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind b0) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b0) t t0) t2))))) with [Abbr
+\Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Bind Abbr) t t0) t2)
+\to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in (let
+H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v))
+(eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind
+Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0
+(lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift
+(S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T (THead
+(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S O) O
+x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))
+(THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind Abbr)
+t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let H5
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O
+x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t0: T).(subst0 O t t0 (lift (S
+O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6
+P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3)))
+(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x)
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3)
+\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2)
+t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O)
+O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind
+Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t
+(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S
+O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2)))
+H1)))) | Abst \Rightarrow (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t
+t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead
+(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda
+(H2: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H3 \def H0 in
+(or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2:
+T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))
+(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead
+(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t
+t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst)
+t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0
+t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to
+(eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
+(Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Abst) t t0) t2))) (\lambda (t2: T).(\lambda (H5: (pr0 (THead
+(Bind Abst) t t0) t2)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0)
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead (Bind
+Abst) x0 x1))).(\lambda (H7: (pr0 t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y
+\def (H4 x1 H8) in (let H_y0 \def (H2 x0 H7) in (let H9 \def (eq_ind_r T x1
+(\lambda (t: T).(pr0 t0 t)) H8 t0 H_y) in (let H10 \def (eq_ind_r T x1
+(\lambda (t: T).(eq T t2 (THead (Bind Abst) x0 t))) H6 t0 H_y) in (let H11
+\def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H7 t H_y0) in (let H12 \def
+(eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Abst) t t0))) H10 t
+H_y0) in (eq_ind_r T (THead (Bind Abst) t t0) (\lambda (t3: T).(eq T (THead
+(Bind Abst) t t0) t3)) (refl_equal T (THead (Bind Abst) t t0)) t2
+H12)))))))))))) (pr0_gen_abst t t0 t2 H5)))))) (\lambda (H4: (ex2 T (\lambda
+(t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0
+t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t
+t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
+T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T
+t0 x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror
+(\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind
+Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2)
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0)
+t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))
+(THead (Bind Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead
+(Bind Abst) t x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef
+_) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) t t0)
+(THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t:
+T).(pr0 t0 t)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t: T).((eq
+T t0 t) \to (\forall (P: Prop).P))) H5 t0 H8) in (H10 (refl_equal T t0)
+P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3)))
+(\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
+t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
+(t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))))
+(\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P:
+Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead
+(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind
+Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x
+t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
+\Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) t t0)
+(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t0:
+T).(pr0 t t0)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).((eq T
+t t0) \to (\forall (P: Prop).P))) H3 t H6) in (H8 (refl_equal T t) P))))))
+(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) | Void
+\Rightarrow (let H_x \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T
+(\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O
+v)))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T
+(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind
+Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Bind Void) t t0) t2)))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0
+(lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift
+(S O) O x)) (eq T t0 (lift (S O) O x)) (or (\forall (t2: T).((pr0 (THead
+(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda
+(H3: (subst0 O t t0 (lift (S O) O x))).(let H4 \def H in (or_ind (\forall
+(t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2)
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2:
+T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
+(\lambda (H5: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H6 \def
+H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T
+(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to
+(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
+(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Void) t t0) t2)))) (\lambda (H7: ((\forall (t2: T).((pr0 t0 t2)
+\to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Void) t
+t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
+T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Void) t t0) t2))) (\lambda (t2: T).(\lambda (H8: (pr0
+(THead (Bind Void) t t0) t2)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (pr0 t0 (lift
+(S O) O t2)) (eq T (THead (Bind Void) t t0) t2) (\lambda (H9: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Void) t t0)
+t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t2 (THead (Bind
+Void) x0 x1))).(\lambda (H11: (pr0 t x0)).(\lambda (H12: (pr0 t0 x1)).(let
+H_y \def (H7 x1 H12) in (let H_y0 \def (H5 x0 H11) in (let H13 \def (eq_ind_r
+T x1 (\lambda (t: T).(pr0 t0 t)) H12 t0 H_y) in (let H14 \def (eq_ind_r T x1
+(\lambda (t: T).(eq T t2 (THead (Bind Void) x0 t))) H10 t0 H_y) in (let H15
+\def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H11 t H_y0) in (let H16 \def
+(eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Void) t t0))) H14 t
+H_y0) in (eq_ind_r T (THead (Bind Void) t t0) (\lambda (t3: T).(eq T (THead
+(Bind Void) t t0) t3)) (refl_equal T (THead (Bind Void) t t0)) t2
+H16)))))))))))) H9)) (\lambda (H9: (pr0 t0 (lift (S O) O t2))).(let H_y \def
+(H7 (lift (S O) O t2) H9) in (let H10 \def (eq_ind T t0 (\lambda (t0:
+T).(subst0 O t t0 (lift (S O) O x))) H3 (lift (S O) O t2) H_y) in (eq_ind_r T
+(lift (S O) O t2) (\lambda (t3: T).(eq T (THead (Bind Void) t t3) t2))
+(subst0_gen_lift_false t2 t (lift (S O) O x) (S O) O O (le_n O) (eq_ind_r nat
+(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S
+O)) (plus_comm O (S O))) H10 (eq T (THead (Bind Void) t (lift (S O) O t2))
+t2)) t0 H_y)))) (pr0_gen_void t t0 t2 H8)))))) (\lambda (H7: (ex2 T (\lambda
+(t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0
+t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t
+t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq
+T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x0: T).(\lambda (H8: (((eq
+T t0 x0) \to (\forall (P: Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror
+(\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind
+Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2)
+\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0)
+t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))
+(THead (Bind Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead
+(Bind Void) t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda
+(e: T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Void)
+t t0) (THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda
+(t: T).(pr0 t0 t)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t:
+T).((eq T t0 t) \to (\forall (P: Prop).P))) H8 t0 H11) in (H13 (refl_equal T
+t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) H6)))
+(\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T
+t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall
+(t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))))
+(\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P:
+Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0
+(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind
+Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void)
+x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match
+e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _)
+\Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Void) t t0)
+(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t0:
+T).(pr0 t t0)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t0: T).((eq
+T t t0) \to (\forall (P: Prop).P))) H6 t H9) in (H11 (refl_equal T t) P))))))
+(pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) (\lambda
+(H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda (t:
+T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2:
+T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))))
+H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(or
+(\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead (Bind
+Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t t2) t3)
+\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) t t2)
+t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S O) O
+x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S
+O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t
+(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead
+(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y
+(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void not_void_abst x x (pr0_refl x)
+t))) t0 H3))) H2))) H1)))]) | (Flat f) \Rightarrow (match f return (\lambda
+(f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) t2) \to (eq T
+(THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat f0)
+t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat f0)
+t t0) t2))))) with [Appl \Rightarrow (let H_x \def (binder_dec t0) in (let H1
+\def H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u:
+T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P:
+Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq
+T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda
+(w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T
+(\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w
+u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T
+(THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0
+(\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t t2))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind
+x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t
+t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq
+T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3:
+T).(pr0 (THead (Flat Appl) t t2) t3))))) ((match x0 return (\lambda (b:
+B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead
+(Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2)
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2)
+t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1
+x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead
+(Bind b) x1 x2)) t2)))))) with [Abbr \Rightarrow (\lambda (_: (or (\forall
+(t2: T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq T (THead (Bind Abbr) x1
+x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) x1 x2) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) x1 x2)
+t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind
+Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2))
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr)
+x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat
+Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1
+x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2))
+(\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (THead
+(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P:
+Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2))
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t) \Rightarrow
+(match t return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])])) I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2))
+H6) in (False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t)
+x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) | Abst \Rightarrow (\lambda (_:
+(or (\forall (t2: T).((pr0 (THead (Bind Abst) x1 x2) t2) \to (eq T (THead
+(Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) x1
+x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst)
+x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead
+(Bind Abst) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abst) x1
+x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind
+Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex_intro2 T (\lambda (t2:
+T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst)
+x1 x2)) t2)) (THead (Bind Abbr) t x2) (\lambda (H6: (eq T (THead (Flat Appl)
+t (THead (Bind Abst) x1 x2)) (THead (Bind Abbr) t x2))).(\lambda (P:
+Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abst) x1 x2))
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P
+H7)))) (pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) | Void
+\Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Void) x1 x2)
+t2) \to (eq T (THead (Bind Void) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Bind Void) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Bind Void) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0
+(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (eq T (THead (Flat
+Appl) t (THead (Bind Void) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1
+x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead
+(Bind Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2)) (THead (Bind Void) x1
+(THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat
+Appl) t (THead (Bind Void) x1 x2)) (THead (Bind Void) x1 (THead (Flat Appl)
+(lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead
+(Flat Appl) t (THead (Bind Void) x1 x2)) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ t) \Rightarrow (match t return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind Void)
+x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7))))
+(pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2
+(pr0_refl x2)))))]) H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b:
+B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to
+(\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0
+t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead
+(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda
+(H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def H0 in
+(or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2:
+T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))
+(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead
+(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t
+t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl)
+t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0
+t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to
+(eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead
+(Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0
+(THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 (THead
+(Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2)
+(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t0 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0
+t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t
+x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def
+(H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H11
+t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Flat
+Appl) x0 t))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t0:
+T).(pr0 t t0)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t:
+T).(eq T t2 (THead (Flat Appl) t t0))) H13 t H_y0) in (eq_ind_r T (THead
+(Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3))
+(refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8:
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead
+(Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead
+(Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t: T).(\forall
+(t2: T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind Abst) x0 x1) H9) in
+(let H14 \def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w:
+T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P:
+Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind
+Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind
+Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl
+(THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0
+x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8:
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0:
+B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T
+t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4
+(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda
+(_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4
+(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat
+Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t: T).(\forall (t2:
+T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16
+\def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: T).(\forall
+(u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) H2
+(THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda
+(t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl)
+(lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl
+(THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2))
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2
+H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T
+(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead
+(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x:
+T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8:
+(pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2)
+\to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T
+(THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat Appl) t x) (\lambda (H9:
+(eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t x))).(\lambda (P:
+Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in
+(let H11 \def (eq_ind_r T x (\lambda (t: T).(pr0 t0 t)) H8 t0 H10) in (let
+H12 \def (eq_ind_r T x (\lambda (t: T).((eq T t0 t) \to (\forall (P:
+Prop).P))) H7 t0 H10) in (H12 (refl_equal T t0) P)))))) (pr0_comp t t
+(pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T
+(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead
+(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T
+(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x:
+T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0
+t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq
+T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat
+Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead
+(Flat Appl) t t0) t2)) (THead (Flat Appl) x t0) (\lambda (H7: (eq T (THead
+(Flat Appl) t t0) (THead (Flat Appl) x t0))).(\lambda (P: Prop).(let H8 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t]))
+(THead (Flat Appl) t t0) (THead (Flat Appl) x t0) H7) in (let H9 \def
+(eq_ind_r T x (\lambda (t0: T).(pr0 t t0)) H6 t H8) in (let H10 \def
+(eq_ind_r T x (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H5 t
+H8) in (H10 (refl_equal T t) P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0)
+(Flat Appl))))))) H4)) H3))) H1))) | Cast \Rightarrow (or_intror (\forall
+(t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq T (THead (Flat Cast) t t0)
+t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Cast) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Cast) t t0) t2)))
+(ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Cast) t t0) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Cast) t t0) t2))
+t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) t0)).(\lambda (P:
+Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_epsilon t0 t0 (pr0_refl t0)
+t)))])])))))) t1).
-axiom csubst0_drop_lt: \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))))))))))))) .
+inductive pr1: T \to (T \to Prop) \def
+| pr1_r: \forall (t: T).(pr1 t t)
+| pr1_u: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3:
+T).((pr1 t2 t3) \to (pr1 t1 t3))))).
+
+theorem pr1_pr0:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_u t2 t1 H t2
+(pr1_r t2)))).
+
+theorem pr1_t:
+ \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2
+t3) \to (pr1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3)))))
+(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda
+(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t3: T).((pr1 t4 t3) \to (pr1 t0
+t3))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_u t0 t3 H0 t5 (H2 t5
+H3)))))))))) t1 t2 H))).
+
+theorem pr1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall
+(k: K).(pr1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t:
+T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k
+t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_r (THead k t0 t))) (\lambda
+(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_u
+(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k
+t3 t) H2))))))) u1 u2 H))))).
+
+theorem pr1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall
+(k: K).(pr1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u:
+T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u
+t) (THead k u t0)))) (\lambda (t: T).(pr1_r (THead k u t))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_u (THead
+k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k u t4)
+H2))))))) t1 t2 H))))).
+
+theorem pr1_strip:
+ \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0
+t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
+(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4:
+T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
+(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
+(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_r t2))))) (\lambda
+(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda
+(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t3: T).((pr0 t2 t3) \to (ex2 T
+(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t3 t))))))).(\lambda (t5:
+T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda
+(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
+t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2
+x)).(ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T
+(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0:
+T).(\lambda (H6: (pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda
+(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_u x t5 H4 x0
+H7))))) (H2 x H5))))) (pr0_confluence t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
+
+theorem pr1_confluence:
+ \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0
+t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda
+(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4:
+T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda
+(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3))
+(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_r t2))))) (\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2
+t4)).(\lambda (H2: ((\forall (t3: T).((pr1 t2 t3) \to (ex2 T (\lambda (t:
+T).(pr1 t4 t)) (\lambda (t: T).(pr1 t3 t))))))).(\lambda (t5: T).(\lambda
+(H3: (pr1 t3 t5)).(ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1
+t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)))
+(\lambda (x: T).(\lambda (H4: (pr1 t5 x)).(\lambda (H5: (pr1 t2 x)).(ex2_ind
+T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T (\lambda (t:
+T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0: T).(\lambda (H6:
+(pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4
+t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_t x t5 H4 x0 H7))))) (H2 x H5)))))
+(pr1_strip t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
-axiom csubst0_drop_eq: \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) .
+inductive wcpr0: C \to (C \to Prop) \def
+| wcpr0_refl: \forall (c: C).(wcpr0 c c)
+| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall
+(u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k
+u1) (CHead c2 k u2)))))))).
+
+theorem wcpr0_gen_sort:
+ \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
+n))))
+\def
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0
+\def (match H return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (wcpr0 c
+c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort n))))))) with
+[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort n))).(\lambda (H1:
+(eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0 x) \to (eq C x
+(CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C (CSort n) (\lambda
+(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x H2)) c (sym_eq C c
+(CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k) \Rightarrow (\lambda
+(H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq C (CHead c2 k u2)
+x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e: C).(match e return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
+\Rightarrow True])) I (CSort n) H2) in (False_ind ((eq C (CHead c2 k u2) x)
+\to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort n))))) H4)) H3 H0
+H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x))))).
+
+theorem wcpr0_gen_head:
+ \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
+(CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
+\def
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
+(H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (c:
+C).(\lambda (c0: C).(\lambda (_: (wcpr0 c c0)).((eq C c (CHead c1 k u1)) \to
+((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
+C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))) with
+[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda
+(H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to
+(or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq
+C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead
+c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead
+c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k
+u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead
+c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1)
+(CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda
+(_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2))
+c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0)
+\Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda
+(H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e:
+C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead
+_ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let H5 \def
+(f_equal C K (\lambda (e: C).(match e return (\lambda (_: C).K) with [(CSort
+_) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c1
+k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in (eq_ind C c1 (\lambda (c:
+C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c2 k0 u2) x) \to ((wcpr0
+c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda
+(c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda
+(_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))))
+(\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to
+((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u0 u2) \to (or (eq C
+x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x
+(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))) (\lambda (H8: (eq T u0
+u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) \to ((wcpr0 c1
+c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3:
+C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
+T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))
+(\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k u2) (\lambda
+(c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C c (CHead c1 k u1))
+(ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c (CHead c3 k u3))))
+(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1 c2)).(\lambda (H11: (pr0
+u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T (\lambda
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2
+k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C (CHead c2
+k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0 u1 H8))) k0 (sym_eq K k0 k H7))) c0
+(sym_eq C c0 c1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead c1
+k u1)) (refl_equal C x))))))).
+
+theorem wcpr0_drop:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c0 k u1) (CHead
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
+c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c0 k u1)
+(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead
+e1 k0 u0) H3) return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CHead
+e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead
+c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))))) with [refl_equal
+\Rightarrow (\lambda (H3: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H4
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u1)
+(CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _)
+\Rightarrow k])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in ((let H6 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u1) (CHead e1
+k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u1 u0) \to
+(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u0 u2))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0
+(\lambda (k: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))))
+(\lambda (H8: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda
+(e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u0 u2))))) (let H9 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2))
+H2 u0 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wcpr0 c c3)) H0 e1
+H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3
+k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))) c3 u2 (drop_refl (CHead c3 k0
+u2)) H10 H9))) u1 (sym_eq T u1 u0 H8))) k (sym_eq K k k0 H7))) c0 (sym_eq C
+c0 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind
+(\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3:
+T).(\forall (k: K).((drop n O (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C
+T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 u2) (CHead e2 k
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3:
+T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead e1 k1 u3)) \to
+(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c3 k0 u2)
+(CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n
+O (CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2 k u4))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead e1 k0
+u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c3 (CHead e2
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c3 (CHead x0
+k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro
+C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u2)
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Bind b) n c3 (CHead
+x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c0 (CHead
+e1 k0 u0) u1 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
+((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 (Flat
+f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
+T).(drop n O (CHead c3 (Flat f) u2) (CHead e2 k u4)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3
+u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(drop (S n) O (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S
+n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c3 (CHead x0 k0
+x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro C
+T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u2)
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Flat f) n c3 (CHead
+x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c0
+(CHead e1 k0 u0) u1 n H4)))))))))) k) h)))))))))) c1 c2 H))).
+
+theorem wcpr0_drop_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead
+e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2
+(CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0
+u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
+e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u2) (CHead
+e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
+c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u2)
+(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead
+e1 k0 u0) H3) return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CHead
+e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead
+c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))))) with [refl_equal
+\Rightarrow (\lambda (H3: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H4
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u2)
+(CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _)
+\Rightarrow k])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in ((let H6 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2) (CHead e1
+k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u2 u0) \to
+(ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead
+e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u0))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0
+(\lambda (k: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2:
+T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))))
+(\lambda (H8: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u2 u0))))) (let H9 \def (eq_ind T u2 (\lambda (t: T).(pr0 u1 t))
+H2 u0 H8) in (let H10 \def (eq_ind C c3 (\lambda (c: C).(wcpr0 c0 c)) H0 e1
+H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0
+k0 u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))) c0 u1 (drop_refl (CHead c0 k0
+u1)) H10 H9))) u2 (sym_eq T u2 u0 H8))) k (sym_eq K k k0 H7))) c3 (sym_eq C
+c3 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind
+(\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3:
+T).(\forall (k: K).((drop n O (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C
+T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 k0 u1) (CHead e2 k
+u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3:
+T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead e1 k1 u3)) \to
+(ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c0 k0 u1)
+(CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n
+O (CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2 k u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead e1 k0
+u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0 (CHead e2
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c0 (CHead x0
+k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro
+C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Bind b) u1)
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Bind b) n c0 (CHead
+x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead
+e1 k0 u0) u2 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
+((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 (Flat
+f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2:
+T).(drop n O (CHead c0 (Flat f) u1) (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
+(drop (S n) O (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0 (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S
+n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c0 (CHead x0 k0
+x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro C
+T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Flat f) u1)
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Flat f) n c0 (CHead
+x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3
+(CHead e1 k0 u0) u2 n H4)))))))))) k) h)))))))))) c2 c1 H))).
+
+theorem wcpr0_getl:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
+u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 k u1) (CHead e1
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
+u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c0 k u1)
+(CHead e1 k0 u0))).((match k return (\lambda (k1: K).((clear (CHead c0 k1 u1)
+(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
+(CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) with [(Bind b)
+\Rightarrow (\lambda (H4: (clear (CHead c0 (Bind b) u1) (CHead e1 k0
+u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead
+e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1
+H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k]))
+(CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0)
+u1 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow
+t])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1
+k0 u0) u1 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1
+c0)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 k1 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3))))) (eq_ind_r T u1 (\lambda (t: T).(ex3_2 C T (\lambda
+(e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b)
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind_r C c0 (\lambda (c: C).(ex3_2 C T
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u1 u3))) c3 u2 (getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0
+H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c0 (Flat
+f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c0 (CHead
+e1 k0 u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e1 k0 u0) u1 H4))) in
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0
+k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u2) (CHead
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f
+u2) H7 H8)))))) H5)))]) (getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3))))))
+(K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3:
+T).(\forall (k: K).((getl n (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k u4))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall
+(k1: K).((getl (S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u2) (CHead e2 k1
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n
+(CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u2) (CHead e2 k u4))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let
+H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 n H4)) in
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0
+k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro
+C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2)
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c3 (CHead
+x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n
+(CHead c0 (Flat f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k u4))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let
+H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 n H4))
+in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
+u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead
+x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0
+x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
+(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat
+f) n c3 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2
+H))).
+
+theorem wcpr0_getl_back:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
+nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1
+k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall
+(u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda
+(u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k
+u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
+k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
+u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
+c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
+(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
+u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
+u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u2) (CHead e1
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k
+u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u2)
+(CHead e1 k0 u0))).((match k return (\lambda (k1: K).((clear (CHead c3 k1 u2)
+(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
+(CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) with [(Bind b)
+\Rightarrow (\lambda (H4: (clear (CHead c3 (Bind b) u2) (CHead e1 k0
+u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead
+e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2
+H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k]))
+(CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0)
+u2 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow
+t])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1
+k0 u0) u2 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1
+c3)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 k1 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0))))) (eq_ind_r T u2 (\lambda (t: T).(ex3_2 C T (\lambda
+(e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b)
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T
+(\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2
+(Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u2))) c0 u1 (getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0
+H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c3 (Flat
+f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c3 (CHead
+e1 k0 u0) c3 (drop_refl c3) (clear_gen_flat f c3 (CHead e1 k0 u0) u2 H4))) in
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c0 (CHead e2 k0
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c0 (CHead x0
+k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro
+C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) u1) (CHead
+e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_flat c0 (CHead x0 k0 x1) O H6 f
+u1) H7 H8)))))) H5)))]) (getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3))))))
+(K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3:
+T).(\forall (k: K).((getl n (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 k0 u1) (CHead e2 k u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall
+(k1: K).((getl (S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c0 k0 u1) (CHead e2 k1
+u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n
+(CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl n (CHead c0 (Bind b) u1) (CHead e2 k u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let
+H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c0 (CHead e2 k0
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c0 (CHead x0
+k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro
+C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1)
+(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c0 (CHead
+x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n:
+nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n
+(CHead c3 (Flat f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0:
+K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let
+H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4))
+in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (CHead e2
+k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
+C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
+u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c0 (CHead
+x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1
+u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0
+(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat
+f) n c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1
+H))).
-axiom csubst0_drop_eq_back: \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) .
+inductive pr2: C \to (T \to (T \to Prop)) \def
+| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to
+(pr2 c t1 t2))))
+| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i:
+nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2:
+T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1
+t)))))))))).
+
+theorem pr2_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort
+n) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t (TSort n)) \to
+((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0 t1 t2 H0)
+\Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TSort
+n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TSort
+n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))) (\lambda
+(H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T t2 x)
+\to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2 x)).(eq_ind
+T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n)))) (\lambda (H6:
+(pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TSort n)
+t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T (TSort n) (\lambda (t:
+T).(eq T t (TSort n))) (refl_equal T (TSort n)) x (pr0_gen_sort x n H6)))) t2
+(sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n) H4))) c0 (sym_eq C c0 c H1) H2
+H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3:
+(eq C c0 c)).(\lambda (H4: (eq T t1 (TSort n))).(\lambda (H5: (eq T t
+x)).(eq_ind C c (\lambda (c: C).((eq T t1 (TSort n)) \to ((eq T t x) \to
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t)
+\to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1 (TSort n))).(eq_ind T
+(TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr)
+u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))
+(\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl i c (CHead d
+(Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0) \to (eq T x
+(TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
+(pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let H11 \def (eq_ind T
+t2 (\lambda (t: T).(subst0 i u t x)) H10 (TSort n) (pr0_gen_sort t2 n H9)) in
+(subst0_gen_sort u x i n H11 (eq T x (TSort n))))))) t (sym_eq T t x H7))) t1
+(sym_eq T t1 (TSort n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in
+(H0 (refl_equal C c) (refl_equal T (TSort n)) (refl_equal T x)))))).
+
+theorem pr2_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
+(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S
+n) O u)))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef
+n) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t (TLRef n)) \to
+((eq T t0 x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(eq T x (lift (S n) O u))))))))))))) with [(pr2_free c0 t1 t2 H0)
+\Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef
+n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef
+n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda
+(_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))) (\lambda (H4: (eq T
+t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t
+t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T
+x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda
+(t: T).((pr0 (TLRef n) t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef
+n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef
+n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T
+t (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d
+(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O
+u))))))) (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d:
+C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef
+n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1
+(TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0
+t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1
+(TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1
+(TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0
+t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T
+t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i
+c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or
+(eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
+(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift
+(S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0:
+T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i
+u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
+T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
+T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind
+Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2
+x)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (TLRef
+n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u))
+(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
+(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift
+(S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S
+n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n: nat).(getl n c (CHead d
+(Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t: T).(pr2
+c (TLRef n) t)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O u)
+(\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda
+(u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
+T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift (S n) O u) (TLRef
+n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
+n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
+(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S
+n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S n) O u)))) x
+H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x H7))) t1 (sym_eq
+T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
+(refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T x)))))).
+
+theorem pr2_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H return
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
+t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x)
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst)
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+t1 t2))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq
+C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq
+T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to
+((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead
+(Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq
+T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
+(\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6:
+(pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0
+x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
+(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead
+(Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t:
+T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind
+Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free
+(CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2
+(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0
+(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2)
+\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind
+Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
+t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind
+Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda
+(H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1)
+(\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t
+x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
+t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
+(pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2
+x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1
+x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t:
+T).(subst0 i u t x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda
+(u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda
+(H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0
+x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1
+t1) t)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst) x2
+x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
+(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1
+(refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2
+H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1
+H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x
+(THead (Bind Abst) x0 t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) u x1
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2:
+T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0
+(s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c
+(THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r T
+(THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2))
+(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c
+(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0)
+t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind Abst) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0
+i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def
+(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead
+(Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda
+(t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3
+(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2
+H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S
+i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3
+H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i
+H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T
+t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))])
+in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal
+T x))))))).
+
+theorem pr2_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H return
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
+t0)).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x)
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
+Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))))) with [(pr2_free c0
+t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0
+(THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda
+(_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2)
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0
+(THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t:
+T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c
+t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
+(Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))
+(\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1
+x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t:
+T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in
+(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0
+x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda
+(H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)
+(pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0
+(sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3
+H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq
+C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq
+T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to
+((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to
+((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))))))
+(\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat
+Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr)
+u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1
+t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3:
+T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1)
+t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1
+x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0
+(THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0
+x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def
+(eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Flat Cast) x0 x1)
+H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead
+(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2:
+T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c
+t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2
+x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c
+d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16:
+(ex2 T (\lambda (t2: T).(eq T x (THead (Flat Cast) x0 t2))) (\lambda (t2:
+T).(subst0 (s (Flat Cast) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x
+(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
+t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda
+(H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat
+Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13)
+(pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) u x1 t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0
+i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8
+t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i
+H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1
+t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1)
+H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
+(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
+
+theorem pr2_gen_csort:
+ \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
+\to (pr0 t1 t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort
+n) t1 t2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (_: (pr2 c t t0)).((eq C c (CSort n)) \to ((eq T
+t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0)
+\Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0
+t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T
+t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4:
+(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
+(pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0
+t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2
+H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) |
+(pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort
+n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort
+n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d
+(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2)))))))
+(\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to
+((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u
+t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda
+(t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
+((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n)
+(CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u
+t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t
+(sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4
+H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1)
+(refl_equal T t2)))))).
+
+theorem pr2_gen_ctail:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
+(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0
+(clen c) u t t2)))))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c)
+(\lambda (c0: C).(pr2 c0 t1 t2)) (or (pr2 c t1 t2) (ex3 T (\lambda (_: T).(eq
+K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen
+c) u t t2)))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (or
+(pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t3:
+T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 t0)))))))) (\lambda
+(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
+(_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) (ex3 T (\lambda (_:
+T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(subst0
+(clen c) u t t4))) (pr2_free c t3 t4 H1))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
+Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0
+(CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda (c: C).(getl i c (CHead d
+(Bind Abbr) u0))) H1 (CTail k u c) H4) in (let H_x \def (getl_gen_tail k Abbr
+u u0 d c i H5) in (let H6 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq C d
+(CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0)))) (ex4
+nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
+Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))))
+(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0:
+T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (H7:
+(ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
+(CHead e (Bind Abbr) u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u
+e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t)
+(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
+(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq
+C d (CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr)
+u0))).(or_introl (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr)))
+(\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))
+(pr2_delta c x u0 i H9 t3 t4 H2 t H3))))) H7)) (\lambda (H7: (ex4 nat
+(\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
+Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort
+n))))).(ex4_ind nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_:
+nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n:
+nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k
+(Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c)
+u t0 t)))) (\lambda (x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda
+(H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d
+(CSort x0))).(let H12 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u0 t4
+t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 (\lambda (t0: T).(subst0
+(clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or
+(pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind Abbr))) (\lambda (t0:
+T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))))) (or_intror (pr2
+c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0:
+T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (ex3_intro T
+(\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
+(\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 (refl_equal K (Bind Abbr)) H2
+H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
+
+theorem pr2_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0)))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0
+t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u
+(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0
+t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i
+H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0
+t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2
+u)))))))))))) c t1 t2 H)))))).
+
+theorem pr2_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0
+(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2
+t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1
+t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c
+u1 u2 H)))))).
+
+theorem pr2_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead
+k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind
+b) u) t1 t2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind b)
+u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1)
+(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
+(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0
+t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_:
+C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind
+b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1
+(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u
+t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
+(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
+u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead
+(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T
+t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1)
+H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda
+(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda
+(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0
+t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0
+t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
+u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
+t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0
+t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind
+b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4:
+T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3)
+\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u
+t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr)
+u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match
+i return (\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr)
+u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind
+b) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c (Bind b) u)
+(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0)
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H13
+\def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c
+(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
+Abbr) u0) H11))) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u)
+(clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b)
+u) (CHead d (Bind Abbr) u0) H11))) in (\lambda (H15: (eq B Abbr b)).(\lambda
+(_: (eq C d c)).(let H17 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t3
+t2)) H12 u H14) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) u
+t1) (THead (Bind b) u t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind
+Abbr) u t2) (pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H17)) b H15))))) H13))
+H)))) | (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Bind b) u)
+(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta
+c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n
+H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u)
+t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r
+(Bind b) n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1
+H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0
+(refl_equal C (CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2)))))
+(\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def
+(match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda
+(_: (pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq
+T t0 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))))) with
+[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f)
+u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead
+c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0
+t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4:
+(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
+(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3
+t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u
+t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead
+(Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6
+(Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0
+(CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t
+H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4:
+(eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u)
+(\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d
+(Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead
+(Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat
+f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to
+(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T
+t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d
+(Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead
+(Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c
+(Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda
+(H10: (subst0 i u0 t3 t2)).((match i return (\lambda (n: nat).((getl n (CHead
+c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c
+(THead (Flat f) u t1) (THead (Flat f) u t2))))) with [O \Rightarrow (\lambda
+(H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H12:
+(subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr)
+u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u0) u
+(getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) (THead (Flat
+f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Flat f))
+(THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) | (S n)
+\Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d (Bind
+Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r
+(Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11) (THead
+(Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9
+(Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f) n)
+H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0
+(sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal
+C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))).
+
+theorem clear_pr2_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H
+return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c t
+t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1
+t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c
+c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2
+(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1
+t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
+t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind
+T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1
+t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1
+H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t
+H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0
+t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1)
+\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3)
+\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d
+(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1
+t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2
+(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1
+t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10:
+(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i
+(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11))))
+t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1
+H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T
+t2)))))))).
+
+theorem pr2_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f:
+F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f:
+F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda
+(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0
+(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))).
+
+theorem pr2_ctail:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0)
+t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail
+Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))).
+
+theorem pr2_gen_cbind:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1)
+(THead (Bind b) v t2)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H
+return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0
+t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to
+(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with [(pr2_free c0
+t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) v))).(\lambda
+(H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) v)
+(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c
+(THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c
+(THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3
+t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v
+t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead
+(Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6
+(Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0
+(CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2)
+\Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T
+t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda
+(c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr)
+u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1)
+(THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1
+(\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d
+(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead
+(Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind
+T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u))
+\to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1)
+(THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead
+d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3
+t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let
+H \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead
+c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j:
+nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1) (THead
+(Bind b) v t2)) (\lambda (H11: (land (eq nat i O) (eq C (CHead d (Bind Abbr)
+u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind Abbr)
+u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))
+(\lambda (H12: (eq nat i O)).(\lambda (H13: (eq C (CHead d (Bind Abbr) u)
+(CHead c (Bind b) v))).(let H14 \def (f_equal C C (\lambda (e: C).(match e
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _)
+\Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let
+H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let H16 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) v) H13) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C
+d c)).(let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 O
+H12) in (let H20 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H19 v
+H16) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) v t1) (THead
+(Bind b) v t2))) (pr2_free c (THead (Bind Abbr) v t1) (THead (Bind Abbr) v
+t2) (pr0_delta v v (pr0_refl v) t1 t3 H9 t2 H20)) b H17)))))) H15)) H14))))
+H11)) (\lambda (H11: (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda
+(j: nat).(getl j c (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j:
+nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))
+(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (x:
+nat).(\lambda (H12: (eq nat i (S x))).(\lambda (H13: (getl x c (CHead d (Bind
+Abbr) u))).(let H14 \def (f_equal nat nat (\lambda (e: nat).e) i (S x) H12)
+in (let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 (S
+x) H14) in (pr2_head_2 c v t1 t2 (Bind b) (pr2_delta (CHead c (Bind b) v) d u
+(S x) (getl_clear_bind b (CHead c (Bind b) v) c v (clear_bind b c v) (CHead d
+(Bind Abbr) u) x H13) t1 t3 H9 t2 H15))))))) H11)) H)))))) t (sym_eq T t t2
+H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5
+H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Bind b) v)) (refl_equal T t1)
+(refl_equal T t2)))))))).
+
+theorem pr2_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
+(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H
+return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0
+t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to
+(pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1:
+(eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq
+T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: C).((eq T t0 t1) \to
+((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) (\lambda (H4: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c t1
+t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t)
+\to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c t1 t2 H6)) t3
+(sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat
+f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) \Rightarrow
+(\lambda (H3: (eq C c0 (CHead c (Flat f) v))).(\lambda (H4: (eq T t0
+t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (c1:
+C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u))
+\to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2))))))) (\lambda (H6:
+(eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead
+c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3
+t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda
+(t4: T).((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t1
+t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 t2))))) (\lambda (H8: (getl i
+(CHead c (Flat f) v) (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 t1
+t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y \def (getl_gen_flat f c
+(CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i H_y t1 t3 H9 t2
+H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0
+(CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c
+(Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+
+theorem pr2_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1
+t2)).(let H1 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1) \to ((eq T
+t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with [(pr2_free c0 t0
+t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: (eq T t0
+t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T t0 t1)
+\to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d
+t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
+t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6:
+(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d
+t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1)
+(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T
+t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
+t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0
+t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1)
+\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3)
+\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda
+(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e
+(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c
+(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2
+(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to
+((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9:
+(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
+(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2))
+(\lambda (H0: (lt i d)).(let H \def (drop_getl_trans_le i d (le_S_n i d (le_S
+(S i) d H0)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C (\lambda
+(e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda (e1:
+C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1
+(CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda (x0:
+C).(\lambda (x1: C).(\lambda (H12: (drop i O c x0)).(\lambda (H13: (drop h
+(minus d i) x0 x1)).(\lambda (H14: (clear x1 (CHead d0 (Bind Abbr) u))).(let
+H15 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 x1)) H13 (S
+(minus d (S i))) (minus_x_Sy d i H0)) in (let H16 \def (drop_clear_S x1 x0 h
+(minus d (S i)) H15 Abbr d0 u H14) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abbr) (lift h (minus d (S i)) u)))) (\lambda (c1: C).(drop h
+(minus d (S i)) c1 d0)) (pr2 c (lift h d t1) (lift h d t2)) (\lambda (x:
+C).(\lambda (H17: (clear x0 (CHead x (Bind Abbr) (lift h (minus d (S i))
+u)))).(\lambda (_: (drop h (minus d (S i)) x d0)).(pr2_delta c x (lift h
+(minus d (S i)) u) i (getl_intro i c (CHead x (Bind Abbr) (lift h (minus d (S
+i)) u)) x0 H12 H17) (lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d)
+(lift h d t2) (subst0_lift_lt t3 t2 u i H11 d H0 h))))) H16)))))))) H)))
+(\lambda (H0: (le d i)).(pr2_delta c d0 u (plus i h) (drop_getl_trans_ge i c
+e d h H (CHead d0 (Bind Abbr) u) H9 H0) (lift h d t1) (lift h d t3) (pr0_lift
+t1 t3 H10 h d) (lift h d t2) (subst0_lift_ge t3 t2 u i h H11 d H0))))))) t
+(sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1
+H2 H3))))]) in (H1 (refl_equal C e) (refl_equal T t1) (refl_equal T
+t2)))))))))).
+
+theorem pr2_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H return
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
+t0)).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x)
+\to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
+Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))))) with [(pr2_free c0
+t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0
+(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda
+(_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2)
+\to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H4: (eq T t0
+(THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t:
+T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))) (\lambda
+(H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1)
+t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))) (\lambda (H6: (pr0 (THead
+(Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H7: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x
+(THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1
+x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0
+x1)) (pr2_free c u1 x0 H8) (pr2_free c t1 x1 H9))) x H)))))) H7)) (\lambda
+(H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda
+(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr)
+x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r
+T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T
+(THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free
+c u1 x2 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) x1
+x3 H10))))) t1 H) x H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b)
+v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
+(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b)
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat
+Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1
+(THead (Bind x0) x1 x2))).(\lambda (H9: (eq T x (THead (Bind x0) x4 (THead
+(Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda
+(H11: (pr0 x1 x4)).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
+x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x5 x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1
+x3 H10) (pr2_free c x1 x4 H11) (pr2_free (CHead c (Bind x0) x4) x2 x5 H12)))
+t1 H8) x H9))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x
+H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1)
+H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda
+(H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda
+(H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1
+t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0
+t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind
+T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c
+(CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))) (\lambda (H7: (eq T t
+x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T
+T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H8: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1)
+t2)).(\lambda (H10: (subst0 i u t2 x)).(or3_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))
+(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr0
+u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t:
+T).(subst0 i u t x)) H10 (THead (Flat Appl) x0 x1) H) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H15: (ex2 T (\lambda (u2:
+T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H16: (eq T x
+(THead (Flat Appl) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(eq_ind_r T
+(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1
+t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O
+u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1
+x0 H12 x2 H17) (pr2_free c t1 x1 H13))) x H16)))) H15)) (\lambda (H15: (ex2 T
+(\lambda (t2: T).(eq T x (THead (Flat Appl) x0 t2))) (\lambda (t2: T).(subst0
+(s (Flat Appl) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead
+(Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
+(H16: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H17: (subst0 (s (Flat
+Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat
+Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr)
+u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2
+(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0
+x2)) (pr2_free c u1 x0 H12) (pr2_delta c d u i H8 t1 x1 H13 x2 H17))) x
+H16)))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
+x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u x1
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H16: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H17:
+(subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Flat Appl) i) u x1
+x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2
+(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2
+x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (pr2_delta c d u i H8 t1 x1 H13
+x3 H18))) x H16)))))) H15)) (subst0_gen_head (Flat Appl) u x0 x1 x i
+H14)))))))) H11)) (\lambda (H11: (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind
+Abst) x0 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda
+(H13: (pr0 u1 x2)).(\lambda (H14: (pr0 x1 x3)).(let H15 \def (eq_ind T t2
+(\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Abbr) x2 x3) H12) in
+(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
+u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda
+(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead
+(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T
+(\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
+i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
+(H17: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H18: (subst0 i u x2
+x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t1: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H13 x4
+H18) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3
+H14))))) x H17)))) H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x
+(THead (Bind Abbr) x2 t2))) (\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr)
+x2 x4))).(\lambda (H18: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
+(THead (Bind Abbr) x2 x4) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H13) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i)
+(getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d
+(Bind Abbr) u) i H8) x1 x3 H14 x4 H18))))) x H17)))) H16)) (\lambda (H16:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H17: (eq T x
+(THead (Bind Abbr) x4 x5))).(\lambda (H18: (subst0 i u x2 x4)).(\lambda (H19:
+(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5)
+(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H13 x4
+H18) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S
+i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d
+(Bind Abbr) u) i H8) x1 x3 H14 x5 H19))))) x H17)))))) H16)) (subst0_gen_head
+(Bind Abbr) u x2 x3 x i H15)) t1 H)))))))))) H11)) (\lambda (H11: (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H12: (eq T t1 (THead (Bind
+x0) x1 x2))).(\lambda (H13: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
+(lift (S O) O x3) x5)))).(\lambda (H14: (pr0 u1 x3)).(\lambda (H15: (pr0 x1
+x4)).(\lambda (H16: (pr0 x2 x5)).(let H17 \def (eq_ind T t2 (\lambda (t:
+T).(subst0 i u t x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O
+x3) x5)) H13) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t1: T).(or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
+x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x
+(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead
+(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))) (\lambda (H18: (ex2 T (\lambda (u2: T).(eq T x (THead
+(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2:
+T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind x0)
+u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u
+x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(H19: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
+x5)))).(\lambda (H20: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
+(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t1: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
+O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
+(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H (refl_equal T (THead
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
+(S O) O x3) x5))) (pr2_free c u1 x3 H14) (pr2_delta c d u i H8 x1 x4 H15 x6
+H20) (pr2_free (CHead c (Bind x0) x6) x2 x5 H16))) x H19)))) H18)) (\lambda
+(H18: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind x0) x4 t2))) (\lambda (t2:
+T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda
+(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(H19: (eq T x (THead (Bind x0) x4 x6))).(\lambda (H20: (subst0 (s (Bind x0)
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind
+x0) x4 x6) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0)
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat
+Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
+u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3)
+t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
+O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
+x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H21: (ex2 T
+(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2:
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda (H22: (eq T x6
+(THead (Flat Appl) x x5))).(\lambda (H23: (subst0 (s (Bind x0) i) u (lift (S
+O) O x3) x)).(eq_ind_r T (THead (Flat Appl) x x5) (\lambda (t1: T).(or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1)
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
+x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x4 (THead (Flat Appl) x x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x x5)) (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) x x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H24: (eq T x (lift (S O) O
+x7))).(\lambda (H25: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x7)).(let H26
+\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
+x3 x7)) H25 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x7) (\lambda (t1:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) t1 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x5)) (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) t1 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x7) x5)) (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x7) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x7) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
+(Flat Appl) (lift (S O) O x7) x5)) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x7 x4 H (refl_equal T (THead
+(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
+(S O) O x7) x5))) (pr2_delta c d u i H8 u1 x3 H14 x7 H26) (pr2_free c x1 x4
+H15) (pr2_free (CHead c (Bind x0) x4) x2 x5 H16))) x H24)))))
+(subst0_gen_lift_ge u x3 x (s (Bind x0) i) (S O) O H23 (le_S_n (S O) (S i)
+(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x6
+H22)))) H21)) (\lambda (H21: (ex2 T (\lambda (t2: T).(eq T x6 (THead (Flat
+Appl) (lift (S O) O x3) t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) (s
+(Bind x0) i)) u x5 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s
+(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda
+(H22: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x))).(\lambda (H23:
+(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x)).(eq_ind_r T (THead (Flat
+Appl) (lift (S O) O x3) x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x3) x)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x)) (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x))) (pr2_free c u1
+x3 H14) (pr2_free c x1 x4 H15) (pr2_delta (CHead c (Bind x0) x4) d u (S i)
+(getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d
+(Bind Abbr) u) i H8) x2 x5 H16 x H23))) x6 H22)))) H21)) (\lambda (H21:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x6 (THead (Flat Appl) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
+O x3) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) (s (Bind
+x0) i)) u x5 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
+T).(\lambda (H22: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H23: (subst0
+(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H24: (subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
+(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
+i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda
+(H25: (eq T x7 (lift (S O) O x))).(\lambda (H26: (subst0 (minus (s (Bind x0)
+i) (S O)) u x3 x)).(let H27 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
+(\lambda (n: nat).(subst0 n u x3 x)) H26 i (s_arith1 x0 i)) in (eq_ind_r T
+(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8)) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) t1 x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
+O x) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))) x0 x1 x2 x8 x x4 H (refl_equal T (THead (Bind x0) x1 x2))
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)))
+(pr2_delta c d u i H8 u1 x3 H14 x H27) (pr2_free c x1 x4 H15) (pr2_delta
+(CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x4) c
+x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x8 H24))) x7
+H25))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H23 (le_S_n (S
+O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n
+i))))))) x6 H22)))))) H21)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3)
+x5 x6 (s (Bind x0) i) H20)) x H19)))) H18)) (\lambda (H18: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind x0) u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda
+(_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S
+O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(x7: T).(\lambda (H19: (eq T x (THead (Bind x0) x6 x7))).(\lambda (H20:
+(subst0 i u x4 x6)).(\lambda (H21: (subst0 (s (Bind x0) i) u (THead (Flat
+Appl) (lift (S O) O x3) x5) x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda
+(t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2))) (ex2 T (\lambda (t3: T).(eq T x7
+(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda
+(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H22: (ex2 T (\lambda (u2:
+T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0)
+i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x7 (THead
+(Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O
+x3) u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
+x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda
+(H23: (eq T x7 (THead (Flat Appl) x x5))).(\lambda (H24: (subst0 (s (Bind x0)
+i) u (lift (S O) O x3) x)).(eq_ind_r T (THead (Flat Appl) x x5) (\lambda (t1:
+T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+T).(eq T x (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
+i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x x5)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x x5)) (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H25: (eq T x
+(lift (S O) O x8))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3
+x8)).(let H27 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n:
+nat).(subst0 n u x3 x8)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O
+x8) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) t1 x5)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) t1 x5)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
+Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x5 x8 x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5))) (pr2_delta c d
+u i H8 u1 x3 H14 x8 H27) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_free
+(CHead c (Bind x0) x6) x2 x5 H16))) x H25))))) (subst0_gen_lift_ge u x3 x (s
+(Bind x0) i) (S O) O H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S
+O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H23)))) H22)) (\lambda (H22: (ex2
+T (\lambda (t2: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) t2)))
+(\lambda (t2: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t2)))).(ex2_ind
+T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)) (or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7)
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda (H23: (eq T x7
+(THead (Flat Appl) (lift (S O) O x3) x))).(\lambda (H24: (subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 x)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O
+x3) x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T (THead (Bind x0) x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
+Appl) (lift (S O) O x3) x)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x x3 x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x))) (pr2_free c u1
+x3 H14) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_delta (CHead c (Bind x0)
+x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0
+c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x H24))) x7 H23)))) H22))
+(\lambda (H22: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x7 (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0)
+i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s
+(Flat Appl) (s (Bind x0) i)) u x5 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda
+(_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6
+x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
+(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (x9:
+T).(\lambda (H23: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H24: (subst0
+(s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H25: (subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
+(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
+i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda
+(H26: (eq T x8 (lift (S O) O x))).(\lambda (H27: (subst0 (minus (s (Bind x0)
+i) (S O)) u x3 x)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
+(\lambda (n: nat).(subst0 n u x3 x)) H27 i (s_arith1 x0 i)) in (eq_ind_r T
+(lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9)) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) t1 x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9))
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
+O x) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
+(ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat
+Appl) (lift (S O) O x) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))) x0 x1 x2 x9 x x6 H (refl_equal T (THead (Bind x0) x1 x2))
+(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)))
+(pr2_delta c d u i H8 u1 x3 H14 x H28) (pr2_delta c d u i H8 x1 x4 H15 x6
+H20) (pr2_delta (CHead c (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c
+(Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5
+H16 x9 H25))) x8 H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O
+H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm
+O i (le_O_n i))))))) x7 H23)))))) H22)) (subst0_gen_head (Flat Appl) u (lift
+(S O) O x3) x5 x7 (s (Bind x0) i) H21)) x H19)))))) H18)) (subst0_gen_head
+(Bind x0) u x4 (THead (Flat Appl) (lift (S O) O x3) x5) x i H17)) t1
+H12)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 H9))))) t (sym_eq T t x H7)))
+t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5
+H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Flat Appl) u1
+t1)) (refl_equal T x))))))).
+
+theorem pr2_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T
+T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H return
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
+t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x)
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
+c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
+x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0
+c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2
+x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq
+T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
+(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
+T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
+(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind
+Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2
+x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
+(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
+T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
+(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
+(\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
+t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind
+Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c
+(Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda
+(_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))
+(\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
+t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
+T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T
+T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
+O u2 y t2)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
+t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
+t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H: (eq T x (THead (Bind Abbr) x0
+x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
+(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind
+(pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y
+x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
+c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
+x))))) (\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1)
+(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u:
+T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda
+(_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
+T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1)
+z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
+(lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
+t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind
+Abbr) x0 x1))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
+t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x0 x1
+(refl_equal T (THead (Bind Abbr) x0 x1)) (pr2_free c u1 x0 H8) (or3_intro0
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 x1))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
+x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b:
+B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H9)))))) x H))
+(\lambda (H_x0: (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O
+x0 y x1)))).(ex2_ind T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O
+x0 y x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u:
+T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda
+(_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
+T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1)
+z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
+(lift (S O) O x))))) (\lambda (x2: T).(\lambda (H9: (pr0 t1 x2)).(\lambda
+(H10: (subst0 O x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t:
+T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
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+(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H17
+(pr2_delta c d u i H8 u1 x0 H12 x3 H18) (or3_intro2 (\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
+u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T
+(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y:
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
+T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
+(Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O
+(getl_refl Abbr c u1) t1 x2 H13 x5 H20) H21 (pr2_delta (CHead c (Bind Abbr)
+u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1
+(pr0_refl x1) x4 H19)))))))) (pr0_subst0_back x0 x2 x1 O H14 u1 H12)))))))
+H16)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H15)))))) H_x0)) H_x))))))
+H11)) (\lambda (H: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
+u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
+(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i)
+(getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H
+(lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i)))))))
+(pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead
+(Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
+(refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T
+x))))))).
+
+theorem pr2_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H return
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
+t0)).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x)
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
+(lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda
+(H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda
+(H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1
+t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
+(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind
+Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
+x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
+(Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead
+(Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O)
+O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
+(lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H:
+(eq T x (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9:
+(pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
+t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1)))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void)
+x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind
+Void) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free
+(CHead c (Bind b) u) t1 x1 H9))))) x H)))))) H7)) (\lambda (H: (pr0 t1 (lift
+(S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead
+c (Bind b) u) t1 (lift (S O) O x) H))))) (pr0_gen_void u1 t1 x H6))) t2
+(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H4))) c0
+(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2)
+\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind
+Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
+t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind
+Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))))))))
+(\lambda (H6: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind
+Void) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr)
+u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))))
+(\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d
+(Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1 t1) t2) \to ((subst0 i u t2
+t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind
+Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
+t1 (lift (S O) O x))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1) t2)).(\lambda (H10: (subst0
+i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0)
+t1 (lift (S O) O x))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c
+(Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H12: (pr0 u1
+x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t:
+T).(subst0 i u t x)) H10 (THead (Bind Void) x0 x1) H) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
+(\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x
+(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
+(S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Void) x2
+x1))).(\lambda (H17: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+u0) t1 t3))))) x2 x1 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b:
+B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H13)))))))) H15))
+(\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Void) x0 t2)))
+(\lambda (t2: T).(subst0 (s (Bind Void) i) u x1 t2)))).(ex2_ind T (\lambda
+(t3: T).(eq T x (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
+Void) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c
+(Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x
+(THead (Bind Void) x0 x2))).(\lambda (H17: (subst0 (s (Bind Void) i) u x1
+x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
+(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c
+(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x0 x2 H16
+(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c
+(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0)
+t1 x1 H13 x2 H17)))))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s (Bind Void) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Bind Void)
+x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind
+Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
+t3))))) x2 x3 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 H18)))))))))) H15))
+(subst0_gen_head (Bind Void) u x0 x1 x i H14)))))))) H11)) (\lambda (H: (pr0
+t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda
+(u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c
+(CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H (lift (S O) O x)
+(subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2
+H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6)))
+c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
+(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
+
+theorem pr2_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0:
+(drop h d c e)).(let H1 \def (match H return (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
+(lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
+t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1)
+\Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d
+t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift
+h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T
+x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0
+(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to
+((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
+T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t:
+T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3)))
+(\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1)
+x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0
+t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2
+e t1 t3))) (\lambda (x0: T).(\lambda (H: (eq T x (lift h d x0))).(\lambda
+(H8: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda
+(t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2
+T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2
+e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H8)) x H))))
+(pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h
+d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
+t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0
+(lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c: C).((eq T
+t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c (CHead d0 (Bind Abbr) u))
+\to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x
+(lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0
+(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to
+((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t)
+\to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e
+t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i
+c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2
+t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2
+e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda
+(H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
+T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x0: T).(\lambda (H: (eq T t2 (lift h d x0))).(\lambda (H12: (pr0 t1
+x0)).(let H13 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H11 (lift h
+d x0) H) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d t3)))
+(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H14: (lt i d)).(let H15 \def
+(eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S
+i)))) (lt_plus_minus i d H14)) in (let H16 \def (eq_ind nat d (\lambda (n:
+nat).(subst0 i u (lift h n x0) x)) H13 (S (plus i (minus d (S i))))
+(lt_plus_minus i d H14)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d
+t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
+C).(\lambda (H0: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H17: (getl i
+e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
+x2)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 i t (lift h (S (plus i
+(minus d (S i)))) x0) x)) H16 (lift h (minus d (S i)) x1) H0) in (ex2_ind T
+(\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3))) (\lambda
+(t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3)))
+(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H20: (eq T x
+(lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H21: (subst0 i x1 x0
+x3)).(let H22 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
+nat).(eq T x (lift h n x3))) H20 d (lt_plus_minus i d H14)) in (ex_intro2 T
+(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
+H22 (pr2_delta e x2 x1 i H17 t1 x0 H12 x3 H21)))))) (subst0_gen_lift_lt x1 x0
+x i h (minus d (S i)) H19)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h
+(minus d (S i)) H15))))) (\lambda (H14: (le d i)).(lt_le_e i (plus d h) (ex2
+T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (H15: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H14 H15
+H13 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e
+t1 t3))))) (\lambda (H15: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq
+T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
+(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x1: T).(\lambda (H16: (eq T x (lift h d x1))).(\lambda (H17:
+(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d
+t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H16 (pr2_delta e d0 u (minus i h)
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H15) t1 x0 H12 x1
+H17))))) (subst0_gen_lift_ge u x0 x i h d H13 H15)))))))))) (pr0_gen_lift t1
+t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0
+(sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T
+(lift h d t1)) (refl_equal T x)))))))))).
+
+theorem pr2_confluence__pr2_free_free:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0
+t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0
+t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t))
+(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2
+x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
+(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1)))))
+(pr0_confluence t0 t2 H0 t1 H))))))).
+
+theorem pr2_confluence__pr2_free_delta:
+ \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2)
+\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))))))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0
+t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0
+t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4
+t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
+(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4:
+(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
+(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2
+c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2
+x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2:
+T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda
+(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7:
+(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0
+H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u))))))
+(pr0_confluence t0 t4 H1 t1 H))))))))))))).
+
+theorem pr2_confluence__pr2_delta_delta:
+ \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall
+(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u:
+T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d
+(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c
+(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to
+(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))))))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u:
+T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i
+c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0
+i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda
+(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t:
+T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4
+x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0
+t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2
+x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda
+(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda
+(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0
+u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x
+x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))
+x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8))
+(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T
+(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind
+T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0:
+T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0
+t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x
+w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t))
+(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H
+t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2))
+(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2
+w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1
+t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2
+x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t:
+T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i
+i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0
+i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15:
+(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d
+u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9
+(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def
+(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15
+\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0)))
+H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0:
+C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind
+Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead
+d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind
+Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono
+c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda
+(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x
+x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c
+(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0
+(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind
+(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t:
+T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda
+(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1
+x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in
+(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0
+(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T
+(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0
+t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i
+u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))
+(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i
+u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c
+t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2
+x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1
+x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u
+x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2
+t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11)))
+(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10))
+(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x
+H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3
+H0))))))))))))))))))).
+
+theorem pr2_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t:
+T).(pr2 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
+t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H
+return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: (pr2 c0
+t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T (\lambda
+(t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))))) with
+[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3:
+(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T
+t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c
+t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind
+T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t0:
+T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H6: (eq T t4
+t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0:
+T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H7: (pr0 t0
+t1)).(let H8 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t3: T).(\lambda (_: (pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T
+t3 t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0)))))))))) with [(pr2_free c1 t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1
+c)).(\lambda (H8: (eq T t5 t0)).(\lambda (H9: (eq T t6 t2)).(eq_ind C c
+(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda
+(H10: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t
+t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0)))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0
+t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0))))) (\lambda (H12: (pr0 t0 t2)).(pr2_confluence__pr2_free_free c t0 t1 t2
+H7 H12)) t6 (sym_eq T t6 t2 H11))) t5 (sym_eq T t5 t0 H10))) c1 (sym_eq C c1
+c H6) H8 H9 H5)))) | (pr2_delta c1 d u i H5 t5 t6 H6 t H7) \Rightarrow
+(\lambda (H8: (eq C c1 c)).(\lambda (H9: (eq T t5 t0)).(\lambda (H10: (eq T t
+t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i
+c0 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2
+T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))))
+(\lambda (H11: (eq T t5 t0)).(eq_ind T t0 (\lambda (t0: T).((eq T t t2) \to
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t)
+\to (ex2 T (\lambda (t2: T).(pr2 c t1 t2)) (\lambda (t1: T).(pr2 c t2
+t1)))))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (t3: T).((getl i
+c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t3) \to (ex2
+T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))
+(\lambda (H13: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H14: (pr0 t0
+t6)).(\lambda (H15: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d
+t0 t1 t2 t6 u i H7 H13 H14 H15)))) t (sym_eq T t t2 H12))) t5 (sym_eq T t5 t0
+H11))) c1 (sym_eq C c1 c H8) H9 H10 H5 H6 H7))))]) in (H8 (refl_equal C c)
+(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T
+t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4
+H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3
+t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0)
+\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4)
+\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda
+(t0: T).(pr2 c t2 t0))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0
+(\lambda (t0: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 t0 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t2: T).(pr2 c t1
+t2)) (\lambda (t1: T).(pr2 c t2 t1)))))))) (\lambda (H8: (eq T t t1)).(eq_ind
+T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4)
+\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda
+(t0: T).(pr2 c t2 t0))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12
+\def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3:
+T).(\lambda (_: (pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3
+t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0)))))))))) with [(pr2_free c1 t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c1
+c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t6 t2)).(eq_ind C c
+(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda
+(H14: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t
+t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0)))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0
+t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0))))) (\lambda (H16: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1)
+(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H16 H9 H10 H11))) t6
+(sym_eq T t6 t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 (sym_eq C c1 c H8) H12
+H13 H7)))) | (pr2_delta c1 d0 u0 i0 H7 t5 t6 H8 t7 H9) \Rightarrow (\lambda
+(H12: (eq C c1 c)).(\lambda (H13: (eq T t5 t0)).(\lambda (H14: (eq T t7
+t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl
+i0 c0 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7)
+\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))))
+(\lambda (H15: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t7 t2) \to
+((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t t6) \to ((subst0 i0 u0 t6
+t7) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2
+t0)))))))) (\lambda (H16: (eq T t7 t2)).(eq_ind T t2 (\lambda (t: T).((getl
+i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0 t6 t) \to
+(ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))
+(\lambda (H17: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (pr0 t0
+t6)).(\lambda (H19: (subst0 i0 u0 t6 t2)).(pr2_confluence__pr2_delta_delta c
+d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 H11 H17 H18 H19)))) t7 (sym_eq T t7 t2
+H16))) t5 (sym_eq T t5 t0 H15))) c1 (sym_eq C c1 c H12) H13 H14 H7 H8
+H9))))]) in (H12 (refl_equal C c) (refl_equal T t0) (refl_equal T t2)))))) t
+(sym_eq T t t1 H8))) t3 (sym_eq T t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1
+H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T t0) (refl_equal T
+t1)))))))).
+
+theorem pr2_delta1:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2)
+\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2
+t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0)
+(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2
+H0 t0 H2))) t H1)))))))))).
+
+theorem pr2_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr2 c t1 t2)).(let H1 \def (match H0 return (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c)
+\to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1 w1)
+\to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2
+w2)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C
+c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c
+(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall
+(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
+(\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall
+(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
+(\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3
+t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i
+v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1
+i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H0:
+(subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
+T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 x)).(\lambda
+(H9: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda
+(w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H8) H9)))) (pr0_subst1 t1 t2
+H7 v w1 i H0 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1
+H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i0 H1 t0 t3 H2 t
+H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0
+t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T t0 t1)
+\to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3)
+\to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T
+(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))
+(\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to
+((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i0 u t3 t)
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))) (\lambda (H8: (eq T t
+t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d (Bind Abbr) u)) \to
+((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1: T).((subst1 i v t1
+w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v
+t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind Abbr) u))).(\lambda
+(H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3 t2)).(\lambda (w1:
+T).(\lambda (H0: (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2))
+(\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
+(\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H12: (pr0 w1
+x)).(\lambda (H13: (subst1 i v t3 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2:
+T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (H14: (not
+(eq nat i i0))).(ex2_ind T (\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1:
+T).(subst1 i0 u x t1)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H15: (subst1 i v t2
+x0)).(\lambda (H16: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c d u i0 H9 w1 x
+H12 x0 H16) H15)))) (subst1_confluence_neq t3 t2 u i0 (subst1_single i0 u t3
+t2 H11) x v i H13 (sym_not_eq nat i i0 H14)))) (\lambda (H14: (eq nat i
+i0)).(let H15 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t3 t2)) H11
+i H14) in (let H16 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d
+(Bind Abbr) u))) H9 i H14) in (let H17 \def (eq_ind C (CHead e (Bind Abbr) v)
+(\lambda (c0: C).(getl i c c0)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead
+e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (let H18 \def (f_equal
+C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind Abbr) v) (CHead
+d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind
+Abbr) u) H16)) in ((let H19 \def (f_equal C T (\lambda (e0: C).(match e0
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t)
+\Rightarrow t])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c
+(CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (\lambda (H20:
+(eq C e d)).(let H21 \def (eq_ind_r T u (\lambda (t: T).(getl i c (CHead d
+(Bind Abbr) t))) H17 v H19) in (let H22 \def (eq_ind_r T u (\lambda (t:
+T).(subst0 i t t3 t2)) H15 v H19) in (let H23 \def (eq_ind_r C d (\lambda
+(c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H21 e H20) in (ex2_ind T
+(\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: T).(subst1 i v x t1)) (ex2
+T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))
+(\lambda (x0: T).(\lambda (H24: (subst1 i v t2 x0)).(\lambda (H25: (subst1 i
+v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t2 w2)) x0 (pr2_delta1 c e v i H23 w1 x H12 x0 H25) H24))))
+(subst1_confluence_eq t3 t2 v i (subst1_single i v t3 t2 H22) x H13)))))))
+H18)))))))))) (pr0_subst1 t1 t3 H10 v w1 i H0 v (pr0_refl v)))))))) t (sym_eq
+T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2
+H3))))]) in (H1 (refl_equal C c) (refl_equal T t1) (refl_equal T t2)))))))))).
+
+theorem pr2_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e:
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to
+(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0
+a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda
+(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1
+x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
+d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d
+x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2:
+T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
+x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0
+(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda
+(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T
+(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
+x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda
+(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t))
+H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4
+(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0
+H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S
+O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1:
+T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda
+(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2))
+(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2:
+T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1)
+x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x
+(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2:
+T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2)))
+(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10:
+(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t: T).(subst1 d0 u0 t4 t))
+H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1
+d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12:
+(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0:
+T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0
+t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2:
+T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O)
+d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d
+(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3:
+T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3)))
+(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr)
+u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i)
+(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0
+(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr)
+d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
+(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_:
+C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
+C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0
+t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4:
+T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr)
+x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1
+(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c: C).(getl i
+a0 c)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0
+(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i))))
+(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0
+u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6:
+T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i))
+x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop
+(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t:
+T).(subst1 (minus d0 (S i)) u0 u t)) H20 (lift (S O) (minus d0 (S i)) x6)
+H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i))
+x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S
+i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0
+x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28:
+(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda
+(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind
+nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S
+O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in
+(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S
+i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9:
+T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9)))
+(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S
+i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8
+(\lambda (t: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t)) H29 (lift (S
+O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat (S
+(plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S
+i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let
+H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n:
+nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12))
+in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10)))
+(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O)
+d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32))))))))
+(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30))))))
+(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S
+i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i
+H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12
+c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i
+(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12))))
+(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n:
+nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def
+(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15
+\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in
+(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind
+Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2:
+T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let
+H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c: C).(getl i c0 c)) H0
+(CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e
+(Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match
+e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _)
+\Rightarrow c])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono
+c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in ((let H19
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0
+(CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d e)).(let H21 \def
+(eq_ind_r T u0 (\lambda (t: T).(getl i c0 (CHead e (Bind Abbr) t))) H17 u
+H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t: T).(subst1 i t t4 (lift (S
+O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r T u0 (\lambda (t:
+T).(csubst1 i t c0 a0)) H15 u H19) in (eq_ind T u (\lambda (t0: T).(ex2 T
+(\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) (\lambda (x2: T).(pr2 a
+x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c: C).(getl i c0 (CHead c
+(Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda (t0: T).(subst1 i u t t0))
+(\lambda (t0: T).(subst1 i u (lift (S O) i x0) t0)) (ex2 T (\lambda (x2:
+T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: T).(pr2 a x1 x2)))
+(\lambda (x2: T).(\lambda (H25: (subst1 i u t x2)).(\lambda (H26: (subst1 i u
+(lift (S O) i x0) x2)).(let H27 \def (eq_ind T x2 (\lambda (t0: T).(subst1 i
+u t t0)) H25 (lift (S O) i x0) (subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i)
+(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O)
+i)) (plus i (S O)) (plus_comm i (S O))) H26)) in (ex_intro2 T (\lambda (x3:
+T).(subst1 i u t (lift (S O) i x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27
+(pr2_free a x1 x0 H10)))))) (subst1_confluence_eq t4 t u i (subst1_single i u
+t4 t H2) (lift (S O) i x0) H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda
+(H12: (lt d0 i)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda
+(t0: T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1
+d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2:
+T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O)
+d0 x0) x2)).(ex2_ind T (\lambda (t5: T).(eq T x2 (lift (S O) d0 t5)))
+(\lambda (t5: T).(subst1 (minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3:
+T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3)))
+(\lambda (x3: T).(\lambda (H15: (eq T x2 (lift (S O) d0 x3))).(\lambda (H16:
+(subst1 (minus i (S O)) u x0 x3)).(let H17 \def (eq_ind T x2 (\lambda (t0:
+T).(subst1 d0 u0 t t0)) H13 (lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda
+(x4: T).(subst1 d0 u0 t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))
+x3 H17 (pr2_delta1 a d u (minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind
+Abbr) u) a0 (csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0
+H4 (CHead d (Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0)
+(\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0
+H10 x3 H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat
+(plus (S O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0
+(S O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2)
+(lift (S O) d0 x0) u0 d0 H11 (sym_not_equal nat d0 i (lt_neq d0 i
+H12)))))))))) (pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0
+(lift (S O) d0 x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).
-axiom csubst0_clear_O: \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to (\forall (c: C).((clear c1 c) \to (clear c2 c)))))) .
+inductive pr3 (c:C): T \to (T \to Prop) \def
+| pr3_refl: \forall (t: T).(pr3 c t t)
+| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3:
+T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
+
+theorem pr3_gen_sort:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to
+(eq T x (TSort n)))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (eq T x (TSort n))
+(\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda (t:
+T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 (TSort n))))) (\lambda
+(t: T).(\lambda (H1: (eq T t (TSort n))).H1)) (\lambda (t2: T).(\lambda (t1:
+T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
+t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 (TSort n))))).(\lambda
+(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t
+t2)) H1 (TSort n) H4) in (H3 (pr2_gen_sort c t2 n H5)))))))))) y x H0)))
+H)))).
+
+theorem pr3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
+T).(pr3 (CHead c (Bind b) u) t1 t2))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
+t1) (\lambda (t: T).(pr3 c t x)) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u:
+T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c
+y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Abst) u1 t)) \to
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead
+(Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x0 t2))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0:
+T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind Abst) x0 x1)) \to
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t2))))))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H1: (eq T t (THead (Bind Abst) x0 x1))).(ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) x0 x1 H1
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b)
+u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3
+t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x:
+T).(\forall (x0: T).((eq T t2 (THead (Bind Abst) x x0)) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Abst) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0
+t2))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead
+(Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2))
+H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 t2 H5)
+in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
+Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
+Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t2 (THead
+(Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 x2)).(\lambda (H9: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H10 \def (eq_ind
+T t2 (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Bind
+Abst) x x0)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead
+(Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x0 t2)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11
+\def (H10 x2 x3 (refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5)))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead
+(Bind Abst) x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))
+x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u:
+T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u))))))))))
+H11)))))))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_cast:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c
+t1 x))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
+t1) (\lambda (t: T).(pr3 c t x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c
+t1 x)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t:
+T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
+t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y
+(THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (pr3 c x0 x)))))
+(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1:
+T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1
+t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 x1))).(eq_ind_r T (THead (Flat
+Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c
+x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
+(Flat Cast) x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c
+x1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1
+t2))) x0 x1 (refl_equal T (THead (Flat Cast) x0 x1)) (pr3_refl c x0)
+(pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1:
+(pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3:
+((\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Cast) x x0)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Flat Cast) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x0 t2)))) (pr3 c x0 t4))))))).(\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Flat Cast) x0 x1))).(let
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Flat Cast) x0
+x1) H4) in (let H6 \def (pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr2 c x1 t5)))) (pr2 c x1 t2) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x1
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T
+t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10:
+(pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(\forall (x:
+T).(\forall (x0: T).((eq T t (THead (Flat Cast) x x0)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Flat Cast) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr3 c x0 t2)))) (pr3 c x0 t4)))))) H3 (THead (Flat Cast) x2 x3) H8) in
+(let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 x3))) in (or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t4 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x3
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T
+t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16:
+(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat
+Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead
+(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5
+(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15)
+(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3
+t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c
+x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4
+H2))) H6)))))))))))) y x H0))))) H))))).
+
+theorem clear_pr3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(pr3_ind c2 (\lambda (t:
+T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3
+t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
+
+theorem pr3_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
+
+theorem pr3_t:
+ \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall
+(t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (c: C).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((pr3 c t0
+t3) \to (pr3 c t t3))))) (\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr3
+c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall
+(t3: T).((pr3 c t4 t3) \to (pr3 c t0 t3))))).(\lambda (t5: T).(\lambda (H3:
+(pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+theorem pr3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(pr3_ind c (\lambda (t: T).(\lambda (t0:
+T).(pr3 c (THead (Flat f) u t) (THead (Flat f) u t0)))) (\lambda (t:
+T).(pr3_refl c (THead (Flat f) u t))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
+t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u
+t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c
+t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (k: K).(\forall
+(t1: T).(pr3 c (THead k t t1) (THead k t0 t1)))))) (\lambda (t: T).(\lambda
+(k: K).(\lambda (t0: T).(pr3_refl c (THead k t t0))))) (\lambda (t2:
+T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
+(_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (k: K).(\forall (t: T).(pr3 c
+(THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing
+c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t)
+(H2 k t)))))))))) u1 u2 H)))).
+
+theorem pr3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr3 (CHead c k u) t1 t2)).(pr3_ind (CHead c k u)
+(\lambda (t: T).(\lambda (t0: T).(pr3 c (THead k u t) (THead k u t0))))
+(\lambda (t: T).(pr3_refl c (THead k u t))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u
+t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0)
+(THead k u t4) H2))))))) t1 t2 H)))))).
+
+theorem pr3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
+(CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c
+u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pr3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
+(CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c
+u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pr3_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t:
+T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0
+t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c:
+C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))).
+
+theorem pr3_cflat:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (f: F).(\forall (v:
+T).(pr3 (CHead c (Flat f) v) t t0))))) (\lambda (t: T).(\lambda (f:
+F).(\lambda (v: T).(pr3_refl (CHead c (Flat f) v) t)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead
+c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c
+(Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
+
+theorem pr3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
+(CHead c k u2) t1 t2)).(let H1 \def (match H0 return (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
+(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1
+t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
+(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
+t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
+\to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3
+(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
+(t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
+t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
+(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
+H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
+(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
+T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
+t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
+u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
+(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3
+(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
+(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
+((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
+(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
+(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
+(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1
+t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((getl O
+(CHead c k u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k u1) t1 t2)))
+(\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind
+Abbr) u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c]))
+(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d
+(Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)
+H14))) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b)
+u2) (CHead d (Bind Abbr) u) H14))) in ((let H16 \def (f_equal C T (\lambda
+(e: C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b)
+u2) (CHead d (Bind Abbr) u) H14))) in (\lambda (H17: (eq B Abbr b)).(\lambda
+(_: (eq C d c)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2))
+H13 u2 H16) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1
+t2)) (ex2_ind T (\lambda (t1: T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0
+t1 t2)) (pr3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H20:
+(subst0 O u1 t3 x)).(\lambda (H21: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr)
+u1) x t1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1
+t3 H10 x H20) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c
+(Bind Abbr) u1) x t2 H21)))))) (pr0_subst0_back u2 t3 t2 O H19 u1 H)) b
+H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat
+f) u2) (CHead d (Bind Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2
+(pr2_cflat c t1 t2 (pr2_delta c d u O (getl_intro O c (CHead d (Bind Abbr) u)
+c (drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O
+(CHead c (Flat f) u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f
+u1)))) k H12))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2)
+(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1
+t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: K).((getl (S
+i0) (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k u2)
+(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1
+t2)))) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (H14: (getl
+(S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl
+i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2)
+\to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b) u1) t1
+t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c
+(CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2 i0
+H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0)
+(CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0
+(CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to
+(pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2
+(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c
+(CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi)))))
+i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0
+(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2))
+(refl_equal T t1) (refl_equal T t2)))))))))).
+
+theorem pr3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
+u2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1) \to ((eq T
+t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k
+u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with [(pr2_free c0 t1 t2
+H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1
+u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T t1 u1)
+\to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
+T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
+t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T
+t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
+K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda
+(H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3:
+T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3
+(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3
+t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H)))))) t2 (sym_eq T t2 u2 H5))) t1
+(sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
+i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
+T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1
+u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
+t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
+K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))))
+(\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
+\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
+t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
+u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
+T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
+t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
+(pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda
+(t0: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 t0)).(let H11
+\def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t1:
+T).(\lambda (_: (pr2 c0 t t1)).((eq C c0 (CHead c k u2)) \to ((eq T t t3) \to
+((eq T t1 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c0 t3 t4
+H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T
+t3 t3)).(\lambda (H6: (eq T t4 t0)).(eq_ind C (CHead c k u2) (\lambda (_:
+C).((eq T t3 t3) \to ((eq T t4 t0) \to ((pr0 t3 t4) \to (pr3 (CHead c k u1)
+t3 t0))))) (\lambda (H7: (eq T t3 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4
+t0) \to ((pr0 t t4) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H8: (eq T t4
+t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pr3 (CHead c k u1) t3
+t0))) (\lambda (H9: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1) t3 t0 (pr2_free
+(CHead c k u1) t3 t0 H9))) t4 (sym_eq T t4 t0 H8))) t3 (sym_eq T t3 t3 H7)))
+c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3
+t3 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda
+(H7: (eq T t3 t3)).(\lambda (H11: (eq T t t0)).(eq_ind C (CHead c k u2)
+(\lambda (c1: C).((eq T t3 t3) \to ((eq T t t0) \to ((getl i0 c1 (CHead d0
+(Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c
+k u1) t3 t0))))))) (\lambda (H12: (eq T t3 t3)).(eq_ind T t3 (\lambda (t1:
+T).((eq T t t0) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
+((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c k u1) t3 t0))))))
+(\lambda (H13: (eq T t t0)).(eq_ind T t0 (\lambda (t1: T).((getl i0 (CHead c
+k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t1)
+\to (pr3 (CHead c k u1) t3 t0))))) (\lambda (H14: (getl i0 (CHead c k u2)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t3 t4)).(\lambda (H16:
+(subst0 i0 u0 t4 t0)).((match i0 return (\lambda (n: nat).((getl n (CHead c k
+u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t0) \to (pr3 (CHead c k
+u1) t3 t0)))) with [O \Rightarrow (\lambda (H17: (getl O (CHead c k u2)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H18: (subst0 O u0 t4 t0)).((match k
+return (\lambda (k: K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
+(pr3 (CHead c k u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda (H19: (clear
+(CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d0 | (CHead c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19))
+in ((let H0 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c
+(CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e:
+C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead
+_ _ t) \Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq
+B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t:
+T).(subst0 O t t4 t0)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pr3
+(CHead c (Bind b0) u1) t3 t0)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4
+t0)) (\lambda (t1: T).(subst0 (S (plus i O)) u t1 t0)) (pr3 (CHead c (Bind
+Abbr) u1) t3 t0) (\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda
+(H10: (subst0 (S (plus i O)) u x t0)).(let H23 \def (f_equal nat nat S (plus
+i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat
+(S (plus i O)) (\lambda (n: nat).(subst0 n u x t0)) H10 (S i) H23) in
+(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 x))
+(pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x0: T).(\lambda (H9: (subst0 O
+u1 t4 x0)).(\lambda (H25: (pr0 x0 x)).(pr3_sing (CHead c (Bind Abbr) u1) x0
+t3 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H15
+x0 H9) t0 (pr3_pr2 (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind
+Abbr) u1) d u (S i) (getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1
+(clear_bind Abbr c u1) (CHead d (Bind Abbr) u) i H8) x0 x H25 t0 H24))))))
+(pr0_subst0_back t2 t4 x O H2 u1 H9))))))) (subst0_subst0 t4 t0 u2 O H22 t2 u
+i H10)) b H20))))) H0)) H))) | (Flat f) \Rightarrow (\lambda (H8: (clear
+(CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f)
+u1) t3 t0 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0
+(Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0)
+u2 H8)) t3 t4 H15 t0 H18) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0
+(Bind Abbr) u0) H17)))) | (S n) \Rightarrow (\lambda (H8: (getl (S n) (CHead
+c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H9: (subst0 (S n) u0 t4
+t0)).((match k return (\lambda (k: K).((getl (S n) (CHead c k u2) (CHead d0
+(Bind Abbr) u0)) \to (pr3 (CHead c k u1) t3 t0))) with [(Bind b) \Rightarrow
+(\lambda (H10: (getl (S n) (CHead c (Bind b) u2) (CHead d0 (Bind Abbr)
+u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta (CHead c (Bind b) u1)
+d0 u0 (S n) (getl_head (Bind b) n c (CHead d0 (Bind Abbr) u0) (getl_gen_S
+(Bind b) c (CHead d0 (Bind Abbr) u0) u2 n H10) u1) t3 t4 H15 t0 H9))) | (Flat
+f) \Rightarrow (\lambda (H10: (getl (S n) (CHead c (Flat f) u2) (CHead d0
+(Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0
+(pr2_delta c d0 u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d0 (Bind
+Abbr) u0) u2 n H10) t3 t4 H15 t0 H9) f u1)))]) H8)))]) H14 H16)))) t (sym_eq
+T t t0 H13))) t3 (sym_eq T t3 t3 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7
+H11 H3 H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t3)
+(refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) t1 (sym_eq T t1 u1 H6))) c0
+(sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T
+u1) (refl_equal T u2)))))).
+
+theorem pr3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to
+(pr3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u1 u2) \to (pr3
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
+u1 u2)).(pr3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u1 u2)
+\to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1
+u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pr3_pr3_pr3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (k: K).((pr3 (CHead c k t0) t1 t2) \to (pr3 (CHead c k t) t1
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H0: (pr3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
+t3)).(\lambda (H2: ((\forall (t1: T).(\forall (t4: T).(\forall (k: K).((pr3
+(CHead c k t3) t1 t4) \to (pr3 (CHead c k t2) t1 t4))))))).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0
+t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
+
+theorem pr3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 e t1
+t2)).(pr3_ind e (\lambda (t: T).(\lambda (t0: T).(pr3 c (lift h d t) (lift h
+d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
+(t3: T).(\lambda (H1: (pr2 e t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 e t0
+t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d
+t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2
+H0)))))))).
+
+theorem pr3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1:
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0
+t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2:
+T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1)
+(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t:
+T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
+(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead
+c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3
+(CHead c0 k u1) (let H7 \def (match H4 return (\lambda (c: C).(\lambda (t:
+T).(\lambda (t1: T).(\lambda (_: (pr2 c t t1)).((eq C c (CHead c3 k u1)) \to
+((eq T t t3) \to ((eq T t1 t0) \to (pr3 (CHead c0 k u1) t3 t0)))))))) with
+[(pr2_free c t1 t2 H2) \Rightarrow (\lambda (H3: (eq C c (CHead c3 k
+u1))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T t2 t0)).(eq_ind C
+(CHead c3 k u1) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 t0) \to ((pr0 t1
+t2) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H6: (eq T t1 t3)).(eq_ind T
+t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pr3 (CHead c0 k u1) t3
+t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t)
+\to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H8: (pr0 t3 t0)).(pr3_pr2 (CHead
+c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H8))) t2 (sym_eq T t2 t0 H7)))
+t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c3 k u1) H3) H4 H5 H2)))) |
+(pr2_delta c d u i H2 t1 t2 H3 t H4) \Rightarrow (\lambda (H5: (eq C c (CHead
+c3 k u1))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T t t0)).(eq_ind C
+(CHead c3 k u1) (\lambda (c1: C).((eq T t1 t3) \to ((eq T t t0) \to ((getl i
+c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (pr3
+(CHead c0 k u1) t3 t0))))))) (\lambda (H8: (eq T t1 t3)).(eq_ind T t3
+(\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c3 k u1) (CHead d (Bind
+Abbr) u)) \to ((pr0 t4 t2) \to ((subst0 i u t2 t) \to (pr3 (CHead c0 k u1) t3
+t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda (t4: T).((getl i
+(CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2
+t4) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H10: (getl i (CHead c3 k
+u1) (CHead d (Bind Abbr) u))).(\lambda (H11: (pr0 t3 t2)).(\lambda (H12:
+(subst0 i u t2 t0)).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl i
+(CHead c0 k u1) (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u))) (pr3 (CHead
+c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H1: (getl i
+(CHead c0 k u1) (CHead x0 (Bind Abbr) x1))).(\lambda (_: (wcpr0 x0
+d)).(\lambda (H14: (pr0 x1 u)).(ex2_ind T (\lambda (t0: T).(subst0 i x1 t2
+t0)) (\lambda (t3: T).(pr0 t3 t0)) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x:
+T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 x t0)).(pr3_sing
+(CHead c0 k u1) x t3 (pr2_delta (CHead c0 k u1) x0 x1 i H1 t3 t2 H11 x H15)
+t0 (pr3_pr2 (CHead c0 k u1) x t0 (pr2_free (CHead c0 k u1) x t0 H16))))))
+(pr0_subst0_back u t2 t0 i H12 x1 H14))))))) (wcpr0_getl_back (CHead c3 k u1)
+(CHead c0 k u1) (wcpr0_comp c0 c3 H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr)
+H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c (CHead
+c3 k u1) H5) H6 H7 H2 H3 H4))))]) in (H7 (refl_equal C (CHead c3 k u1))
+(refl_equal T t3) (refl_equal T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2
+t1 t2 k H3 u1 (pr2_free c3 u1 u2 H2)))))))))))))) c2 c1 H))).
+
+theorem pr3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
+(d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to
+(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1
+t2))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1)
+(\lambda (t: T).(pr3 c t x)) (\forall (e: C).((drop h d c e) \to (ex2 T
+(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 t2)))))
+(\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq
+T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) (pr3_ind
+c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h d x0))
+\to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T t0
+(lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t:
+T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e:
+C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h
+d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda
+(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
+T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x: T).((eq T t2 (lift
+h d x)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T
+t4 (lift h d t2))) (\lambda (t2: T).(pr3 e x t2))))))))).(\lambda (x0:
+T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: (drop
+h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (lift h
+d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in (ex2_ind T
+(\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e x0 t5))
+(ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0
+t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d x1))).(\lambda (H9:
+(pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda
+(t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5)))
+(\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda (H10: (eq T t4
+(lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T (\lambda (t5:
+T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 H10 (pr3_sing
+e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x H0)))) H)))))).
+
+theorem pr3_gen_lref:
+ \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to
+(or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T x (lift (S n) O v))))))))))
+\def
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (or (eq T x (TLRef
+n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c
+(CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v:
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T x
+(lift (S n) O v))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c
+(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 (TLRef
+n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c
+(CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v:
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t0
+(lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (H1: (eq T t (TLRef
+n))).(or_introl (eq T t (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) H1))) (\lambda (t2:
+T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
+(H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or (eq T t3
+(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v:
+T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3
+(lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef n))).(let H5 \def
+(eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) in (let H6 \def
+(pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef n)) (ex2_2 C T (\lambda
+(d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T t3 (TLRef n))
+(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead
+d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v))))))) (\lambda (H7: (eq T t2 (TLRef n))).(H3 H7)) (\lambda (H7: (ex2_2 C T
+(\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda
+(_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda
+(d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
+C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n))
+(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead
+d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O
+v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0
+(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def
+(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let
+H11 \def (pr3_gen_lift c x1 t3 (S n) O H10 x0 (getl_drop Abbr c x0 x1 n H8))
+in (ex2_ind T (\lambda (t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4:
+T).(pr3 x0 x1 t4)) (or (eq T t3 (TLRef n)) (ex3_3 C T T (\lambda (d:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u)))))
+(\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))))) (\lambda
+(x2: T).(\lambda (H12: (eq T t3 (lift (S n) O x2))).(\lambda (H13: (pr3 x0 x1
+x2)).(or_intror (eq T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d:
+C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v)))))) (ex3_3_intro C T T
+(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind
+Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v)))))
+x0 x1 x2 H8 H13 H12))))) H11))))))) H7)) H6)))))))))) y x H0))) H)))).
+
+theorem pr3_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
+t1) (\lambda (t: T).(pr3 c t x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1
+(\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t2))))))
+(pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 (\lambda
+(t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2)))))) (pr3
+(CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t:
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind
+Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O
+t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1:
+(eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind Void) x0 x1)
+(\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O
+t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
+(Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1
+(lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr3_refl c x0)
+(\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) u) x1))))) t
+H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3
+t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall
+(x: T).(\forall (x0: T).((eq T t2 (THead (Bind Void) x x0)) \to (or (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2)))))) (pr3
+(CHead c (Bind Void) x) x0 (lift (S O) O t4)))))))).(\lambda (x0: T).(\lambda
+(x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let H5 \def
+(eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 x1) H4)
+in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O t2)))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead
+(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq
+T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O)
+O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind
+Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind
+T t2 (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Bind
+Void) x x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2)))
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x0 t2)))))) (pr3 (CHead c (Bind Void) x) x0 (lift (S O) O
+t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3
+(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c
+(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c
+(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Void) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t2))))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c
+(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14
+(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead
+c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13:
+(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind
+Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift
+(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7:
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O
+t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))
+(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S
+O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void)
+O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1
+t1) (\lambda (t: T).(pr3 c t x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr)
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))) (\lambda
+(y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y
+(THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t
+t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) O x))))) (unintro T u1
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Abbr) t x0)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 (CHead c
+(Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: T).(\lambda
+(t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind Abbr) x0 x1))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c
+(Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0
+x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0)
+x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0
+x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1)))))
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4:
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x: T).(\forall (x0:
+T).((eq T t2 (THead (Bind Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x) x0 t2)))) (pr3 (CHead c (Bind Abbr) x) x0 (lift (S O)
+O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead
+(Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2))
+H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 t2 H5)
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead
+c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z
+t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift
+(S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
+(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1
+t2))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind
+Abbr) u) x1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c
+(Bind Abbr) x0) x1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda
+(_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z t2))))))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1
+t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind
+Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c
+(Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z t5))))))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 c x0
+x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
+b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead
+c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) x0) x1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
+(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z
+x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c
+(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3))))
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind
+T t2 (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Bind
+Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x) x0 t2)))) (pr3
+(CHead c (Bind Abbr) x) x0 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2
+x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3)))
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c
+(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
+t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x2) x3
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c
+x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T
+(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4
+x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1
+(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4
+H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr)
+x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O)
+O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u:
+T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1
+x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c
+(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4:
+T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4)
+x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x: T).(\forall
+(x0: T).((eq T t (THead (Bind Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x) x0 t2)))) (pr3 (CHead c (Bind Abbr) x) x0 (lift (S O)
+O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind Abbr) x2) x3 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5
+x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr)
+x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0)
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5
+x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0)
+(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3
+x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c
+(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1
+x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O
+t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T
+(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y)))
+(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
+(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq
+T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1
+x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0)
+x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x: T).(\forall
+(x0: T).((eq T t (THead (Bind Abbr) x x0)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind Abbr) x) x0 t2)))) (pr3 (CHead c (Bind Abbr) x) x0 (lift (S O)
+O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind Abbr) x2) x3 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda
+(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6
+x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr)
+x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0)
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S
+O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3
+(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6
+x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1
+H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind
+Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7
+(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17))
+(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr)
+x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4
+(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing
+(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3
+(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10))))))
+H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr)
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing
+(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4)
+(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c
+(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
+(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
+t1) (\lambda (t: T).(pr3 c t x)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y
+(THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t t2)))) (ex4_4 T T T T
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c
+(THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro
+T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl)
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2)
+x))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))) (pr3_ind
+c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t
+(THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))))
+(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t
+(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda
+(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat
+Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0
+x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2
+t4)).(\lambda (H3: ((\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat
+Appl) x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4
+(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
+z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3
+(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c
+t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1
+t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr2 c x1 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2
+x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t
+(THead (Flat Appl) x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t4 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
+(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t:
+T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3
+(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead (Flat
+Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c x3 t2))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4
+x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T
+(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4
+x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl)
+x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c
+x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T
+(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10
+x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15:
+(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda
+(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro
+T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5:
+T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1
+t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10
+(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8:
+T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3
+c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9
+(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2
+x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9)
+x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))
+x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17
+(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7))
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2)))
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq
+T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4
+x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2
+x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T
+t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t
+(THead (Flat Appl) x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T t4 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (ex4_4 T
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u2 t2) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
+(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t:
+T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind
+Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u)
+x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl)
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_:
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5)
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b)
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
+(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6:
+T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T
+x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7
+(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0
+x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7)
+x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4)))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2
+(\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl)
+x x0)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3
+c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3
+(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let
+H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7
+(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda
+(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4))
+H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2)
+x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
+
+theorem pr3_strip:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
+(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2
+t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda
+(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2
+t4)).(\lambda (H2: ((\forall (t3: T).((pr2 c t2 t3) \to (ex2 T (\lambda (t:
+T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t3 t))))))).(\lambda (t5: T).(\lambda
+(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t:
+T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c
+t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t))
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c
+x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0))))))))))
+t0 t1 H)))).
+
+theorem pr3_confluence:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
+(t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t))))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0
+t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t
+t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3
+t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t
+t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2
+t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda
+(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda
+(H2: ((\forall (t3: T).((pr3 c t2 t3) \to (ex2 T (\lambda (t: T).(pr3 c t4
+t)) (\lambda (t: T).(pr3 c t3 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c
+t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2
+t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)))
+(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2
+x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t))
+(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda
+(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T
+(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5
+c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1
+H)))).
+
+theorem pr3_subst1:
+ \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2)
+\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0:
+T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1
+w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1:
+T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1
+w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v
+t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i
+v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T
+(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T
+(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2)))
+(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3
+x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5
+w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5
+w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v
+t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1
+i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c
+e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
+
+theorem pr3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
+(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a
+x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
+(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1
+d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))
+(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
+(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_:
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0
+a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d
+x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2)))
+(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4:
+T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
+(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2:
+T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1
+x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
+(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4:
+(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0
+a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d
+x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda
+(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
+x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1
+d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda
+(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2))
+(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2:
+T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O)
+d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d
+u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x
+x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0
+e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).
+
+theorem pr3_iso_appls_cast:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1
+\def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2:
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
+(THeads (Flat Appl) vs t) u2))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t0: TList).(let u1 \def (THeads (Flat Appl) t0
+(THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 t) u2))))))
+(\lambda (u2: T).(\lambda (H: (pr3 c (THead (Flat Cast) v t) u2)).(\lambda
+(H0: (((iso (THead (Flat Cast) v t) u2) \to (\forall (P: Prop).P)))).(let H1
+\def (pr3_gen_cast c v t u2 H) in (or_ind (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
+t2)))) (pr3 c t u2) (pr3 c t u2) (\lambda (H2: (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c v u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t
+t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c t t2))) (pr3 c t u2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T u2 (THead (Flat Cast) x0
+x1))).(\lambda (_: (pr3 c v x0)).(\lambda (_: (pr3 c t x1)).(let H6 \def
+(eq_ind T u2 (\lambda (t0: T).((iso (THead (Flat Cast) v t) t0) \to (\forall
+(P: Prop).P))) H0 (THead (Flat Cast) x0 x1) H3) in (eq_ind_r T (THead (Flat
+Cast) x0 x1) (\lambda (t0: T).(pr3 c t t0)) (H6 (iso_head (Flat Cast) v x0 t
+x1) (pr3 c t (THead (Flat Cast) x0 x1))) u2 H3))))))) H2)) (\lambda (H2: (pr3
+c t u2)).H2) H1))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H:
+((\forall (u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2)
+\to ((((iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) u2) \to (\forall
+(P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1 t) u2)))))).(\lambda (u2:
+T).(\lambda (H0: (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead
+(Flat Cast) v t))) u2)).(\lambda (H1: (((iso (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Flat Cast) v t))) u2) \to (\forall (P:
+Prop).P)))).(let H2 \def (pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead
+(Flat Cast) v t)) u2 H0) in (or3_ind (ex3_2 T T (\lambda (u3: T).(\lambda
+(t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c t0 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat
+Appl) t1 (THead (Flat Cast) v t)) t2)))) (ex4_4 T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (_: T).(pr3 c t0 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Cast) v t)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u3) z2))
+u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2)
+(\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Appl) u3 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t0 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Cast) v t)) t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
+(THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Cast) v t)) t2))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat Appl)
+x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat Appl)
+t1 (THead (Flat Cast) v t)) x1)).(let H7 \def (eq_ind T u2 (\lambda (t2:
+T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Cast) v
+t))) t2) \to (\forall (P: Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in
+(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 t)) t2)) (H7 (iso_head (Flat Appl) t0 x0
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) x1) (pr3 c (THead (Flat Appl)
+t0 (THeads (Flat Appl) t1 t)) (THead (Flat Appl) x0 x1))) u2 H4))))))) H3))
+(\lambda (H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t0 u2)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1
+t2))))))))).(ex4_4_ind T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))
+(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) u2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (pr3 c
+(THead (Bind Abbr) x2 x3) u2)).(\lambda (H5: (pr3 c t0 x2)).(\lambda (H6:
+(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind Abst) x0
+x1))).(\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
+u) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 t)) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1))
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads
+(Flat Appl) t1 t) (THead (Bind Abst) x0 x1) (H (THead (Bind Abst) x0 x1) H6
+(\lambda (H8: (iso (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead
+(Bind Abst) x0 x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Cast
+Abst x0 v x1 t t1 H8 P)))) t0 Appl) (THead (Bind Abbr) t0 x1) (pr3_pr2 c
+(THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) t0 x1)
+(pr2_free c (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Bind
+Abbr) t0 x1) (pr0_beta x0 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1))))) u2
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t0 x1) c (pr3_head_12 c
+t0 x2 H5 (Bind Abbr) x1 x3 (H7 Abbr x2)) u2 H4)))))))))) H3)) (\lambda (H3:
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b)
+y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t0
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr3 c t0 u3)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))) (pr3 c (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t1 (THead (Flat Cast) v t))
+(THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: (pr3 c t0 x4)).(\lambda
+(H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 t)) c (pr3_t (THead (Bind x0) x1 (THead (Flat
+Appl) (lift (S O) O t0) x2)) (THead (Flat Appl) t0 (THeads (Flat Appl) t1 t))
+c (pr3_t (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 t)) c (pr3_thin_dx c (THeads (Flat Appl) t1 t) (THead
+(Bind x0) x1 x2) (H (THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads
+(Flat Appl) t1 (THead (Flat Cast) v t)) (THead (Bind x0) x1 x2))).(\lambda
+(P: Prop).(iso_flats_flat_bind_false Appl Cast x0 x1 v x2 t t1 H10 P)))) t0
+Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0) x2)) (pr3_pr2
+c (THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead
+(Flat Appl) (lift (S O) O t0) x2)) (pr2_free c (THead (Flat Appl) t0 (THead
+(Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t0)
+x2)) (pr0_upsilon x0 H4 t0 t0 (pr0_refl t0) x1 x1 (pr0_refl x1) x2 x2
+(pr0_refl x2))))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4)
+x2)) (pr3_head_12 c x1 x1 (pr3_refl c x1) (Bind x0) (THead (Flat Appl) (lift
+(S O) O t0) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead
+c (Bind x0) x1) (lift (S O) O t0) (lift (S O) O x4) (pr3_lift (CHead c (Bind
+x0) x1) c (S O) O (drop_drop (Bind x0) O c c (drop_refl c) x1) t0 x4 H7)
+(Flat Appl) x2 x2 (pr3_refl (CHead (CHead c (Bind x0) x1) (Flat Appl) (lift
+(S O) O x4)) x2)))) u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
+O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c
+(pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2)
+(THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5)
+x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))).
-axiom csubst0_clear_O_back: \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to (\forall (c: C).((clear c2 c) \to (clear c1 c)))))) .
+inductive csuba (g:G): C \to (C \to Prop) \def
+| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
+| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
+(k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
+| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall
+(t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u:
+T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind
+Abbr) u))))))))).
+
+theorem csuba_gen_abbr:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
+(CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H return (\lambda
+(c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 (CHead d1
+(Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead
+d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(csuba_sort
+n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abbr)
+u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n)
+(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr)
+u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) |
+(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e:
+C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead
+_ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let
+H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind
+Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2)
+\to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind
+Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to
+((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0
+u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to
+((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2
+(Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba
+g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1
+c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2
+(Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr)
+H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
+H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
+(Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5
+\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow
+(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst
+\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])]))
+I (CHead d1 (Bind Abbr) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0)
+c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0
+a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C
+(CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
+
+theorem csuba_gen_void:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
+(CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H return (\lambda
+(c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 (CHead d1
+(Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead
+d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(csuba_sort
+n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Void)
+u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n)
+(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
+u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c
+(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) |
+(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0)
+(CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0)
+(CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e:
+C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead
+_ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in ((let
+H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0)
+(CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind
+Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2)
+\to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2:
+C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Void))).(eq_ind K (Bind
+Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to
+((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void)
+u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0
+u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to
+((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void)
+u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2
+(Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c: C).((csuba
+g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u)))
+(\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1
+c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2
+(Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void)
+H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
+H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
+(Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5
+\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow
+(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst
+\Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])]))
+I (CHead d1 (Bind Void) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0)
+c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0
+a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda
+(d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C
+(CHead d1 (Bind Void) u)) (refl_equal C c))))))).
+
+theorem csuba_gen_abst:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
+(CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H return
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0
+(CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2:
+C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0:
+(eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n)
+c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C
+(CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))
+H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead
+c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u)
+c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead
+c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K (\lambda
+(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1)
+in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead
+c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K
+k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0
+c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda
+(H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u
+u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to (or (ex2 C
+(\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1
+(\lambda (t: T).((eq C (CHead c2 (Bind Abst) t) c) \to ((csuba g d1 c2) \to
+(or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H8: (eq C (CHead
+c2 (Bind Abst) u1) c)).(eq_ind C (CHead c2 (Bind Abst) u1) (\lambda (c:
+C).((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))
+(\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C (\lambda (d2: C).(eq C
+(CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C
+(\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abst) u1))
+H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Bind Abst) H6))) c1
+(sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2)
+\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind
+Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t)
+(CHead d1 (Bind Abst) u1) H3) in ((let H6 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead
+c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1)
+H3) in (eq_ind C d1 (\lambda (c0: C).((eq T t u1) \to ((eq C (CHead c2 (Bind
+Abbr) u) c) \to ((csuba g c0 c2) \to ((arity g c0 t (asucc g a)) \to ((arity
+g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))))))))))
+(\lambda (H7: (eq T t u1)).(eq_ind T u1 (\lambda (t0: T).((eq C (CHead c2
+(Bind Abbr) u) c) \to ((csuba g d1 c2) \to ((arity g d1 t0 (asucc g a)) \to
+((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
+a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) c)).(eq_ind C
+(CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba g d1 c2) \to ((arity g d1 u1
+(asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc
+g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
+a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10: (arity g d1 u1
+(asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2 C (\lambda
+(d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g
+a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
+a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a
+(refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t
+u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C
+(CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
+
+theorem csuba_gen_flat:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
+(f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H
+return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
+c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2))))))))) with [(csuba_sort n) \Rightarrow
+(\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda (H1: (eq C
+(CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in (False_ind ((eq C (CSort
+n) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) H2))
+H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k
+u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k u) c)).((let H3
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
+(CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K (\lambda (e:
+C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead
+_ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H5
+\def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead d1 (Flat f) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat
+f)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 c2) \to
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) (\lambda (H6:
+(eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u u1) \to ((eq
+C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))))))) (\lambda (H7: (eq T u u1)).(eq_ind
+T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to ((csuba g d1 c2) \to
+(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))) (\lambda (H8:
+(eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2 (Flat f) u1) (\lambda
+(c: C).((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2)))))) (\lambda (H9: (csuba g d1 c2)).(ex2_2_intro C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C (CHead
+c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Flat f)
+H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
+H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
+(Flat f) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def
+(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C
+(CHead c2 (Bind Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g
+a)) \to ((arity g c2 u a) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d1 d2)))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f)
+u1)) (refl_equal C c)))))))).
+
+theorem csuba_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
+(v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
+(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def
+(match H return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba ? c
+c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
+e2)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
+(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def
+(eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I
+(CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2_3 B C
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
+e2)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq
+C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u)
+c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead
+c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda
+(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in
+((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k
+u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind
+b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csuba g c c0) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind
+b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba
+g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind
+T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0)
+\to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1)
+c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2))))))) (\lambda (H9: (csuba g e1 c0)).(let H10 \def
+(eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead
+c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1
+(refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k
+(sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) |
+(csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1
+(Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind
+Abbr) u) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow
+t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort
+_) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1
+(Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t)
+(CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B Abst b1)
+\to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g c c0)
+\to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
+e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_:
+B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0)
+\to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
+e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C
+(CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc
+g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10:
+(eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u)
+(\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity
+g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H11: (csuba g e1
+c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u
+a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1)
+v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1
+(\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u)))
+H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda
+(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u
+(refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1
+H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0
+(refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
+
+theorem csuba_refl:
+ \forall (g: G).(\forall (c: C).(csuba g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csuba g c0 c0))
+(\lambda (n: nat).(csuba_sort g n)) (\lambda (c0: C).(\lambda (H: (csuba g c0
+c0)).(\lambda (k: K).(\lambda (t: T).(csuba_head g c0 c0 H k t))))) c)).
+
+theorem csuba_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2))
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2))
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
+(clear (CHead c3 k u) e1)).((match k return (\lambda (k0: K).((clear (CHead
+c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2:
+C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\lambda (H3:
+(clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda
+(c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead
+c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind
+b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind
+b) u) (csuba_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1
+(clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: (clear
+(CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 u H3))
+in (ex2_ind C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4
+e2)) (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead
+c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csuba g e1 x)).(\lambda
+(H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda
+(e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u)))))
+H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g
+c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda
+(e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (H2: (arity g c3 t (asucc g a))).(\lambda (u:
+T).(\lambda (H3: (arity g c4 u a)).(\lambda (e1: C).(\lambda (H4: (clear
+(CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda
+(c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead
+c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3
+(Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2))
+(CHead c4 (Bind Abbr) u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr
+c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
+
+theorem csuba_drop_abbr:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
+O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
+c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))))
+(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1
+(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0:
+(csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in
+(let H2 \def (csuba_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (d2: C).(eq
+C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind
+Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u)
+(\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(drop
+O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3))))
+H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1:
+C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S
+n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
+O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind
+Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) (\lambda (H2: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (_:
+(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 return
+(\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)))))) with [refl_equal \Rightarrow (\lambda (H4: (eq C
+(CHead d1 (Bind Abbr) u) (CSort n0))).(let H5 \def (eq_ind C (CHead d1 (Bind
+Abbr) u) (\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort
+_) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) H4) in
+(False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2))) H5)))]) in (H5 (refl_equal C (CSort
+n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u) H0)))))))))
+(\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: T).((drop (S n)
+O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
+c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) O (CHead c k t)
+(CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2:
+(csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t)
+c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g
+d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c (Bind b) t)
+c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) u))).(B_ind
+(\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0)
+n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda
+(H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr)
+n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_abbr g c c2 t H5)
+in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda
+(d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda
+(H8: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c
+x)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2:
+C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11:
+(drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let
+H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n
+(\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr)
+n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr)
+t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop
+(Bind Abbr) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8))))
+H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6:
+(drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def
+(csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq
+C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba
+g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2
+(CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x
+(Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def
+(H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead
+x0 (Bind Abbr) u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal
+nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop
+n O x (CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead
+x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0
+(Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t
+(asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind
+Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u
+H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x
+(Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat
+(r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O
+x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0
+(CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7))))
+(\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r
+(Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_void g
+c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t)))
+(\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x:
+C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g c
+x)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2:
+C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11:
+(drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let
+H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n
+(\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr)
+n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void)
+t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop
+(Bind Void) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8))))
+H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f)
+t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr)
+u))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda
+(H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(ex2
+C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 H7) in (ex2_ind C
+(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
+(x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) u))).(\lambda
+(H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x
+(drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) H8)) c2
+H6))))) H5))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n
+H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abst:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
+O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1:
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n
+O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1:
+T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g:
+G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1
+(\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl
+c1 (CHead d1 (Bind Abst) u1) H)) in (let H2 \def (csuba_gen_abst g d1 c2 u1
+H1) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop O O c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst)
+u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abst) u1)
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))
+(or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) u1) (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind Abst) u1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x
+(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
+x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 H4)))) H3)) (\lambda (H3:
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr)
+x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 (asucc g
+x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1)
+(\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))
+(or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) x1) (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))
+x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 H7)) c2 H4)))))))) H3))
+H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1:
+C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n
+O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop
+(S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1:
+C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind
+Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0)
+c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O)
+(eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda
+(_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2
+return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c (CSort n0)) \to (or
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) with
+[refl_equal \Rightarrow (\lambda (H4: (eq C (CHead d1 (Bind Abst) u1) (CSort
+n0))).(let H5 \def (eq_ind C (CHead d1 (Bind Abst) u1) (\lambda (e: C).(match
+e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _
+_) \Rightarrow True])) I (CSort n0) H4) in (False_ind (or (ex2 C (\lambda
+(d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5)))]) in (H5 (refl_equal C
+(CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1)
+H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u1:
+T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall
+(c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda
+(u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst)
+u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t)
+c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n)
+O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c
+(Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst)
+u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop
+(r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g
+(CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead
+d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_abbr g c c2 t H5) in (ex2_ind C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
+g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr)
+t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t)
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead
+x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def
+(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda
+(n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14)
+in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst)
+u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12:
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1
+x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0
+x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def
+(eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H12
+(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr)
+n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2
+H8)))) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda
+(H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def
+(csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
+t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2
+(CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x:
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c
+x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g
+x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C
+(\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14:
+(csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16
+\def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1)))
+H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n)
+O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst)
+n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13:
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1
+x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0
+x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def
+(eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H13
+(r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
+(S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst)
+n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2
+H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind
+Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g
+x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1)
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda
+(d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead
+x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal
+nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n: nat).(drop
+n O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in (or_introl
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr)
+x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O
+(CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x (Bind Abst) u1) H18 x1)
+H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4:
+T).(\lambda (x5: A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abbr)
+x4))).(\lambda (H16: (csuba g d1 x3)).(\lambda (H17: (arity g d1 u1 (asucc g
+x5))).(\lambda (H18: (arity g x3 x4 x5)).(let H19 \def (refl_equal nat (r
+(Bind Abbr) n)) in (let H20 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0
+(CHead x3 (Bind Abbr) x4))) H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))) x3 x4 x5 (drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1)
+H16 H17 H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7)))) (\lambda (H5:
+(csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O
+c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_void g c c2 t H5) in
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2:
+C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void)
+t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t)
+(\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2:
+C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda
+(H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead
+x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def
+(refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda
+(n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14)
+in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst)
+u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12:
+(drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1
+x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0
+x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def
+(eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H12
+(r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop
+(S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x
+(Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Void)
+n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2
+H8)))) H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c
+(Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst)
+u1))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq
+C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C
+(CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S
+n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2
+C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O
+x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C
+(\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat
+f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead
+x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10:
+(drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1
+x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1)
+H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda
+(d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10:
+(drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1
+x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2
+x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f)
+x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0
+(Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
+Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5))))) k H2
+(drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
+
+theorem csuba_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g
+c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
+(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x:
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
+Abbr) u))).((match x return (\lambda (c: C).((drop i O c1 c) \to ((clear c
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2)))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O
+c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr)
+u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n H4 (\forall (c2: C).((csuba g
+c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2)))))))) | (CHead c k t) \Rightarrow (\lambda
+(H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead
+d1 (Bind Abbr) u))).((match k return (\lambda (k0: K).((drop i O c1 (CHead c
+k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2:
+C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
+Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(Bind b) \Rightarrow
+(\lambda (H5: (drop i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead
+c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda
+(e: C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t)
+(clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def (f_equal
+C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d1
+(Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr)
+u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
+t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead
+d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq
+C d1 c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def
+(eq_ind_r T t (\lambda (t: T).(drop i O c1 (CHead c (Bind b) t))) H5 u H9) in
+(let H14 \def (eq_ind_r B b (\lambda (b: B).(drop i O c1 (CHead c (Bind b)
+u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop i O
+c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i
+c1 d1 u H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))
+(\lambda (x0: C).(\lambda (H17: (drop i O c2 (CHead x0 (Bind Abbr)
+u))).(\lambda (H18: (csuba g d1 x0)).(ex_intro2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i
+c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0
+u)) H18)))) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5:
+(drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t)
+(CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0:
+C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2)
+\to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x0:
+C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2)
+\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d1 d2)))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0
+(CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let
+H10 \def (eq_ind C x0 (\lambda (c: C).(csuba g c c2)) H9 (CHead c (Flat f) t)
+(drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c
+(CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f
+t) in (let H11 \def (csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1
+(Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind
+Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl O
+c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda
+(x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13:
+(clear c2 x1)).(let H14 \def (csuba_gen_abbr g d1 x1 u H12) in (ex2_ind C
+(\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba
+g d1 d2)) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: C).(\lambda (H15: (eq C x1
+(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csuba g d1 x2)).(let H17 \def
+(eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15)
+in (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))
+(\lambda (d2: C).(csuba g d1 d2)) x2 (getl_intro O c2 (CHead x2 (Bind Abbr)
+u) c2 (drop_refl c2) H17) H16))))) H14))))) H11)))))))) (\lambda (n:
+nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to
+(\forall (c2: C).((csuba g x c2) \to (ex2 C (\lambda (d2: C).(getl n c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))).(\lambda
+(x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2:
+C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c
+(Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_:
+B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) t))))) (ex2
+C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(csuba g d1 d2))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3:
+T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n
+O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10
+(CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead
+x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3)
+x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4
+x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g x2 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5:
+B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind
+x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda
+(c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2
+H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x8:
+C).(\lambda (H22: (getl n x6 (CHead x8 (Bind Abbr) u))).(\lambda (H23: (csuba
+g d1 x8)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20
+(CHead x8 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17))))) H14)))))))
+H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+
+theorem csuba_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1)))
+(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear
+x (CHead d1 (Bind Abst) u1))).((match x return (\lambda (c: C).((drop i O c1
+c) \to ((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1
+c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) with
+[(CSort n) \Rightarrow (\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4:
+(clear (CSort n) (CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind
+Abst) u1) n H4 (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))))))))) | (CHead c k t) \Rightarrow (\lambda (H3:
+(drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1
+(Bind Abst) u1))).((match k return (\lambda (k0: K).((drop i O c1 (CHead c k0
+t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abst) u1)) \to (\forall (c2:
+C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c
+(Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abst)
+u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead
+d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind
+Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t)
+(clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) u1)
+(CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in
+(\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2:
+C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t:
+T).(drop i O c1 (CHead c (Bind b) t))) H5 u1 H9) in (let H14 \def (eq_ind_r B
+b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) u1))) H13 Abst H10) in (let
+H15 \def (eq_ind_r C c (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst)
+u1))) H14 d1 H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12)
+in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda
+(d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x0: C).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abst)
+u1))).(\lambda (H19: (csuba g d1 x0)).(or_introl (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i c2
+(CHead x0 (Bind Abst) u1) (CHead x0 (Bind Abst) u1) H18 (clear_bind Abst x0
+u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H18: (drop i O c2 (CHead x0
+(Bind Abbr) x1))).(\lambda (H19: (csuba g d1 x0)).(\lambda (H20: (arity g d1
+u1 (asucc g x2))).(\lambda (H21: (arity g x0 x1 x2)).(or_intror (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))
+x0 x1 x2 (getl_intro i c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1)
+H18 (clear_bind Abbr x0 x1)) H19 H20 H21))))))))) H17)) H16)))))))))) H8))
+H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f)
+t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abst)
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead
+c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) \to (or (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
+c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x0:
+C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2)
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda
+(x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2:
+C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c:
+C).(csuba g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat
+f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) u1)
+(clear_gen_flat f c (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def
+(csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1)
+H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1)
+x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abst g d1 x1 u1
+H12) in (or_ind (ex2 C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H15: (ex2 C (\lambda
+(d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2)))).(ex2_ind C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst)
+u1))).(\lambda (H17: (csuba g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c:
+C).(clear c2 c)) H13 (CHead x2 (Bind Abst) u1) H16) in (or_introl (ex2 C
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
+x2 (getl_intro O c2 (CHead x2 (Bind Abst) u1) c2 (drop_refl c2) H18)
+H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2:
+C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H16: (eq C x1 (CHead x2 (Bind
+Abbr) x3))).(\lambda (H17: (csuba g d1 x2)).(\lambda (H18: (arity g d1 u1
+(asucc g x4))).(\lambda (H19: (arity g x2 x3 x4)).(let H20 \def (eq_ind C x1
+(\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) x3) H16) in
+(or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))) x2 x3 x4 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl
+c2) H20) H17 H18 H19)))))))))) H15)) H14))))) H11)))))))) (\lambda (n:
+nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to
+(\forall (c2: C).((csuba g x c2) \to (or (ex2 C (\lambda (d2: C).(getl n c2
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat
+f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def
+(drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b:
+B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v)))))
+(\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f)
+t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1:
+B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind
+x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def
+(csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C
+(\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2:
+C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3)
+x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4
+x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g x2 e2)))) (or (ex2 C (\lambda (d2: C).(getl (S n)
+c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a)))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18:
+(eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20
+\def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7)
+H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2:
+C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6
+(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
+g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2:
+C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
+(or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1)))
+(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x8:
+C).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abst) u1))).(\lambda (H24:
+(csuba g d1 x8)).(or_introl (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
+u1))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20
+(CHead x8 (Bind Abst) u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind
+Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
+d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
+g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
+d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a)))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda
+(x10: A).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H24:
+(csuba g d1 x8)).(\lambda (H25: (arity g d1 u1 (asucc g x10))).(\lambda (H26:
+(arity g x8 x9 x10)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead
+d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
+a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x8 x9 x10 (getl_clear_bind x5 c2
+x6 x7 H20 (CHead x8 (Bind Abbr) x9) n H23) H24 H25 H26))))))))) H22))
+H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2))))
+H0))))))).
+
+theorem csuba_arity:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
+A).(\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
+C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c
+c2)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall
+(c2: C).((csuba g d c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
+(H3: (csuba g c c2)).(let H4 \def (csuba_getl_abbr g c d u i H0 c2 H3) in
+(ex2_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda
+(d2: C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda
+(H5: (getl i c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (csuba g d
+x)).(arity_abbr g c2 x u i H5 a0 (H2 x H6))))) H4)))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
+g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g d c2) \to (arity g c2 u
+(asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c c2)).(let H4 \def
+(csuba_getl_abst g c d u i H0 c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc
+g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+a1))))) (arity g c2 (TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d d2)))).(ex2_ind C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d d2)) (arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6:
+(getl i c2 (CHead x (Bind Abst) u))).(\lambda (H7: (csuba g d x)).(arity_abst
+g c2 x u i H6 a0 (H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d u (asucc g a))))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u (asucc
+g a1))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
+a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(x2: A).(\lambda (H6: (getl i c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
+(csuba g d x0)).(\lambda (H8: (arity g d u (asucc g x2))).(\lambda (H9:
+(arity g x0 x1 x2)).(arity_repl g c2 (TLRef i) x2 (arity_abbr g c2 x0 x1 i H6
+x2 H9) a0 (asucc_inj g x2 a0 (arity_mono g d u (asucc g x2) H8 (asucc g a0)
+H1)))))))))) H5)) H4)))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
+g c u a1)).(\lambda (H2: ((\forall (c2: C).((csuba g c c2) \to (arity g c2 u
+a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba g (CHead c (Bind
+b) u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (csuba g
+c c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u)
+(csuba_head g c c2 H5 (Bind b) u)))))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g
+a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: C).((csuba g (CHead c
+(Bind Abst) u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4:
+(csuba g c c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind
+Abst) u) (csuba_head g c c2 H4 (Bind Abst) u)))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda
+(H1: ((\forall (c2: C).((csuba g c c2) \to (arity g c2 u a1))))).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda
+(H3: ((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 (AHead a1
+a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c c2)).(arity_appl g c2 u a1
+(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
+T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 u (asucc g
+a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
+((\forall (c2: C).((csuba g c c2) \to (arity g c2 t0 a0))))).(\lambda (c2:
+C).(\lambda (H4: (csuba g c c2)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
+H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
+(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c c2) \to (arity
+g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
+C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
+H2)))))))))) c1 t a H))))).
+
+axiom csuba_arity_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
+t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a)))))))
+.
+
+theorem arity_appls_appl:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c
+v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t:
+T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs
+(THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead
+(Flat Appl) v (THead (Bind Abst) u t))) a2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (a1: A).(\lambda (H:
+(arity g c v a1)).(\lambda (u: T).(\lambda (H0: (arity g c u (asucc g
+a1))).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) t0 (THead (Bind Abbr)
+v t)) a2) \to (arity g c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead
+(Bind Abst) u t))) a2)))) (\lambda (a2: A).(\lambda (H1: (arity g c (THead
+(Bind Abbr) v t) a2)).(let H_x \def (arity_gen_bind Abbr (\lambda (H2: (eq B
+Abbr Abst)).(not_abbr_abst H2)) g c v t a2 H1) in (let H2 \def H_x in
+(ex2_ind A (\lambda (a3: A).(arity g c v a3)) (\lambda (_: A).(arity g (CHead
+c (Bind Abbr) v) t a2)) (arity g c (THead (Flat Appl) v (THead (Bind Abst) u
+t)) a2) (\lambda (x: A).(\lambda (_: (arity g c v x)).(\lambda (H4: (arity g
+(CHead c (Bind Abbr) v) t a2)).(arity_appl g c v a1 H (THead (Bind Abst) u t)
+a2 (arity_head g c u a1 H0 t a2 (csuba_arity_rev g (CHead c (Bind Abbr) v) t
+a2 H4 (CHead c (Bind Abst) u) (csuba_abst g c c (csuba_refl g c) u a1 H0 v
+H))))))) H2))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H1:
+((\forall (a2: A).((arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
+a2) \to (arity g c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+Abst) u t))) a2))))).(\lambda (a2: A).(\lambda (H2: (arity g c (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))) a2)).(let H3 \def
+(arity_gen_appl g c t0 (THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) a2 H2)
+in (ex2_ind A (\lambda (a3: A).(arity g c t0 a3)) (\lambda (a3: A).(arity g c
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)) (AHead a3 a2))) (arity g c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
+(Bind Abst) u t)))) a2) (\lambda (x: A).(\lambda (H4: (arity g c t0
+x)).(\lambda (H5: (arity g c (THeads (Flat Appl) t1 (THead (Bind Abbr) v t))
+(AHead x a2))).(arity_appl g c t0 x H4 (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind Abst) u t))) a2 (H1 (AHead x a2) H5))))) H3)))))))
+vs))))))))).
+
+theorem arity_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
+c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1
+t2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (a: A).(\lambda
+(H: (arity g c1 t1 a)).(arity_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(a0: A).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to
+(arity g c2 t2 a0)))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (c2:
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort n)
+t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n)))
+(arity_sort g c2 n) t2 (pr0_gen_sort t2 n H1)))))))) (\lambda (c: C).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d
+(Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda
+(H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to
+(arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i) t2)).(eq_ind_r T (TLRef i)
+(\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T (\lambda (e2: C).(\lambda
+(u2: T).(getl i c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (arity g c2
+(TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl i c2
+(CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u
+x1)).(arity_abbr g c2 x0 x1 i H5 a0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2
+H3 i d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 i H4)))))))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g
+a0))).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2:
+T).((pr0 u t2) \to (arity g c2 t2 (asucc g a0)))))))).(\lambda (c2:
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef i)
+t2)).(eq_ind_r T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u2: T).(getl i c2 (CHead e2 (Bind Abst) u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
+T).(pr0 u u2))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H5: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0
+d x0)).(\lambda (H7: (pr0 u x1)).(arity_abst g c2 x0 x1 i H5 a0 (H2 x0 H6 x1
+H7))))))) (wcpr0_getl c c2 H3 i d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 i
+H4)))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda
+(c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u
+a1)).(\lambda (H2: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
+u t2) \to (arity g c2 t2 a1))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda
+(H3: (arity g (CHead c (Bind b) u) t a2)).(\lambda (H4: ((\forall (c2:
+C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to
+(arity g c2 t2 a2))))))).(\lambda (c2: C).(\lambda (H5: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H6: (pr0 (THead (Bind b) u t) t2)).(insert_eq
+T (THead (Bind b) u t) (\lambda (t0: T).(pr0 t0 t2)) (arity g c2 t2 a2)
+(\lambda (y: T).(\lambda (H7: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda
+(t3: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t3 a2)))) (\lambda
+(t0: T).(\lambda (H8: (eq T t0 (THead (Bind b) u t))).(let H9 \def (f_equal T
+T (\lambda (e: T).e) t0 (THead (Bind b) u t) H8) in (eq_ind_r T (THead (Bind
+b) u t) (\lambda (t3: T).(arity g c2 t3 a2)) (arity_bind g b H0 c2 u a1 (H2
+c2 H5 u (pr0_refl u)) t a2 (H4 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u
+(pr0_refl u) (Bind b)) t (pr0_refl t))) t0 H9)))) (\lambda (u1: T).(\lambda
+(u2: T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u
+t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H10: (pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity
+g c2 t4 a2)))).(\lambda (k: K).(\lambda (H12: (eq T (THead k u1 t3) (THead
+(Bind b) u t))).(let H13 \def (f_equal T K (\lambda (e: T).(match e return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
+(THead k _ _) \Rightarrow k])) (THead k u1 t3) (THead (Bind b) u t) H12) in
+((let H14 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t
+_) \Rightarrow t])) (THead k u1 t3) (THead (Bind b) u t) H12) in ((let H15
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t)
+\Rightarrow t])) (THead k u1 t3) (THead (Bind b) u t) H12) in (\lambda (H16:
+(eq T u1 u)).(\lambda (H17: (eq K k (Bind b))).(eq_ind_r K (Bind b) (\lambda
+(k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H18 \def (eq_ind T t3 (\lambda
+(t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H11 t H15)
+in (let H19 \def (eq_ind T t3 (\lambda (t: T).(pr0 t t4)) H10 t H15) in (let
+H20 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to
+(arity g c2 u2 a2))) H9 u H16) in (let H21 \def (eq_ind T u1 (\lambda (t:
+T).(pr0 t u2)) H8 u H16) in (arity_bind g b H0 c2 u2 a1 (H2 c2 H5 u2 H21) t4
+a2 (H4 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H5 u u2 H21 (Bind b)) t4
+H19)))))) k H17)))) H14)) H13)))))))))))) (\lambda (u0: T).(\lambda (v1:
+T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead
+(Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t))
+\to (arity g c2 t4 a2)))).(\lambda (H12: (eq T (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t3)) (THead (Bind b) u t))).(let H13 \def (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind b) u t) H12) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4)
+a2) H13)))))))))))) (\lambda (b0: B).(\lambda (_: (not (eq B b0
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda
+(_: (((eq T v1 (THead (Bind b) u t)) \to (arity g c2 v2 a2)))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead
+(Bind b) u t)) \to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind b) u t))
+\to (arity g c2 t4 a2)))).(\lambda (H15: (eq T (THead (Flat Appl) v1 (THead
+(Bind b0) u1 t3)) (THead (Bind b) u t))).(let H16 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind b0) u1 t3)) (\lambda (ee: T).(match ee return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t)
+H15) in (False_ind (arity g c2 (THead (Bind b0) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) a2) H16))))))))))))))))) (\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (((eq T u1 (THead (Bind b) u t))
+\to (arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H10:
+(pr0 t3 t4)).(\lambda (H11: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2
+t4 a2)))).(\lambda (w: T).(\lambda (H12: (subst0 O u2 t4 w)).(\lambda (H13:
+(eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u t))).(let H14 \def (f_equal
+T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in
+((let H15 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t
+_) \Rightarrow t])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in
+((let H16 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
+t) \Rightarrow t])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u t) H13) in
+(\lambda (H17: (eq T u1 u)).(\lambda (H18: (eq B Abbr b)).(let H19 \def
+(eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t)) \to (arity g c2
+t4 a2))) H11 t H16) in (let H20 \def (eq_ind T t3 (\lambda (t: T).(pr0 t t4))
+H10 t H16) in (let H21 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0 (THead
+(Bind b) u t)) \to (arity g c2 u2 a2))) H9 u H17) in (let H22 \def (eq_ind T
+u1 (\lambda (t: T).(pr0 t u2)) H8 u H17) in (let H23 \def (eq_ind_r B b
+(\lambda (b: B).((eq T t (THead (Bind b) u t)) \to (arity g c2 t4 a2))) H19
+Abbr H18) in (let H24 \def (eq_ind_r B b (\lambda (b: B).((eq T u (THead
+(Bind b) u t)) \to (arity g c2 u2 a2))) H21 Abbr H18) in (let H25 \def
+(eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u)
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a2)))))) H4 Abbr H18)
+in (let H26 \def (eq_ind_r B b (\lambda (b: B).(arity g (CHead c (Bind b) u)
+t a2)) H3 Abbr H18) in (let H27 \def (eq_ind_r B b (\lambda (b: B).(not (eq B
+b Abst))) H0 Abbr H18) in (arity_bind g Abbr H27 c2 u2 a1 (H2 c2 H5 u2 H22) w
+a2 (arity_subst0 g (CHead c2 (Bind Abbr) u2) t4 a2 (H25 (CHead c2 (Bind Abbr)
+u2) (wcpr0_comp c c2 H5 u u2 H22 (Bind Abbr)) t4 H20) c2 u2 O (getl_refl Abbr
+c2 u2) w H12)))))))))))))) H15)) H14))))))))))))) (\lambda (b0: B).(\lambda
+(H8: (not (eq B b0 Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9:
+(pr0 t3 t4)).(\lambda (H10: (((eq T t3 (THead (Bind b) u t)) \to (arity g c2
+t4 a2)))).(\lambda (u0: T).(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S
+O) O t3)) (THead (Bind b) u t))).(let H12 \def (f_equal T B (\lambda (e:
+T).(match e return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef
+_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead
+(Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in ((let H13 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Bind b0) u0 (lift (S O) O t3)) (THead (Bind b) u t) H11) in ((let H14
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t:
+T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
+(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O
+t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat)
+(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef
+i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
+(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O
+t3) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t3))
+(THead (Bind b) u t) H11) in (\lambda (_: (eq T u0 u)).(\lambda (H16: (eq B
+b0 b)).(let H17 \def (eq_ind B b0 (\lambda (b: B).(not (eq B b Abst))) H8 b
+H16) in (let H18 \def (eq_ind_r T t (\lambda (t: T).((eq T t3 (THead (Bind b)
+u t)) \to (arity g c2 t4 a2))) H10 (lift (S O) O t3) H14) in (let H19 \def
+(eq_ind_r T t (\lambda (t: T).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u)
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a2)))))) H4 (lift (S
+O) O t3) H14) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(arity g (CHead
+c (Bind b) u) t a2)) H3 (lift (S O) O t3) H14) in (arity_gen_lift g (CHead c2
+(Bind b) u) t4 a2 (S O) O (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H5 u u
+(pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t3 t4 H9 (S O) O)) c2
+(drop_drop (Bind b) O c2 c2 (drop_refl c2) u))))))))) H13)) H12))))))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_:
+(((eq T t3 (THead (Bind b) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0:
+T).(\lambda (H10: (eq T (THead (Flat Cast) u0 t3) (THead (Bind b) u t))).(let
+H11 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind b) u t) H10) in (False_ind (arity g c2 t4 a2) H11)))))))) y t2
+H7))) H6)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (arity g c2 t2 (asucc g
+a1)))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c
+(Bind Abst) u) t a2)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind
+Abst) u) c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2
+a2))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2:
+T).(\lambda (H5: (pr0 (THead (Bind Abst) u t) t2)).(insert_eq T (THead (Bind
+Abst) u t) (\lambda (t0: T).(pr0 t0 t2)) (arity g c2 t2 (AHead a1 a2))
+(\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind (\lambda (t0: T).(\lambda
+(t3: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 t3 (AHead a1
+a2))))) (\lambda (t0: T).(\lambda (H7: (eq T t0 (THead (Bind Abst) u
+t))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind Abst) u t)
+H7) in (eq_ind_r T (THead (Bind Abst) u t) (\lambda (t3: T).(arity g c2 t3
+(AHead a1 a2))) (arity_head g c2 u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3
+(CHead c2 (Bind Abst) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abst)) t
+(pr0_refl t))) t0 H8)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0
+u1 u2)).(\lambda (H8: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2
+(AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3
+t4)).(\lambda (H10: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4
+(AHead a1 a2))))).(\lambda (k: K).(\lambda (H11: (eq T (THead k u1 t3) (THead
+(Bind Abst) u t))).(let H12 \def (f_equal T K (\lambda (e: T).(match e return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
+(THead k _ _) \Rightarrow k])) (THead k u1 t3) (THead (Bind Abst) u t) H11)
+in ((let H13 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t
+_) \Rightarrow t])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in ((let H14
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t)
+\Rightarrow t])) (THead k u1 t3) (THead (Bind Abst) u t) H11) in (\lambda
+(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Bind Abst))).(eq_ind_r K (Bind
+Abst) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) (AHead a1 a2))) (let H17
+\def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Bind Abst) u t)) \to
+(arity g c2 t4 (AHead a1 a2)))) H10 t H14) in (let H18 \def (eq_ind T t3
+(\lambda (t: T).(pr0 t t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda
+(t0: T).((eq T t0 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1
+a2)))) H8 u H15) in (let H20 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7
+u H15) in (arity_head g c2 u2 a1 (H1 c2 H4 u2 H20) t4 a2 (H3 (CHead c2 (Bind
+Abst) u2) (wcpr0_comp c c2 H4 u u2 H20 (Bind Abst)) t4 H18)))))) k H16))))
+H13)) H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u t))
+\to (arity g c2 v2 (AHead a1 a2))))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t))
+\to (arity g c2 t4 (AHead a1 a2))))).(\lambda (H11: (eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u0 t3)) (THead (Bind Abst) u t))).(let H12 \def (eq_ind
+T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Abst) u t) H11) in (False_ind (arity g c2 (THead (Bind Abbr) v2
+t4) (AHead a1 a2)) H12)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda
+(_: (((eq T v1 (THead (Bind Abst) u t)) \to (arity g c2 v2 (AHead a1
+a2))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (THead (Bind Abst) u t)) \to (arity g c2 u2 (AHead a1
+a2))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda
+(_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1
+a2))))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(THead (Bind Abst) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u
+t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) (AHead a1 a2)) H15))))))))))))))))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead
+(Bind Abst) u t)) \to (arity g c2 u2 (AHead a1 a2))))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead
+(Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))).(\lambda (w:
+T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind Abbr)
+u1 t3) (THead (Bind Abst) u t))).(let H13 \def (eq_ind T (THead (Bind Abbr)
+u1 t3) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow
+(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
+\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
+False])])) I (THead (Bind Abst) u t) H12) in (False_ind (arity g c2 (THead
+(Bind Abbr) u2 w) (AHead a1 a2)) H13))))))))))))) (\lambda (b: B).(\lambda
+(H7: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
+t3 t4)).(\lambda (H9: (((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4
+(AHead a1 a2))))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind b) u0
+(lift (S O) O t3)) (THead (Bind Abst) u t))).(let H11 \def (f_equal T B
+(\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in
+((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t
+_) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t3)) (THead (Bind Abst)
+u t) H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x (S O))) O t3) | (THead _ _ t) \Rightarrow t])) (THead (Bind
+b) u0 (lift (S O) O t3)) (THead (Bind Abst) u t) H10) in (\lambda (_: (eq T
+u0 u)).(\lambda (H15: (eq B b Abst)).(let H16 \def (eq_ind B b (\lambda (b:
+B).(not (eq B b Abst))) H7 Abst H15) in (let H17 \def (eq_ind_r T t (\lambda
+(t: T).((eq T t3 (THead (Bind Abst) u t)) \to (arity g c2 t4 (AHead a1 a2))))
+H9 (lift (S O) O t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t:
+T).(\forall (c2: C).((wcpr0 (CHead c (Bind Abst) u) c2) \to (\forall (t2:
+T).((pr0 t t2) \to (arity g c2 t2 a2)))))) H3 (lift (S O) O t3) H13) in (let
+H19 \def (eq_ind_r T t (\lambda (t: T).(arity g (CHead c (Bind Abst) u) t
+a2)) H2 (lift (S O) O t3) H13) in (let H20 \def (match (H16 (refl_equal B
+Abst)) return (\lambda (_: False).(arity g c2 t4 (AHead a1 a2))) with []) in
+H20)))))))) H12)) H11)))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda
+(_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Bind Abst) u t)) \to (arity
+g c2 t4 (AHead a1 a2))))).(\lambda (u0: T).(\lambda (H9: (eq T (THead (Flat
+Cast) u0 t3) (THead (Bind Abst) u t))).(let H10 \def (eq_ind T (THead (Flat
+Cast) u0 t3) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t) H9) in
+(False_ind (arity g c2 t4 (AHead a1 a2)) H10)))))))) y t2 H6)))
+H5)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
+(_: (arity g c u a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to
+(\forall (t2: T).((pr0 u t2) \to (arity g c2 t2 a1))))))).(\lambda (t:
+T).(\lambda (a2: A).(\lambda (H2: (arity g c t (AHead a1 a2))).(\lambda (H3:
+((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g
+c2 t2 (AHead a1 a2)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) u t)
+t2)).(insert_eq T (THead (Flat Appl) u t) (\lambda (t0: T).(pr0 t0 t2))
+(arity g c2 t2 a2) (\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind
+(\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Flat Appl) u t)) \to
+(arity g c2 t3 a2)))) (\lambda (t0: T).(\lambda (H7: (eq T t0 (THead (Flat
+Appl) u t))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat
+Appl) u t) H7) in (eq_ind_r T (THead (Flat Appl) u t) (\lambda (t3: T).(arity
+g c2 t3 a2)) (arity_appl g c2 u a1 (H1 c2 H4 u (pr0_refl u)) t a2 (H3 c2 H4 t
+(pr0_refl t))) t0 H8)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0
+u1 u2)).(\lambda (H8: (((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2
+a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda
+(H10: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda
+(k: K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Flat Appl) u t))).(let
+H12 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H13
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _)
+\Rightarrow t])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in ((let H14
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t)
+\Rightarrow t])) (THead k u1 t3) (THead (Flat Appl) u t) H11) in (\lambda
+(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Flat Appl))).(eq_ind_r K (Flat
+Appl) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) a2)) (let H17 \def
+(eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to (arity g
+c2 t4 a2))) H10 t H14) in (let H18 \def (eq_ind T t3 (\lambda (t: T).(pr0 t
+t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0
+(THead (Flat Appl) u t)) \to (arity g c2 u2 a2))) H8 u H15) in (let H20 \def
+(eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u H15) in (arity_appl g c2 u2 a1
+(H1 c2 H4 u2 H20) t4 a2 (H3 c2 H4 t4 H18)))))) k H16)))) H13))
+H12)))))))))))) (\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H7: (pr0 v1 v2)).(\lambda (H8: (((eq T v1 (THead (Flat Appl) u t)) \to
+(arity g c2 v2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3
+t4)).(\lambda (H10: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4
+a2)))).(\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
+(THead (Flat Appl) u t))).(let H12 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t3)) (THead (Flat Appl) u t) H11) in ((let H13 \def (f_equal T
+T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow (THead (Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind
+Abst) u0 t3) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t3)) (THead (Flat Appl) u t) H11) in (\lambda (H14: (eq T v1
+u)).(let H15 \def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl)
+u t)) \to (arity g c2 v2 a2))) H8 u H14) in (let H16 \def (eq_ind T v1
+(\lambda (t: T).(pr0 t v2)) H7 u H14) in (let H17 \def (eq_ind_r T t (\lambda
+(t: T).((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H10 (THead
+(Bind Abst) u0 t3) H13) in (let H18 \def (eq_ind_r T t (\lambda (t: T).((eq T
+u (THead (Flat Appl) u t)) \to (arity g c2 v2 a2))) H15 (THead (Bind Abst) u0
+t3) H13) in (let H19 \def (eq_ind_r T t (\lambda (t: T).(\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead
+a1 a2))))))) H3 (THead (Bind Abst) u0 t3) H13) in (let H20 \def (eq_ind_r T t
+(\lambda (t: T).(arity g c t (AHead a1 a2))) H2 (THead (Bind Abst) u0 t3)
+H13) in (let H21 \def (H1 c2 H4 v2 H16) in (let H22 \def (H19 c2 H4 (THead
+(Bind Abst) u0 t4) (pr0_comp u0 u0 (pr0_refl u0) t3 t4 H9 (Bind Abst))) in
+(let H23 \def (arity_gen_abst g c2 u0 t4 (AHead a1 a2) H22) in (ex3_2_ind A A
+(\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4))))
+(\lambda (a3: A).(\lambda (_: A).(arity g c2 u0 (asucc g a3)))) (\lambda (_:
+A).(\lambda (a4: A).(arity g (CHead c2 (Bind Abst) u0) t4 a4))) (arity g c2
+(THead (Bind Abbr) v2 t4) a2) (\lambda (x0: A).(\lambda (x1: A).(\lambda
+(H24: (eq A (AHead a1 a2) (AHead x0 x1))).(\lambda (H25: (arity g c2 u0
+(asucc g x0))).(\lambda (H26: (arity g (CHead c2 (Bind Abst) u0) t4 x1)).(let
+H27 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: A).A) with
+[(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a2)
+(AHead x0 x1) H24) in ((let H28 \def (f_equal A A (\lambda (e: A).(match e
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a)
+\Rightarrow a])) (AHead a1 a2) (AHead x0 x1) H24) in (\lambda (H29: (eq A a1
+x0)).(let H30 \def (eq_ind_r A x1 (\lambda (a: A).(arity g (CHead c2 (Bind
+Abst) u0) t4 a)) H26 a2 H28) in (let H31 \def (eq_ind_r A x0 (\lambda (a:
+A).(arity g c2 u0 (asucc g a))) H25 a1 H29) in (arity_bind g Abbr
+not_abbr_abst c2 v2 a1 H21 t4 a2 (csuba_arity g (CHead c2 (Bind Abst) u0) t4
+a2 H30 (CHead c2 (Bind Abbr) v2) (csuba_abst g c2 c2 (csuba_refl g c2) u0 a1
+H31 v2 H21))))))) H27))))))) H23)))))))))))) H12)))))))))))) (\lambda (b:
+B).(\lambda (H7: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (H8: (pr0 v1 v2)).(\lambda (H9: (((eq T v1 (THead (Flat Appl) u
+t)) \to (arity g c2 v2 a2)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(H10: (pr0 u1 u2)).(\lambda (H11: (((eq T u1 (THead (Flat Appl) u t)) \to
+(arity g c2 u2 a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H12: (pr0
+t3 t4)).(\lambda (H13: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4
+a2)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(THead (Flat Appl) u t))).(let H15 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind b) u1 t3)) (THead (Flat Appl) u t) H14) in ((let H16 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1
+t3) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t3)) (THead (Flat Appl) u t) H14) in (\lambda (H17: (eq T v1 u)).(let H18
+\def (eq_ind T v1 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) u t)) \to
+(arity g c2 v2 a2))) H9 u H17) in (let H19 \def (eq_ind T v1 (\lambda (t:
+T).(pr0 t v2)) H8 u H17) in (let H20 \def (eq_ind_r T t (\lambda (t: T).((eq
+T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2))) H13 (THead (Bind b) u1
+t3) H16) in (let H21 \def (eq_ind_r T t (\lambda (t: T).((eq T u1 (THead
+(Flat Appl) u t)) \to (arity g c2 u2 a2))) H11 (THead (Bind b) u1 t3) H16) in
+(let H22 \def (eq_ind_r T t (\lambda (t: T).((eq T u (THead (Flat Appl) u t))
+\to (arity g c2 v2 a2))) H18 (THead (Bind b) u1 t3) H16) in (let H23 \def
+(eq_ind_r T t (\lambda (t: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall
+(t2: T).((pr0 t t2) \to (arity g c2 t2 (AHead a1 a2))))))) H3 (THead (Bind b)
+u1 t3) H16) in (let H24 \def (eq_ind_r T t (\lambda (t: T).(arity g c t
+(AHead a1 a2))) H2 (THead (Bind b) u1 t3) H16) in (let H25 \def (H1 c2 H4 v2
+H19) in (let H26 \def (H23 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H10
+t3 t4 H12 (Bind b))) in (let H27 \def (arity_gen_bind b H7 g c2 u2 t4 (AHead
+a1 a2) H26) in (ex2_ind A (\lambda (a3: A).(arity g c2 u2 a3)) (\lambda (_:
+A).(arity g (CHead c2 (Bind b) u2) t4 (AHead a1 a2))) (arity g c2 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) a2) (\lambda (x:
+A).(\lambda (H28: (arity g c2 u2 x)).(\lambda (H29: (arity g (CHead c2 (Bind
+b) u2) t4 (AHead a1 a2))).(arity_bind g b H7 c2 u2 x H28 (THead (Flat Appl)
+(lift (S O) O v2) t4) a2 (arity_appl g (CHead c2 (Bind b) u2) (lift (S O) O
+v2) a1 (arity_lift g c2 v2 a1 H25 (CHead c2 (Bind b) u2) (S O) O (drop_drop
+(Bind b) O c2 c2 (drop_refl c2) u2)) t4 a2 H29))))) H27)))))))))))))
+H15))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1
+u2)).(\lambda (_: (((eq T u1 (THead (Flat Appl) u t)) \to (arity g c2 u2
+a2)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda
+(_: (((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda
+(w: T).(\lambda (_: (subst0 O u2 t4 w)).(\lambda (H12: (eq T (THead (Bind
+Abbr) u1 t3) (THead (Flat Appl) u t))).(let H13 \def (eq_ind T (THead (Bind
+Abbr) u1 t3) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t) H12) in
+(False_ind (arity g c2 (THead (Bind Abbr) u2 w) a2) H13))))))))))))) (\lambda
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Appl) u t))
+\to (arity g c2 t4 a2)))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind
+b) u0 (lift (S O) O t3)) (THead (Flat Appl) u t))).(let H11 \def (eq_ind T
+(THead (Bind b) u0 (lift (S O) O t3)) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Appl) u t) H10) in (False_ind (arity g c2 t4 a2) H11))))))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda (_:
+(((eq T t3 (THead (Flat Appl) u t)) \to (arity g c2 t4 a2)))).(\lambda (u0:
+T).(\lambda (H9: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl) u
+t))).(let H10 \def (eq_ind T (THead (Flat Cast) u0 t3) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
+(match f return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast
+\Rightarrow True])])])) I (THead (Flat Appl) u t) H9) in (False_ind (arity g
+c2 t4 a2) H10)))))))) y t2 H6))) H5)))))))))))))) (\lambda (c: C).(\lambda
+(u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda
+(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to
+(arity g c2 t2 (asucc g a0)))))))).(\lambda (t: T).(\lambda (_: (arity g c t
+a0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
+t t2) \to (arity g c2 t2 a0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Cast) u t)
+t2)).(insert_eq T (THead (Flat Cast) u t) (\lambda (t0: T).(pr0 t0 t2))
+(arity g c2 t2 a0) (\lambda (y: T).(\lambda (H6: (pr0 y t2)).(pr0_ind
+(\lambda (t0: T).(\lambda (t3: T).((eq T t0 (THead (Flat Cast) u t)) \to
+(arity g c2 t3 a0)))) (\lambda (t0: T).(\lambda (H7: (eq T t0 (THead (Flat
+Cast) u t))).(let H8 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Flat
+Cast) u t) H7) in (eq_ind_r T (THead (Flat Cast) u t) (\lambda (t3: T).(arity
+g c2 t3 a0)) (arity_cast g c2 u a0 (H1 c2 H4 u (pr0_refl u)) t (H3 c2 H4 t
+(pr0_refl t))) t0 H8)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H7: (pr0
+u1 u2)).(\lambda (H8: (((eq T u1 (THead (Flat Cast) u t)) \to (arity g c2 u2
+a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H9: (pr0 t3 t4)).(\lambda
+(H10: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda
+(k: K).(\lambda (H11: (eq T (THead k u1 t3) (THead (Flat Cast) u t))).(let
+H12 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u1 t3) (THead (Flat Cast) u t) H11) in ((let H13
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _)
+\Rightarrow t])) (THead k u1 t3) (THead (Flat Cast) u t) H11) in ((let H14
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t)
+\Rightarrow t])) (THead k u1 t3) (THead (Flat Cast) u t) H11) in (\lambda
+(H15: (eq T u1 u)).(\lambda (H16: (eq K k (Flat Cast))).(eq_ind_r K (Flat
+Cast) (\lambda (k0: K).(arity g c2 (THead k0 u2 t4) a0)) (let H17 \def
+(eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat Cast) u t)) \to (arity g
+c2 t4 a0))) H10 t H14) in (let H18 \def (eq_ind T t3 (\lambda (t: T).(pr0 t
+t4)) H9 t H14) in (let H19 \def (eq_ind T u1 (\lambda (t0: T).((eq T t0
+(THead (Flat Cast) u t)) \to (arity g c2 u2 a0))) H8 u H15) in (let H20 \def
+(eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u H15) in (arity_cast g c2 u2 a0
+(H1 c2 H4 u2 H20) t4 (H3 c2 H4 t4 H18)))))) k H16)))) H13)) H12))))))))))))
+(\lambda (u0: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast) u t)) \to (arity g c2 v2
+a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).(\lambda
+(_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda
+(H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat
+Cast) u t))).(let H12 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst)
+u0 t3)) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t)
+H11) in (False_ind (arity g c2 (THead (Bind Abbr) v2 t4) a0) H12))))))))))))
+(\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
+(v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Flat Cast)
+u t)) \to (arity g c2 v2 a0)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead (Flat Cast) u t)) \to (arity
+g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
+t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4
+a0)))).(\lambda (H14: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(THead (Flat Cast) u t))).(let H15 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_:
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
+(Flat Cast) u t) H14) in (False_ind (arity g c2 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) a0) H15))))))))))))))))) (\lambda (u1:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (THead
+(Flat Cast) u t)) \to (arity g c2 u2 a0)))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (pr0 t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t))
+\to (arity g c2 t4 a0)))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t4
+w)).(\lambda (H12: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Cast) u
+t))).(let H13 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Cast) u t) H12) in (False_ind (arity g c2 (THead
+(Bind Abbr) u2 w) a0) H13))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq
+B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3
+t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) u t)) \to (arity g c2 t4
+a0)))).(\lambda (u0: T).(\lambda (H10: (eq T (THead (Bind b) u0 (lift (S O) O
+t3)) (THead (Flat Cast) u t))).(let H11 \def (eq_ind T (THead (Bind b) u0
+(lift (S O) O t3)) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t)
+H10) in (False_ind (arity g c2 t4 a0) H11)))))))))) (\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H7: (pr0 t3 t4)).(\lambda (H8: (((eq T t3 (THead (Flat
+Cast) u t)) \to (arity g c2 t4 a0)))).(\lambda (u0: T).(\lambda (H9: (eq T
+(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t))).(let H10 \def (f_equal T
+T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in ((let H11 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Cast) u0 t3) (THead (Flat Cast) u t) H9) in (\lambda (_: (eq T
+u0 u)).(let H13 \def (eq_ind T t3 (\lambda (t0: T).((eq T t0 (THead (Flat
+Cast) u t)) \to (arity g c2 t4 a0))) H8 t H11) in (let H14 \def (eq_ind T t3
+(\lambda (t: T).(pr0 t t4)) H7 t H11) in (H3 c2 H4 t4 H14))))) H10)))))))) y
+t2 H6))) H5))))))))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (a1:
+A).(\lambda (_: (arity g c t a1)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
+c2) \to (\forall (t2: T).((pr0 t t2) \to (arity g c2 t2 a1))))))).(\lambda
+(a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (wcpr0 c
+c2)).(\lambda (t2: T).(\lambda (H4: (pr0 t t2)).(arity_repl g c2 t2 a1 (H1 c2
+H3 t2 H4) a2 H2)))))))))))) c1 t1 a H))))).
+
+theorem arity_sred_wcpr0_pr1:
+ \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
+(c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1
+c2) \to (arity g c2 t2 a)))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
+(t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c1: C).(\forall (a:
+A).((arity g c1 t a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t0
+a))))))))) (\lambda (t: T).(\lambda (g: G).(\lambda (c1: C).(\lambda (a:
+A).(\lambda (H0: (arity g c1 t a)).(\lambda (c2: C).(\lambda (H1: (wcpr0 c1
+c2)).(arity_sred_wcpr0_pr0 g c1 t a H0 c2 H1 t (pr0_refl t))))))))) (\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c1: C).(\forall (a:
+A).((arity g c1 t3 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t5
+a))))))))).(\lambda (g: G).(\lambda (c1: C).(\lambda (a: A).(\lambda (H3:
+(arity g c1 t4 a)).(\lambda (c2: C).(\lambda (H4: (wcpr0 c1 c2)).(H2 g c2 a
+(arity_sred_wcpr0_pr0 g c1 t4 a H3 c2 H4 t3 H0) c2 (wcpr0_refl
+c2)))))))))))))) t1 t2 H))).
+
+theorem arity_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
+G).(\forall (a: A).((arity g c0 t a) \to (arity g c0 t0 a))))))) (\lambda
+(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda
+(g: G).(\lambda (a: A).(\lambda (H1: (arity g c0 t3 a)).(arity_sred_wcpr0_pr0
+g c0 t3 a H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
+G).(\lambda (a: A).(\lambda (H3: (arity g c0 t3 a)).(arity_subst0 g c0 t4 a
+(arity_sred_wcpr0_pr0 g c0 t3 a H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem arity_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (a:
+A).((arity g c t a) \to (arity g c t0 a)))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (a: A).(\lambda (H0: (arity g c t a)).H0)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (a: A).((arity g c
+t3 a) \to (arity g c t5 a)))))).(\lambda (g: G).(\lambda (a: A).(\lambda (H3:
+(arity g c t4 a)).(H2 g a (arity_sred_pr2 c t4 t3 H0 g a H3))))))))))) t1 t2
+H)))).
+
+definition nf2:
+ C \to (T \to Prop)
+\def
+ \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1
+t2)))).
+
+theorem nf2_gen_base__aux:
+ \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to
+(\forall (P: Prop).P))))
+\def
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).((eq
+T (THead k u t0) t0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda
+(u: T).(\lambda (H: (eq T (THead k u (TSort n)) (TSort n))).(\lambda (P:
+Prop).(let H0 \def (eq_ind T (THead k u (TSort n)) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H) in
+(False_ind P H0)))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (H: (eq T
+(THead k u (TLRef n)) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind T
+(THead k u (TLRef n)) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n) H) in (False_ind P H0)))))) (\lambda (k0:
+K).(\lambda (t0: T).(\lambda (_: ((\forall (u: T).((eq T (THead k u t0) t0)
+\to (\forall (P: Prop).P))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u:
+T).((eq T (THead k u t1) t1) \to (\forall (P: Prop).P))))).(\lambda (u:
+T).(\lambda (H1: (eq T (THead k u (THead k0 t0 t1)) (THead k0 t0
+t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u (THead k0 t0 t1))
+(THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k u (THead k0 t0 t1))
+(THead k0 t0 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 t0 t1) |
+(TLRef _) \Rightarrow (THead k0 t0 t1) | (THead _ _ t) \Rightarrow t]))
+(THead k u (THead k0 t0 t1)) (THead k0 t0 t1) H1) in (\lambda (_: (eq T u
+t0)).(\lambda (H6: (eq K k k0)).(let H7 \def (eq_ind K k (\lambda (k:
+K).(\forall (u: T).((eq T (THead k u t1) t1) \to (\forall (P: Prop).P)))) H0
+k0 H6) in (H7 t0 H4 P))))) H3)) H2)))))))))) t)).
+
+theorem nf2_gen_lref:
+ \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2
+c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P:
+Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
+(lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef
+i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
+
+theorem nf2_gen_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u
+t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
+T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t)
+t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2:
+T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2:
+T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef
+_) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) u t)
+(THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2 H0
+(Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t: T).(pr2 c u t))
+H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u) t2
+H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t
+t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t)
+\Rightarrow t])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H (THead
+(Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in H_y))) in
+(let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind Abst) u) t
+t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0)) (refl_equal T t)
+t2 H1))))))))).
+
+theorem nf2_gen_cast:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
+(Flat Cast) u t))).(\lambda (P: Prop).(nf2_gen_base__aux (Flat Cast) t u (H t
+(pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u)))
+P))))).
+
+theorem nf2_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
+(THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
+u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
+(t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
+u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Flat f) u t) (THead (Flat f) t2 t) (H (THead (Flat
+f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1))) (\lambda (t2:
+T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef
+_) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t) (THead
+(Flat f) u t2) (H (THead (Flat f) u t2) (pr2_head_2 c u t t2 (Flat f)
+(pr2_cflat c t t2 H0 f u)))) in H1)))))))).
+
+theorem nf2_sort:
+ \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
+\def
+ \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort
+n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal
+T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
+
+theorem nf2_abst:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v:
+T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind
+Abst) u t))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
+\to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda
+(H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t
+t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t)
+t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead
+(Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2
+(THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5:
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t
+x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead
+(Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t
+x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3))))))
+H2)))))))))).
+
+theorem nf2_pr3_unfold:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c
+t1) \to (eq T t1 t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t
+t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t
+(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0)
+\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def
+(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def
+(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T
+t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
+
+theorem nf2_pr3_confluence:
+ \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2)
+\to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2:
+T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t
+t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0))
+(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3:
+(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1
+x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t: T).(pr3 c t1 t)) H4 t1
+H_y) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr3 c t2 t)) H3 t1 H_y)
+in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2
+(\lambda (t: T).(pr3 c t t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0:
+T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2
+t1 H1))))))))).
+
+theorem nf2_appl_lref:
+ \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c
+(TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
+\to (eq T u t2))))).(\lambda (i: nat).(\lambda (H0: ((\forall (t2: T).((pr2 c
+(TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (t2: T).(\lambda (H1: (pr2
+c (THead (Flat Appl) u (TLRef i)) t2)).(let H2 \def (pr2_gen_appl c u (TLRef
+i) t2 H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
+(eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (H3: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c
+(TLRef i) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))) (eq T (THead (Flat
+Appl) u (TLRef i)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T
+t2 (THead (Flat Appl) x0 x1))).(\lambda (H5: (pr2 c u x0)).(\lambda (H6: (pr2
+c (TLRef i) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(eq T
+(THead (Flat Appl) u (TLRef i)) t)) (let H7 \def (eq_ind_r T x1 (\lambda (t:
+T).(pr2 c (TLRef i) t)) H6 (TLRef i) (H0 x1 H6)) in (eq_ind T (TLRef i)
+(\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead (Flat Appl) x0
+t))) (let H8 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c u t)) H5 u (H x0 H5))
+in (eq_ind T u (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead
+(Flat Appl) t (TLRef i)))) (refl_equal T (THead (Flat Appl) u (TLRef i))) x0
+(H x0 H5))) x1 (H0 x1 H6))) t2 H4)))))) H3)) (\lambda (H3: (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1
+t2))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
+b) u0) z1 t3))))))) (eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T
+(TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H5: (eq T t2 (THead (Bind
+Abbr) x2 x3))).(\lambda (_: (pr2 c u x2)).(\lambda (_: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(eq_ind_r T (THead
+(Bind Abbr) x2 x3) (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) t))
+(let H8 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0 x1) H4) in
+(False_ind (eq T (THead (Flat Appl) u (TLRef i)) (THead (Bind Abbr) x2 x3))
+H8)) t2 H5))))))))) H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i)
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c u u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c u u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
+(eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (x0: B).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H5: (eq T (TLRef i) (THead
+(Bind x0) x1 x2))).(\lambda (H6: (eq T t2 (THead (Bind x0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c u x4)).(\lambda (_: (pr2 c
+x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(eq_ind_r T (THead
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (\lambda (t: T).(eq T
+(THead (Flat Appl) u (TLRef i)) t)) (let H10 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind x0) x1 x2) H5) in (False_ind (eq T (THead (Flat Appl)
+u (TLRef i)) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)))
+H10)) t2 H6))))))))))))) H3)) H2)))))))).
+
+theorem nf2_lref_abst:
+ \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c
+(CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c
+(TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2
+(TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d
+(Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O
+u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T
+(TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2
+H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
+i) O u)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead
+d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O
+u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i) O
+x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t)) (let
+H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c c0)) H
+(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0
+(Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst) u)
+(\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1)
+(getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1) H3)) in
+(False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2))
+H1)))))))).
+
+theorem nf2_lift:
+ \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2)
+\to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i:
+nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c
+(lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind
+T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3))
+(eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i
+x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq
+T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x
+(\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq
+T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3))))
+H2)))))))))).
+
+theorem nf2_lift1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1
+hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p
+t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c
+e)).(\lambda (H0: (nf2 e t)).(let H1 \def (match H return (\lambda (p:
+PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq
+PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c t)))))))) with
+[(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2:
+(eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C
+c1 e) \to (nf2 c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c:
+C).(nf2 c t)) H0 c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) |
+(drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList
+(PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3
+e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match
+e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _
+_) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e)
+\to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c t))))) H6)) H4 H5 H1
+H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C
+e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c
+(lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1
+(PCons n n0 p) c e)).(\lambda (H1: (nf2 e t)).(let H2 \def (match H0 return
+(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0
+c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to
+(nf2 c (lift n n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow
+(\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0
+c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0:
+PList).(match e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True
+| (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq
+C c0 c) \to ((eq C c0 e) \to (nf2 c (lift n n0 (lift1 p t))))) H5)) H3 H4))))
+| (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList
+(PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq
+C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0
+return (\lambda (_: PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p)
+\Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal
+PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: PList).nat) with
+[PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n
+n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0
+return (\lambda (_: PList).nat) with [PNil \Rightarrow h | (PCons n _ _)
+\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda
+(n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3
+e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1
+p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1:
+nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1
+c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 p t))))))))) (\lambda
+(H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to
+((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (nf2 c (lift n
+n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0:
+C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (nf2 c (lift
+n n0 (lift1 p t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0:
+C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (nf2 c (lift n n0 (lift1 p
+t)))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2
+e)).(nf2_lift c2 (lift1 p t) (H c2 t H15 H1) c n n0 H14))) c3 (sym_eq C c3 e
+H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat
+d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2
+(refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e)))))))))))
+hds)).
+
+theorem nf2_iso_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
+TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso
+(THeads (Flat Appl) vs (TLRef i)) u))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(vs: TList).(TList_ind (\lambda (t: TList).(\forall (u: T).((pr3 c (THeads
+(Flat Appl) t (TLRef i)) u) \to (iso (THeads (Flat Appl) t (TLRef i)) u))))
+(\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i) u)).(let H_y \def
+(nf2_pr3_unfold c (TLRef i) u H0 H) in (let H1 \def (eq_ind_r T u (\lambda
+(t: T).(pr3 c (TLRef i) t)) H0 (TLRef i) H_y) in (eq_ind T (TLRef i) (\lambda
+(t: T).(iso (TLRef i) t)) (iso_lref i i) u H_y))))) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat Appl) t0
+(TLRef i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) u))))).(\lambda (u:
+T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
+i))) u)).(let H2 \def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u
+H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
+T).(pr3 (CHead c (Bind b) u0) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2)))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
+i))) u) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u
+(THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead (Flat
+Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) t2))) (iso
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H4: (eq T u (THead (Flat Appl) x0
+x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0
+(TLRef i)) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(iso
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t1)) (iso_head (Flat
+Appl) t x0 (THeads (Flat Appl) t0 (TLRef i)) x1) u H4)))))) H3)) (\lambda
+(H3: (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(pr3 c (THead (Bind Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
+Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1
+t2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
+(_: (pr3 c (THead (Bind Abbr) x2 x3) u)).(\lambda (_: (pr3 c t x2)).(\lambda
+(H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0
+x1))).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+x1 x3))))).(let H_y \def (H0 (THead (Bind Abst) x0 x1) H6) in
+(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y (iso (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i))) u))))))))))) H3)) (\lambda (H3: (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
+(CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat
+Appl) t0 (TLRef i)) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
+T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))
+u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b)
+y2) z1 z2))))))) (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
+u) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
+Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift
+(S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1
+x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let H_y \def (H0
+(THead (Bind x0) x1 x2) H5) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0
+H_y (iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i)))
+u))))))))))))))) H3)) H2))))))) vs)))).
+
+theorem nf2_dec:
+ \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq
+T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
+\def
+ \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall
+(t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1
+t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda
+(n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in
+(or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to
+(eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2
+(CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2
+H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))
+(or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to
+(\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2:
+T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T
+t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x
+H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or
+(\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1
+t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H
+t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0)
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1:
+((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(match k return
+(\lambda (k0: K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail k0 t c0) t1 t2))))) with [(Bind b) \Rightarrow
+(match b return (\lambda (b0: B).(or (\forall (t2: T).((pr2 (CTail (Bind b0)
+t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1
+t2))))) with [Abbr \Rightarrow (let H_x0 \def (dnf_dec t t1 (clen c0)) in
+(let H2 \def H_x0 in (ex_ind T (\lambda (v: T).(or (subst0 (clen c0) t t1
+(lift (S O) (clen c0) v)) (eq T t1 (lift (S O) (clen c0) v)))) (or (\forall
+(t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x: T).(\lambda (H3: (or
+(subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S O) (clen
+c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1
+(lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0)
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2))))
+(\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) (clen c0) x))).(let H_x1
+\def (getl_ctail_clen Abbr t c0) in (let H5 \def H_x1 in (ex_ind nat (\lambda
+(n: nat).(getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort n) (Bind
+Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq
+T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x0:
+nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort x0)
+(Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0)
+t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))
+(ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)) (lift (S O) (clen c0)
+x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) x))).(\lambda (P: Prop).(let
+H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 (clen c0) t t0 (lift (S O)
+(clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in (subst0_gen_lift_false x t
+(lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r
+nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S
+O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S O))) H8 P))))
+(pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t (clen c0) H6 t1 t1 (pr0_refl
+t1) (lift (S O) (clen c0) x) H4))))) H5)))) (\lambda (H4: (eq T t1 (lift (S
+O) (clen c0) x))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
+T).((pr2 c0 t t2) \to (eq T t t2)))) H1 (lift (S O) (clen c0) x) H4) in
+(eq_ind_r T (lift (S O) (clen c0) x) (\lambda (t0: T).(or (\forall (t2:
+T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda
+(t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2
+(CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall (t2: T).((pr2 (CTail
+(Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T (lift (S O) (clen
+c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) (clen c0) x) t2) \to
+(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) (lift
+(S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda (H6: (pr2 (CTail (Bind
+Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 \def (pr2_gen_ctail (Bind
+Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let H7 \def H_x1 in (or_ind
+(pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda (_: T).(eq K (Bind Abbr)
+(Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda
+(t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift (S O) (clen c0) x) t2)
+(\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) t2)).(H5 t2 H8)) (\lambda (H8:
+(ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t: T).(pr0
+(lift (S O) (clen c0) x) t)) (\lambda (t0: T).(subst0 (clen c0) t t0
+t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
+(t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0)
+t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x0: T).(\lambda (_:
+(eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: (pr0 (lift (S O) (clen c0) x)
+x0)).(\lambda (H11: (subst0 (clen c0) t x0 t2)).(ex2_ind T (\lambda (t3:
+T).(eq T x0 (lift (S O) (clen c0) t3))) (\lambda (t3: T).(pr0 x t3)) (eq T
+(lift (S O) (clen c0) x) t2) (\lambda (x1: T).(\lambda (H12: (eq T x0 (lift
+(S O) (clen c0) x1))).(\lambda (_: (pr0 x x1)).(let H14 \def (eq_ind T x0
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)) H11 (lift (S O) (clen c0) x1)
+H12) in (subst0_gen_lift_false x1 t t2 (S O) (clen c0) (clen c0) (le_n (clen
+c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n))
+(le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S
+O))) H14 (eq T (lift (S O) (clen c0) x) t2)))))) (pr0_gen_lift x x0 (S O)
+(clen c0) H10)))))) H8)) H7)))))) t1 H4))) H3))) H2))) | Abst \Rightarrow
+(or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda (t2:
+T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def
+(pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr)))
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
+(\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t))
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
+K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
+(eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee:
+K).(match ee return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abbr)
+H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) | Void \Rightarrow
+(or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T
+t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
+(\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2:
+T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def
+(pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
+(pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr)))
+(\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
+(eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
+(\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t))
+(\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
+K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
+(eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
+(subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee:
+K).(match ee return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])) I (Bind Abbr)
+H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))]) | (Flat f)
+\Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) t1 t2)
+\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda (t2:
+T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def
+(pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2
+c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0:
+T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2)
+(\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_:
+T).(eq K (Flat f) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0:
+T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f)
+(Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen
+c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f)
+(Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0
+t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4))
+H3))))))])) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))
+(or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x)
+\to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall
+(t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
+T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t
+c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
+Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1
+x H3 k t)))))) H1)) H0)))))))) c).
-axiom csubst0_clear_S: \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 (S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))) .
+inductive sn3 (c:C): T \to Prop \def
+| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
+
+definition sns3:
+ C \to (TList \to Prop)
+\def
+ let rec sns3 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))])
+in sns3.
+
+theorem sn3_gen_flat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
+(THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t))))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Flat f) u t))).(insert_eq T (THead (Flat f) u t) (\lambda (t0:
+T).(sn3 c t0)) (land (sn3 c u) (sn3 c t)) (\lambda (y: T).(\lambda (H0: (sn3
+c y)).(unintro T t (\lambda (t0: T).((eq T y (THead (Flat f) u t0)) \to (land
+(sn3 c u) (sn3 c t0)))) (unintro T u (\lambda (t0: T).(\forall (x: T).((eq T
+y (THead (Flat f) t0 x)) \to (land (sn3 c t0) (sn3 c x))))) (sn3_ind c
+(\lambda (t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat f) x
+x0)) \to (land (sn3 c x) (sn3 c x0)))))) (\lambda (t1: T).(\lambda (H1:
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1
+t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).(\forall (x0:
+T).((eq T t2 (THead (Flat f) x x0)) \to (land (sn3 c x) (sn3 c
+x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1 (THead
+(Flat f) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2:
+T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall
+(x: T).(\forall (x0: T).((eq T t2 (THead (Flat f) x x0)) \to (land (sn3 c x)
+(sn3 c x0))))))))) H2 (THead (Flat f) x x0) H3) in (let H5 \def (eq_ind T t1
+(\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P)))
+\to ((pr3 c t t2) \to (sn3 c t2))))) H1 (THead (Flat f) x x0) H3) in (conj
+(sn3 c x) (sn3 c x0) (sn3_sing c x (\lambda (t2: T).(\lambda (H6: (((eq T x
+t2) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr3 c x t2)).(let H8 \def (H4
+(THead (Flat f) t2 x0) (\lambda (H3: (eq T (THead (Flat f) x x0) (THead (Flat
+f) t2 x0))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef
+_) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat f) x x0)
+(THead (Flat f) t2 x0) H3) in (let H5 \def (eq_ind_r T t2 (\lambda (t:
+T).(pr3 c x t)) H7 x H4) in (let H6 \def (eq_ind_r T t2 (\lambda (t: T).((eq
+T x t) \to (\forall (P: Prop).P))) H6 x H4) in (H6 (refl_equal T x) P))))))
+(pr3_head_12 c x t2 H7 (Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2
+x0 (refl_equal T (THead (Flat f) t2 x0))) in (and_ind (sn3 c t2) (sn3 c x0)
+(sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8))))))
+(sn3_sing c x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x
+t2) (\lambda (H3: (eq T (THead (Flat f) x x0) (THead (Flat f) x
+t2))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t) \Rightarrow t])) (THead (Flat f) x x0) (THead
+(Flat f) x t2) H3) in (let H5 \def (eq_ind_r T t2 (\lambda (t: T).(pr3 c x0
+t)) H7 x0 H4) in (let H6 \def (eq_ind_r T t2 (\lambda (t: T).((eq T x0 t) \to
+(\forall (P: Prop).P))) H6 x0 H4) in (H6 (refl_equal T x0) P))))))
+(pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f) x t2))) in
+(and_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c x)).(\lambda
+(H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
+
+theorem sn3_nf2:
+ \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t
+(\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P:
+Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2
+H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y)
+in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P:
+Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3
+(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
+
+theorem sn3_pr3_trans:
+ \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
+t2) \to (sn3 c t2)))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: (sn3 c t1)).(sn3_ind c (\lambda
+(t: T).(\forall (t2: T).((pr3 c t t2) \to (sn3 c t2)))) (\lambda (t2:
+T).(\lambda (H0: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H1: ((\forall
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
+(\forall (t2: T).((pr3 c t3 t2) \to (sn3 c t2)))))))).(\lambda (t3:
+T).(\lambda (H2: (pr3 c t2 t3)).(sn3_sing c t3 (\lambda (t0: T).(\lambda (H3:
+(((eq T t3 t0) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 c t3 t0)).(let
+H_x \def (term_dec t2 t3) in (let H5 \def H_x in (or_ind (eq T t2 t3) ((eq T
+t2 t3) \to (\forall (P: Prop).P)) (sn3 c t0) (\lambda (H6: (eq T t2 t3)).(let
+H7 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t t0)) H4 t2 H6) in (let H8
+\def (eq_ind_r T t3 (\lambda (t: T).((eq T t t0) \to (\forall (P: Prop).P)))
+H3 t2 H6) in (let H9 \def (eq_ind_r T t3 (\lambda (t: T).(pr3 c t2 t)) H2 t2
+H6) in (H0 t0 H8 H7))))) (\lambda (H6: (((eq T t2 t3) \to (\forall (P:
+Prop).P)))).(H1 t3 H6 H2 t0 H4)) H5)))))))))))) t1 H))).
+
+theorem sn3_pr2_intro:
+ \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to
+(\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (H: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c
+t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H0: (((eq T t1 t2) \to
+(\forall (P: Prop).P)))).(\lambda (H1: (pr3 c t1 t2)).(let H2 \def H0 in
+((let H3 \def H in (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(((\forall
+(t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr2 c t t2) \to (sn3
+c t2))))) \to ((((eq T t t0) \to (\forall (P: Prop).P))) \to (sn3 c t0)))))
+(\lambda (t: T).(\lambda (H4: ((\forall (t2: T).((((eq T t t2) \to (\forall
+(P: Prop).P))) \to ((pr2 c t t2) \to (sn3 c t2)))))).(\lambda (H5: (((eq T t
+t) \to (\forall (P: Prop).P)))).(H4 t H5 (pr2_free c t t (pr0_refl t))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H4: (pr2 c t4 t3)).(\lambda (t5:
+T).(\lambda (H5: (pr3 c t3 t5)).(\lambda (H6: ((((\forall (t2: T).((((eq T t3
+t2) \to (\forall (P: Prop).P))) \to ((pr2 c t3 t2) \to (sn3 c t2))))) \to
+((((eq T t3 t5) \to (\forall (P: Prop).P))) \to (sn3 c t5))))).(\lambda (H7:
+((\forall (t2: T).((((eq T t4 t2) \to (\forall (P: Prop).P))) \to ((pr2 c t4
+t2) \to (sn3 c t2)))))).(\lambda (H8: (((eq T t4 t5) \to (\forall (P:
+Prop).P)))).(let H_x \def (term_dec t4 t3) in (let H9 \def H_x in (or_ind (eq
+T t4 t3) ((eq T t4 t3) \to (\forall (P: Prop).P)) (sn3 c t5) (\lambda (H10:
+(eq T t4 t3)).(let H11 \def (eq_ind T t4 (\lambda (t: T).((eq T t t5) \to
+(\forall (P: Prop).P))) H8 t3 H10) in (let H12 \def (eq_ind T t4 (\lambda (t:
+T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr2 c t
+t2) \to (sn3 c t2))))) H7 t3 H10) in (let H13 \def (eq_ind T t4 (\lambda (t:
+T).(pr2 c t t3)) H4 t3 H10) in (H6 H12 H11))))) (\lambda (H10: (((eq T t4 t3)
+\to (\forall (P: Prop).P)))).(sn3_pr3_trans c t3 (H7 t3 H10 H4) t5 H5))
+H9))))))))))) t1 t2 H1 H3)) H2)))))))).
+
+theorem sn3_cast:
+ \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to
+(sn3 c (THead (Flat Cast) u t))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c u)).(sn3_ind c (\lambda
+(t: T).(\forall (t0: T).((sn3 c t0) \to (sn3 c (THead (Flat Cast) t t0)))))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) t2
+t))))))))).(\lambda (t: T).(\lambda (H2: (sn3 c t)).(sn3_ind c (\lambda (t0:
+T).(sn3 c (THead (Flat Cast) t1 t0))) (\lambda (t0: T).(\lambda (H3:
+((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0
+t2) \to (sn3 c t2)))))).(\lambda (H4: ((\forall (t2: T).((((eq T t0 t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c (THead (Flat Cast) t1
+t2))))))).(sn3_pr2_intro c (THead (Flat Cast) t1 t0) (\lambda (t2:
+T).(\lambda (H5: (((eq T (THead (Flat Cast) t1 t0) t2) \to (\forall (P:
+Prop).P)))).(\lambda (H6: (pr2 c (THead (Flat Cast) t1 t0) t2)).(let H7 \def
+(pr2_gen_cast c t1 t0 t2 H6) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3)))) (pr2 c
+t0 t2) (sn3 c t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t0
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c t0 t3))) (sn3 c t2) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Cast) x0
+x1))).(\lambda (H10: (pr2 c t1 x0)).(\lambda (H11: (pr2 c t0 x1)).(let H12
+\def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Cast) t1 t0) t) \to
+(\forall (P: Prop).P))) H5 (THead (Flat Cast) x0 x1) H9) in (eq_ind_r T
+(THead (Flat Cast) x0 x1) (\lambda (t3: T).(sn3 c t3)) (let H_x \def
+(term_dec x0 t1) in (let H13 \def H_x in (or_ind (eq T x0 t1) ((eq T x0 t1)
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) x0 x1)) (\lambda (H14:
+(eq T x0 t1)).(let H15 \def (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat
+Cast) t1 t0) (THead (Flat Cast) t x1)) \to (\forall (P: Prop).P))) H12 t1
+H14) in (let H16 \def (eq_ind T x0 (\lambda (t: T).(pr2 c t1 t)) H10 t1 H14)
+in (eq_ind_r T t1 (\lambda (t3: T).(sn3 c (THead (Flat Cast) t3 x1))) (let
+H_x0 \def (term_dec t0 x1) in (let H17 \def H_x0 in (or_ind (eq T t0 x1) ((eq
+T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Cast) t1 x1))
+(\lambda (H18: (eq T t0 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t:
+T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) t1 t)) \to (\forall
+(P: Prop).P))) H15 t0 H18) in (let H20 \def (eq_ind_r T x1 (\lambda (t:
+T).(pr2 c t0 t)) H11 t0 H18) in (eq_ind T t0 (\lambda (t3: T).(sn3 c (THead
+(Flat Cast) t1 t3))) (H19 (refl_equal T (THead (Flat Cast) t1 t0)) (sn3 c
+(THead (Flat Cast) t1 t0))) x1 H18)))) (\lambda (H18: (((eq T t0 x1) \to
+(\forall (P: Prop).P)))).(H4 x1 H18 (pr3_pr2 c t0 x1 H11))) H17))) x0 H14))))
+(\lambda (H14: (((eq T x0 t1) \to (\forall (P: Prop).P)))).(H1 x0 (\lambda
+(H15: (eq T t1 x0)).(\lambda (P: Prop).(let H16 \def (eq_ind_r T x0 (\lambda
+(t: T).((eq T t t1) \to (\forall (P: Prop).P))) H14 t1 H15) in (let H17 \def
+(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat
+Cast) t x1)) \to (\forall (P: Prop).P))) H12 t1 H15) in (let H18 \def
+(eq_ind_r T x0 (\lambda (t: T).(pr2 c t1 t)) H10 t1 H15) in (H16 (refl_equal
+T t1) P)))))) (pr3_pr2 c t1 x0 H10) x1 (let H_x0 \def (term_dec t0 x1) in
+(let H15 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall (P:
+Prop).P)) (sn3 c x1) (\lambda (H16: (eq T t0 x1)).(let H17 \def (eq_ind_r T
+x1 (\lambda (t: T).((eq T (THead (Flat Cast) t1 t0) (THead (Flat Cast) x0 t))
+\to (\forall (P: Prop).P))) H12 t0 H16) in (let H18 \def (eq_ind_r T x1
+(\lambda (t: T).(pr2 c t0 t)) H11 t0 H16) in (eq_ind T t0 (\lambda (t3:
+T).(sn3 c t3)) (sn3_sing c t0 H3) x1 H16)))) (\lambda (H16: (((eq T t0 x1)
+\to (\forall (P: Prop).P)))).(H3 x1 H16 (pr3_pr2 c t0 x1 H11))) H15)))))
+H13))) t2 H9))))))) H8)) (\lambda (H8: (pr2 c t0 t2)).(sn3_pr3_trans c t0
+(sn3_sing c t0 H3) t2 (pr3_pr2 c t0 t2 H8))) H7))))))))) t H2)))))) u H))).
+
+theorem nf2_sn3:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c
+t u)) (\lambda (u: T).(nf2 c u)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda
+(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u))))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let
+H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2
+c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u))
+(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
+T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall
+(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c
+t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1
+x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1
+x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u:
+T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1
+u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x
+x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u))
+(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3))
+H2)))))) t H))).
+
+theorem sn3_appl_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v:
+T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(v: T).(\lambda (H0: (sn3 c v)).(sn3_ind c (\lambda (t: T).(sn3 c (THead
+(Flat Appl) t (TLRef i)))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c (THead (Flat Appl) t2 (TLRef
+i)))))))).(sn3_pr2_intro c (THead (Flat Appl) t1 (TLRef i)) (\lambda (t2:
+T).(\lambda (H3: (((eq T (THead (Flat Appl) t1 (TLRef i)) t2) \to (\forall
+(P: Prop).P)))).(\lambda (H4: (pr2 c (THead (Flat Appl) t1 (TLRef i))
+t2)).(let H5 \def (pr2_gen_appl c t1 (TLRef i) t2 H4) in (or3_ind (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
+(sn3 c t2) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t1
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c (TLRef i) t2))))).(ex3_2_ind T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr2 c t1
+x0)).(\lambda (H9: (pr2 c (TLRef i) x1)).(let H10 \def (eq_ind T t2 (\lambda
+(t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to (\forall (P: Prop).P)))
+H3 (THead (Flat Appl) x0 x1) H7) in (eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t: T).(sn3 c t)) (let H11 \def (eq_ind_r T x1 (\lambda (t: T).((eq
+T (THead (Flat Appl) t1 (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
+Prop).P))) H10 (TLRef i) (H x1 H9)) in (let H12 \def (eq_ind_r T x1 (\lambda
+(t: T).(pr2 c (TLRef i) t)) H9 (TLRef i) (H x1 H9)) in (eq_ind T (TLRef i)
+(\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H_x \def (term_dec t1
+x0) in (let H13 \def H_x in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x0 (TLRef i))) (\lambda (H14: (eq T
+t1 x0)).(let H15 \def (eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat
+Appl) t1 (TLRef i)) (THead (Flat Appl) t (TLRef i))) \to (\forall (P:
+Prop).P))) H11 t1 H14) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c
+t1 t)) H8 t1 H14) in (eq_ind T t1 (\lambda (t: T).(sn3 c (THead (Flat Appl) t
+(TLRef i)))) (H15 (refl_equal T (THead (Flat Appl) t1 (TLRef i))) (sn3 c
+(THead (Flat Appl) t1 (TLRef i)))) x0 H14)))) (\lambda (H14: (((eq T t1 x0)
+\to (\forall (P: Prop).P)))).(H2 x0 H14 (pr3_pr2 c t1 x0 H8))) H13))) x1 (H
+x1 H9)))) t2 H7))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T
+T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))
+(sn3 c t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H7: (eq T (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H8:
+(eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c t1 x2)).(\lambda (_:
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let
+H11 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i))
+t) \to (\forall (P: Prop).P))) H3 (THead (Bind Abbr) x2 x3) H8) in (eq_ind_r
+T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t)) (let H12 \def (eq_ind
+T (TLRef i) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (THead (Bind Abst) x0 x1) H7) in (False_ind (sn3 c
+(THead (Bind Abbr) x2 x3)) H12)) t2 H8)))))))))) H6)) (\lambda (H6: (ex6_6 B
+T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
+(_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
+T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c t1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
+Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2) (\lambda (x0: B).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H8: (eq T (TLRef i) (THead
+(Bind x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c t1 x4)).(\lambda (_: (pr2
+c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2 x3)).(let H13 \def
+(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Appl) t1 (TLRef i)) t) \to
+(\forall (P: Prop).P))) H3 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O)
+O x4) x3)) H9) in (eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S
+O) O x4) x3)) (\lambda (t: T).(sn3 c t)) (let H14 \def (eq_ind T (TLRef i)
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind x0) x1 x2) H8) in (False_ind (sn3 c (THead (Bind x0)
+x5 (THead (Flat Appl) (lift (S O) O x4) x3))) H14)) t2 H9)))))))))))))) H6))
+H5))))))))) v H0))))).
+
+theorem sn3_appl_cast:
+ \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v
+u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
+(Flat Appl) v (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (u: T).(\lambda (H: (sn3 c (THead
+(Flat Appl) v u))).(insert_eq T (THead (Flat Appl) v u) (\lambda (t: T).(sn3
+c t)) (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead
+(Flat Appl) v (THead (Flat Cast) u t))))) (\lambda (y: T).(\lambda (H0: (sn3
+c y)).(unintro T u (\lambda (t: T).((eq T y (THead (Flat Appl) v t)) \to
+(\forall (t0: T).((sn3 c (THead (Flat Appl) v t0)) \to (sn3 c (THead (Flat
+Appl) v (THead (Flat Cast) t t0))))))) (unintro T v (\lambda (t: T).(\forall
+(x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (t0: T).((sn3 c (THead
+(Flat Appl) t t0)) \to (sn3 c (THead (Flat Appl) t (THead (Flat Cast) x
+t0)))))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T
+t (THead (Flat Appl) x x0)) \to (\forall (t0: T).((sn3 c (THead (Flat Appl) x
+t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 t0)))))))))
+(\lambda (t1: T).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H2: ((\forall
+(t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to
+(\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to
+(\forall (t: T).((sn3 c (THead (Flat Appl) x t)) \to (sn3 c (THead (Flat
+Appl) x (THead (Flat Cast) x0 t))))))))))))).(\lambda (x: T).(\lambda (x0:
+T).(\lambda (H3: (eq T t1 (THead (Flat Appl) x x0))).(\lambda (t: T).(\lambda
+(H4: (sn3 c (THead (Flat Appl) x t))).(insert_eq T (THead (Flat Appl) x t)
+(\lambda (t0: T).(sn3 c t0)) (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
+x0 t))) (\lambda (y0: T).(\lambda (H5: (sn3 c y0)).(unintro T t (\lambda (t0:
+T).((eq T y0 (THead (Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead
+(Flat Cast) x0 t0))))) (sn3_ind c (\lambda (t0: T).(\forall (x1: T).((eq T t0
+(THead (Flat Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast)
+x0 x1)))))) (\lambda (t0: T).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda
+(H7: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3
+c t0 t2) \to (\forall (x1: T).((eq T t2 (THead (Flat Appl) x x1)) \to (sn3 c
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)))))))))).(\lambda (x1:
+T).(\lambda (H8: (eq T t0 (THead (Flat Appl) x x1))).(let H9 \def (eq_ind T
+t0 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t t2) \to (\forall (x1: T).((eq T t2 (THead (Flat
+Appl) x x1)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
+x1))))))))) H7 (THead (Flat Appl) x x1) H8) in (let H10 \def (eq_ind T t0
+(\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P)))
+\to ((pr3 c t t2) \to (sn3 c t2))))) H6 (THead (Flat Appl) x x1) H8) in (let
+H11 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall (x: T).(\forall (x0:
+T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (t0: T).((sn3 c (THead
+(Flat Appl) x t0)) \to (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0
+t0)))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H12 \def (eq_ind T t1
+(\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P)))
+\to ((pr3 c t t2) \to (sn3 c t2))))) H1 (THead (Flat Appl) x x0) H3) in
+(sn3_pr2_intro c (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (\lambda
+(t2: T).(\lambda (H13: (((eq T (THead (Flat Appl) x (THead (Flat Cast) x0
+x1)) t2) \to (\forall (P: Prop).P)))).(\lambda (H14: (pr2 c (THead (Flat
+Appl) x (THead (Flat Cast) x0 x1)) t2)).(let H15 \def (pr2_gen_appl c x
+(THead (Flat Cast) x0 x1) t2 H14) in (or3_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Flat Cast) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1)
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T
+T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (sn3 c t2) (\lambda (H16: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c
+(THead (Flat Cast) x0 x1) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Flat Cast)
+x0 x1) t3))) (sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq
+T t2 (THead (Flat Appl) x2 x3))).(\lambda (H18: (pr2 c x x2)).(\lambda (H19:
+(pr2 c (THead (Flat Cast) x0 x1) x3)).(let H20 \def (eq_ind T t2 (\lambda (t:
+T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t) \to (\forall (P:
+Prop).P))) H13 (THead (Flat Appl) x2 x3) H17) in (eq_ind_r T (THead (Flat
+Appl) x2 x3) (\lambda (t3: T).(sn3 c t3)) (let H21 \def (pr2_gen_cast c x0 x1
+x3 H19) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3)))) (pr2 c x1 x3) (sn3 c (THead
+(Flat Appl) x2 x3)) (\lambda (H22: (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x3 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x1
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x3 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c x1 t3))) (sn3 c (THead (Flat Appl) x2
+x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H23: (eq T x3 (THead (Flat
+Cast) x4 x5))).(\lambda (H24: (pr2 c x0 x4)).(\lambda (H25: (pr2 c x1
+x5)).(let H26 \def (eq_ind T x3 (\lambda (t: T).((eq T (THead (Flat Appl) x
+(THead (Flat Cast) x0 x1)) (THead (Flat Appl) x2 t)) \to (\forall (P:
+Prop).P))) H20 (THead (Flat Cast) x4 x5) H23) in (eq_ind_r T (THead (Flat
+Cast) x4 x5) (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 t3))) (let H_x
+\def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) in (let
+H27 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
+x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4)) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
+(\lambda (H28: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x2
+x4))).(let H29 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4) H28) in
+((let H30 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x2 x4) H28)
+in (\lambda (H31: (eq T x x2)).(let H32 \def (eq_ind_r T x4 (\lambda (t:
+T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
+x2 (THead (Flat Cast) t x5))) \to (\forall (P: Prop).P))) H26 x0 H30) in (let
+H33 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H24 x0 H30) in (eq_ind
+T x0 (\lambda (t3: T).(sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) t3
+x5)))) (let H34 \def (eq_ind_r T x2 (\lambda (t: T).((eq T (THead (Flat Appl)
+x (THead (Flat Cast) x0 x1)) (THead (Flat Appl) t (THead (Flat Cast) x0 x5)))
+\to (\forall (P: Prop).P))) H32 x H31) in (let H35 \def (eq_ind_r T x2
+(\lambda (t: T).(pr2 c x t)) H18 x H31) in (eq_ind T x (\lambda (t3: T).(sn3
+c (THead (Flat Appl) t3 (THead (Flat Cast) x0 x5)))) (let H_x0 \def (term_dec
+(THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) in (let H36 \def H_x0 in
+(or_ind (eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) ((eq T
+(THead (Flat Appl) x x1) (THead (Flat Appl) x x5)) \to (\forall (P: Prop).P))
+(sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x5))) (\lambda (H37: (eq T
+(THead (Flat Appl) x x1) (THead (Flat Appl) x x5))).(let H38 \def (f_equal T
+T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Appl) x x1) (THead (Flat Appl) x x5) H37) in (let H39 \def
+(eq_ind_r T x5 (\lambda (t: T).((eq T (THead (Flat Appl) x (THead (Flat Cast)
+x0 x1)) (THead (Flat Appl) x (THead (Flat Cast) x0 t))) \to (\forall (P:
+Prop).P))) H34 x1 H38) in (let H40 \def (eq_ind_r T x5 (\lambda (t: T).(pr2 c
+x1 t)) H25 x1 H38) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead (Flat Appl)
+x (THead (Flat Cast) x0 t3)))) (H39 (refl_equal T (THead (Flat Appl) x (THead
+(Flat Cast) x0 x1))) (sn3 c (THead (Flat Appl) x (THead (Flat Cast) x0 x1))))
+x5 H38))))) (\lambda (H37: (((eq T (THead (Flat Appl) x x1) (THead (Flat
+Appl) x x5)) \to (\forall (P: Prop).P)))).(H9 (THead (Flat Appl) x x5) H37
+(pr3_pr2 c (THead (Flat Appl) x x1) (THead (Flat Appl) x x5) (pr2_thin_dx c
+x1 x5 H25 x Appl)) x5 (refl_equal T (THead (Flat Appl) x x5)))) H36))) x2
+H31))) x4 H30))))) H29))) (\lambda (H28: (((eq T (THead (Flat Appl) x x0)
+(THead (Flat Appl) x2 x4)) \to (\forall (P: Prop).P)))).(let H_x0 \def
+(term_dec (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) in (let H29
+\def H_x0 in (or_ind (eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x2
+x5)) ((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x2 (THead (Flat Cast) x4 x5)))
+(\lambda (H30: (eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x2
+x5))).(let H31 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30) in
+((let H32 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5) H30)
+in (\lambda (H33: (eq T x x2)).(let H34 \def (eq_ind_r T x5 (\lambda (t:
+T).(pr2 c x1 t)) H25 x1 H32) in (eq_ind T x1 (\lambda (t3: T).(sn3 c (THead
+(Flat Appl) x2 (THead (Flat Cast) x4 t3)))) (let H35 \def (eq_ind_r T x2
+(\lambda (t: T).((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) t x4)) \to
+(\forall (P: Prop).P))) H28 x H33) in (let H36 \def (eq_ind_r T x2 (\lambda
+(t: T).(pr2 c x t)) H18 x H33) in (eq_ind T x (\lambda (t3: T).(sn3 c (THead
+(Flat Appl) t3 (THead (Flat Cast) x4 x1)))) (H11 (THead (Flat Appl) x x4) H35
+(pr3_pr2 c (THead (Flat Appl) x x0) (THead (Flat Appl) x x4) (pr2_thin_dx c
+x0 x4 H24 x Appl)) x x4 (refl_equal T (THead (Flat Appl) x x4)) x1 (sn3_sing
+c (THead (Flat Appl) x x1) H10)) x2 H33))) x5 H32)))) H31))) (\lambda (H30:
+(((eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x5)) \to (\forall (P:
+Prop).P)))).(H11 (THead (Flat Appl) x2 x4) H28 (pr3_head_12 c x x2 (pr3_pr2 c
+x x2 H18) (Flat Appl) x0 x4 (pr3_pr2 (CHead c (Flat Appl) x2) x0 x4
+(pr2_cflat c x0 x4 H24 Appl x2))) x2 x4 (refl_equal T (THead (Flat Appl) x2
+x4)) x5 (H10 (THead (Flat Appl) x2 x5) H30 (pr3_head_12 c x x2 (pr3_pr2 c x
+x2 H18) (Flat Appl) x1 x5 (pr3_pr2 (CHead c (Flat Appl) x2) x1 x5 (pr2_cflat
+c x1 x5 H25 Appl x2)))))) H29)))) H27))) x3 H23))))))) H22)) (\lambda (H22:
+(pr2 c x1 x3)).(let H_x \def (term_dec (THead (Flat Appl) x x1) (THead (Flat
+Appl) x2 x3)) in (let H23 \def H_x in (or_ind (eq T (THead (Flat Appl) x x1)
+(THead (Flat Appl) x2 x3)) ((eq T (THead (Flat Appl) x x1) (THead (Flat Appl)
+x2 x3)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x2 x3)) (\lambda
+(H24: (eq T (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3))).(let H25
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24) in
+((let H26 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) x x1) (THead (Flat Appl) x2 x3) H24)
+in (\lambda (H27: (eq T x x2)).(let H28 \def (eq_ind_r T x3 (\lambda (t:
+T).(pr2 c x1 t)) H22 x1 H26) in (let H29 \def (eq_ind_r T x3 (\lambda (t:
+T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat Appl)
+x2 t)) \to (\forall (P: Prop).P))) H20 x1 H26) in (eq_ind T x1 (\lambda (t3:
+T).(sn3 c (THead (Flat Appl) x2 t3))) (let H30 \def (eq_ind_r T x2 (\lambda
+(t: T).((eq T (THead (Flat Appl) x (THead (Flat Cast) x0 x1)) (THead (Flat
+Appl) t x1)) \to (\forall (P: Prop).P))) H29 x H27) in (let H31 \def
+(eq_ind_r T x2 (\lambda (t: T).(pr2 c x t)) H18 x H27) in (eq_ind T x
+(\lambda (t3: T).(sn3 c (THead (Flat Appl) t3 x1))) (sn3_sing c (THead (Flat
+Appl) x x1) H10) x2 H27))) x3 H26))))) H25))) (\lambda (H24: (((eq T (THead
+(Flat Appl) x x1) (THead (Flat Appl) x2 x3)) \to (\forall (P:
+Prop).P)))).(H10 (THead (Flat Appl) x2 x3) H24 (pr3_head_12 c x x2 (pr3_pr2 c
+x x2 H18) (Flat Appl) x1 x3 (pr3_pr2 (CHead c (Flat Appl) x2) x1 x3
+(pr2_cflat c x1 x3 H22 Appl x2))))) H23)))) H21)) t2 H17))))))) H16))
+(\lambda (H16: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast)
+x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))
+(sn3 c t2) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H17: (eq T (THead (Flat Cast) x0 x1) (THead (Bind Abst) x2
+x3))).(\lambda (H18: (eq T t2 (THead (Bind Abbr) x4 x5))).(\lambda (_: (pr2 c
+x x4)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
+u) x3 x5))))).(let H21 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
+Appl) x (THead (Flat Cast) x0 x1)) t) \to (\forall (P: Prop).P))) H13 (THead
+(Bind Abbr) x4 x5) H18) in (eq_ind_r T (THead (Bind Abbr) x4 x5) (\lambda
+(t3: T).(sn3 c t3)) (let H22 \def (eq_ind T (THead (Flat Cast) x0 x1)
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abst) x2 x3) H17) in (False_ind
+(sn3 c (THead (Bind Abbr) x4 x5)) H22)) t2 H18)))))))))) H16)) (\lambda (H16:
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Flat Cast) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
+O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Cast) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t2)
+(\lambda (x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
+(x6: T).(\lambda (x7: T).(\lambda (_: (not (eq B x2 Abst))).(\lambda (H18:
+(eq T (THead (Flat Cast) x0 x1) (THead (Bind x2) x3 x4))).(\lambda (H19: (eq
+T t2 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda
+(_: (pr2 c x x6)).(\lambda (_: (pr2 c x3 x7)).(\lambda (_: (pr2 (CHead c
+(Bind x2) x7) x4 x5)).(let H23 \def (eq_ind T t2 (\lambda (t: T).((eq T
+(THead (Flat Appl) x (THead (Flat Cast) x0 x1)) t) \to (\forall (P:
+Prop).P))) H13 (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5))
+H19) in (eq_ind_r T (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
+x5)) (\lambda (t3: T).(sn3 c t3)) (let H24 \def (eq_ind T (THead (Flat Cast)
+x0 x1) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind x2) x3 x4) H18) in
+(False_ind (sn3 c (THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6)
+x5))) H24)) t2 H19)))))))))))))) H16)) H15))))))))))))))) y0 H5))))
+H4))))))))) y H0))))) H)))).
+
+theorem sn3_appl_appl:
+ \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
+(\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
+(sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2
+u1)))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in
+(\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T
+(THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\forall (v2: T).((sn3 c
+v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t1) u2) \to ((((iso
+(THead (Flat Appl) v1 t1) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
+(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl)
+v1 t1)))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t1 (\lambda
+(t: T).((eq T y (THead (Flat Appl) v1 t)) \to (\forall (v2: T).((sn3 c v2)
+\to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t) u2) \to ((((iso
+(THead (Flat Appl) v1 t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
+(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl)
+v1 t)))))))) (unintro T v1 (\lambda (t: T).(\forall (x: T).((eq T y (THead
+(Flat Appl) t x)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2:
+T).((pr3 c (THead (Flat Appl) t x) u2) \to ((((iso (THead (Flat Appl) t x)
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
+(sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) t x))))))))) (sn3_ind c
+(\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl)
+x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead
+(Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall
+(P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead
+(Flat Appl) v2 (THead (Flat Appl) x x0)))))))))) (\lambda (t2: T).(\lambda
+(H1: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3
+c t2 t3) \to (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3)
+\to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall
+(x0: T).((eq T t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2)
+\to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso
+(THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead
+(Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x
+x0)))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2
+(THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c
+v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c (THead (Flat Appl)
+x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) \to (sn3 c (THead (Flat
+Appl) t (THead (Flat Appl) x x0))))) (\lambda (t0: T).(\lambda (H5: ((\forall
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
+(sn3 c t2)))))).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat
+Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat
+Appl) t2 (THead (Flat Appl) x x0))))))))).(\lambda (H7: ((\forall (u2:
+T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0)
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0
+u2))))))).(let H8 \def (eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T
+t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall (x:
+T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2:
+T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2)
+\to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3
+c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead
+(Flat Appl) x x0))))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H9 \def
+(eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H1 (THead (Flat Appl) x x0)
+H3) in (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
+(\lambda (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) t0 (THead (Flat
+Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead
+(Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H12 \def (pr2_gen_appl c
+t0 (THead (Flat Appl) x x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+(THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
+T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2)))))))) (sn3 c t3) (\lambda (H13: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t3 (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c
+(THead (Flat Appl) x x0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl)
+x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T
+t3 (THead (Flat Appl) x1 x2))).(\lambda (H15: (pr2 c t0 x1)).(\lambda (H16:
+(pr2 c (THead (Flat Appl) x x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t:
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
+Prop).P))) H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat
+Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H18 \def (pr2_gen_appl c x x0 x2
+H16) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c
+(THead (Flat Appl) x1 x2)) (\lambda (H19: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x2 (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x0
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
+(Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1
+x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Flat
+Appl) x3 x4))).(\lambda (H21: (pr2 c x x3)).(\lambda (H22: (pr2 c x0
+x4)).(let H23 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P:
+Prop).P))) H17 (THead (Flat Appl) x3 x4) H20) in (eq_ind_r T (THead (Flat
+Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def
+(term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H24
+\def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))
+((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P:
+Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda
+(H25: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H26
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in
+((let H27 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25)
+in (\lambda (H28: (eq T x x3)).(let H29 \def (eq_ind_r T x4 (\lambda (t:
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
+x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H23 x0 H27) in (let
+H30 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H22 x0 H27) in (eq_ind
+T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))))
+(let H31 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0
+(THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))
+\to (\forall (P: Prop).P))) H29 x H28) in (let H32 \def (eq_ind_r T x3
+(\lambda (t: T).(pr2 c x t)) H21 x H28) in (eq_ind T x (\lambda (t: T).(sn3 c
+(THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0
+x1) in (let H33 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall
+(P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0)))
+(\lambda (H34: (eq T t0 x1)).(let H35 \def (eq_ind_r T x1 (\lambda (t:
+T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl)
+t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H31 t0 H34) in (let
+H36 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H15 t0 H34) in (eq_ind
+T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))
+(H35 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H34)))) (\lambda (H34:
+(((eq T t0 x1) \to (\forall (P: Prop).P)))).(H6 x1 H34 (pr3_pr2 c t0 x1 H15)
+(\lambda (u2: T).(\lambda (H35: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda
+(H36: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 H35 H36) (THead
+(Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1
+u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H33))) x3 H28))) x4
+H27))))) H26))) (\lambda (H25: (((eq T (THead (Flat Appl) x x0) (THead (Flat
+Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H8 (THead (Flat Appl) x3 x4) H25
+(pr3_head_12 c x x3 (pr3_pr2 c x x3 H21) (Flat Appl) x0 x4 (pr3_pr2 (CHead c
+(Flat Appl) x3) x0 x4 (pr2_cflat c x0 x4 H22 Appl x3))) x3 x4 (refl_equal T
+(THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c t0 H5) x1
+(pr3_pr2 c t0 x1 H15)) (\lambda (u2: T).(\lambda (H26: (pr3 c (THead (Flat
+Appl) x3 x4) u2)).(\lambda (H27: (((iso (THead (Flat Appl) x3 x4) u2) \to
+(\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2
+(pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) (pr2_thin_dx c
+x0 x4 H22 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) (THead (Flat Appl)
+x x4) (pr2_head_1 c x x3 H21 (Flat Appl) x4) u2 H26)) (\lambda (H28: (iso
+(THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H27 (iso_trans (THead (Flat
+Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head (Flat Appl) x3 x x4 x0) u2
+H28) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2)
+(THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2))))))))
+H24))) x2 H20))))))) H19)) (\lambda (H19: (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(eq T x2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2
+(CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
+(x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H20: (eq
+T x0 (THead (Bind Abst) x3 x4))).(\lambda (H21: (eq T x2 (THead (Bind Abbr)
+x5 x6))).(\lambda (H22: (pr2 c x x5)).(\lambda (H23: ((\forall (b:
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 x6))))).(let H24 \def (eq_ind
+T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))
+(THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind Abbr)
+x5 x6) H21) in (eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(sn3 c
+(THead (Flat Appl) x1 t))) (let H25 \def (eq_ind T x0 (\lambda (t: T).((eq T
+(THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead
+(Bind Abbr) x5 x6))) \to (\forall (P: Prop).P))) H24 (THead (Bind Abst) x3
+x4) H20) in (let H26 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2:
+T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3
+c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind Abst) x3 x4)
+H20) in (let H27 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T
+(THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead
+(Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead
+(Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2:
+T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0)
+u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
+(sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 (THead
+(Bind Abst) x3 x4) H20) in (let H28 \def (eq_ind T x0 (\lambda (t:
+T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
+(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) t0 u2)))))) H7 (THead (Bind Abst) x3 x4) H20) in (let H29 \def (eq_ind
+T x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat
+Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat
+Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind Abst) x3 x4) H20) in
+(sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (H28 (THead
+(Bind Abbr) x5 x6) (pr3_sing c (THead (Bind Abbr) x x4) (THead (Flat Appl) x
+(THead (Bind Abst) x3 x4)) (pr2_free c (THead (Flat Appl) x (THead (Bind
+Abst) x3 x4)) (THead (Bind Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4
+(pr0_refl x4))) (THead (Bind Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5
+H22) (Bind Abbr) x4 x6 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H23 Abbr
+x5)))) (\lambda (H30: (iso (THead (Flat Appl) x (THead (Bind Abst) x3 x4))
+(THead (Bind Abbr) x5 x6))).(\lambda (P: Prop).(let H31 \def (match H30
+return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t
+(THead (Flat Appl) x (THead (Bind Abst) x3 x4))) \to ((eq T t0 (THead (Bind
+Abbr) x5 x6)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq
+T (TSort n1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1:
+(eq T (TSort n2) (THead (Bind Abbr) x5 x6))).((let H2 \def (eq_ind T (TSort
+n1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (False_ind
+((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H2)) H1))) | (iso_lref i1
+i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4)))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind Abbr) x5
+x6))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
+(THead (Bind Abst) x3 x4)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind
+Abbr) x5 x6)) \to P) H2)) H1))) | (iso_head k v4 v5 t1 t2) \Rightarrow
+(\lambda (H0: (eq T (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst)
+x3 x4)))).(\lambda (H1: (eq T (THead k v5 t2) (THead (Bind Abbr) x5
+x6))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _
+t) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst)
+x3 x4)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4
+| (THead _ t _) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match
+e return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat
+Appl) x (THead (Bind Abst) x3 x4)) H0) in (eq_ind K (Flat Appl) (\lambda (k0:
+K).((eq T v4 x) \to ((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0
+v5 t2) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H5: (eq T v4
+x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq
+T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H6:
+(eq T t1 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4)
+(\lambda (_: T).((eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6))
+\to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5
+x6))).(let H8 \def (eq_ind T (THead (Flat Appl) v5 t2) (\lambda (e: T).(match
+e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind Abbr) x5 x6) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead
+(Bind Abst) x3 x4) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl)
+H4))) H3)) H2)) H1)))]) in (H31 (refl_equal T (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 x6))))))) (THead
+(Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead (Flat Appl) t0
+(THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))
+(pr2_head_1 c t0 x1 H15 (Flat Appl) (THead (Bind Abbr) x5 x6))))))))) x2
+H21)))))))))) H19)) (\lambda (H19: (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c (THead
+(Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H20: (not (eq
+B x3 Abst))).(\lambda (H21: (eq T x0 (THead (Bind x3) x4 x5))).(\lambda (H22:
+(eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6)))).(\lambda (H23: (pr2 c x x7)).(\lambda (H24: (pr2 c x4 x8)).(\lambda
+(H25: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H26 \def (eq_ind T x2 (\lambda
+(t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat
+Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind x3) x8 (THead (Flat
+Appl) (lift (S O) O x7) x6)) H22) in (eq_ind_r T (THead (Bind x3) x8 (THead
+(Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c (THead (Flat Appl)
+x1 t))) (let H27 \def (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl)
+t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead
+(Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: Prop).P))) H26 (THead
+(Bind x3) x4 x5) H21) in (let H28 \def (eq_ind T x0 (\lambda (t: T).(\forall
+(t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to
+((pr3 c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind x3) x4
+x5) H21) in (let H29 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2:
+T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3
+c (THead (Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2
+(THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall
+(u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x
+x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2))))))
+\to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8
+(THead (Bind x3) x4 x5) H21) in (let H30 \def (eq_ind T x0 (\lambda (t:
+T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead
+(Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) t0 u2)))))) H7 (THead (Bind x3) x4 x5) H21) in (let H31 \def (eq_ind T
+x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat
+Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P:
+Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat
+Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind x3) x4 x5) H21) in
+(sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl)
+(lift (S O) O x7) x6))) (H30 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S
+O) O x7) x6)) (pr3_sing c (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O)
+O x) x5)) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead
+(Flat Appl) x (THead (Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl)
+(lift (S O) O x) x5)) (pr0_upsilon x3 H20 x x (pr0_refl x) x4 x4 (pr0_refl
+x4) x5 x5 (pr0_refl x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O)
+O x7) x6)) (pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H24) (Bind x3) (THead (Flat
+Appl) (lift (S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6)
+(pr3_head_12 (CHead c (Bind x3) x8) (lift (S O) O x) (lift (S O) O x7)
+(pr3_lift (CHead c (Bind x3) x8) c (S O) O (drop_drop (Bind x3) O c c
+(drop_refl c) x8) x x7 (pr3_pr2 c x x7 H23)) (Flat Appl) x5 x6 (pr3_pr2
+(CHead (CHead c (Bind x3) x8) (Flat Appl) (lift (S O) O x7)) x5 x6 (pr2_cflat
+(CHead c (Bind x3) x8) x5 x6 H25 Appl (lift (S O) O x7)))))) (\lambda (H32:
+(iso (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (THead (Bind x3) x8 (THead
+(Flat Appl) (lift (S O) O x7) x6)))).(\lambda (P: Prop).(let H33 \def (match
+H32 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t
+(THead (Flat Appl) x (THead (Bind x3) x4 x5))) \to ((eq T t0 (THead (Bind x3)
+x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1
+n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat Appl) x (THead
+(Bind x3) x4 x5)))).(\lambda (H1: (eq T (TSort n2) (THead (Bind x3) x8 (THead
+(Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (eq_ind T (TSort n1)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (False_ind
+((eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6))) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T
+(TLRef i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq
+T (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6)))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x
+(THead (Bind x3) x4 x5)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind x3)
+x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H2)) H1))) | (iso_head k
+v4 v5 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v4 t1) (THead (Flat
+Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (THead k v5 t2) (THead
+(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t]))
+(THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in ((let H3
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _)
+\Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4
+x5)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
+(THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat Appl) x (THead
+(Bind x3) x4 x5)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x)
+\to ((eq T t1 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t2) (THead
+(Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda
+(H5: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind x3) x4
+x5)) \to ((eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat
+Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H6: (eq T t1 (THead (Bind
+x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead
+(Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7)
+x6))) \to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind x3)
+x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H8 \def (eq_ind T (THead
+(Flat Appl) v5 t2) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind x3) x8 (THead (Flat
+Appl) (lift (S O) O x7) x6)) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead
+(Bind x3) x4 x5) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl)
+H4))) H3)) H2)) H1)))]) in (H33 (refl_equal T (THead (Flat Appl) x (THead
+(Bind x3) x4 x5))) (refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift
+(S O) O x7) x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat
+Appl) (lift (S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind
+x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1
+(THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c
+t0 x1 H15 (Flat Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O
+x7) x6)))))))))) x2 H22)))))))))))))) H19)) H18)) t3 H14))))))) H13))
+(\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
+T).(eq T t3 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat
+Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))
+(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H14: (eq T (THead (Flat Appl) x x0) (THead (Bind Abst) x1
+x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
+t0 x3)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
+u) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat
+Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead
+(Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t:
+T).(sn3 c t)) (let H19 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) x1 x2) H14) in (False_ind (sn3 c (THead (Bind
+Abbr) x3 x4)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3)
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15:
+(eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T
+t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
+(_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c
+(Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t: T).((eq T
+(THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P:
+Prop).P))) H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
+H16) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat Appl) x
+x0) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) H15) in (False_ind
+(sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))) H21))
+t3 H16)))))))))))))) H13)) H12)))))))))))) v2 H4))))))))) y H0))))) H))))).
+
+theorem sn3_appl_appls:
+ \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads
+(Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall
+(v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2)
+\to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to
+(sn3 c (THead (Flat Appl) v2 u1))))))))))
+\def
+ \lambda (v1: T).(\lambda (t1: T).(\lambda (vs: TList).(let u1 \def (THeads
+(Flat Appl) (TCons v1 vs) t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead
+(Flat Appl) v1 (THeads (Flat Appl) vs t1)))).(\lambda (v2: T).(\lambda (H0:
+(sn3 c v2)).(\lambda (H1: ((\forall (u2: T).((pr3 c (THead (Flat Appl) v1
+(THeads (Flat Appl) vs t1)) u2) \to ((((iso (THead (Flat Appl) v1 (THeads
+(Flat Appl) vs t1)) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat
+Appl) v2 u2))))))).(sn3_appl_appl v1 (THeads (Flat Appl) vs t1) c H v2 H0
+H1))))))))).
+
+theorem sn3_appls_lref:
+ \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us:
+TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
+(us: TList).(TList_ind (\lambda (t: TList).((sns3 c t) \to (sn3 c (THeads
+(Flat Appl) t (TLRef i))))) (\lambda (_: True).(sn3_nf2 c (TLRef i) H))
+(\lambda (t: T).(\lambda (t0: TList).(TList_ind (\lambda (t1: TList).((((sns3
+c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t)
+(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
+i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
+(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
+in (and_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
+TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
+c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
+(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
+(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
+(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
+(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
+c (TCons t1 t2)))).(let H3 \def H2 in (and_ind (sn3 c t) (land (sn3 c t1)
+(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
+(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
+t2))).(and_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
+(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
+(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
+(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
+Appl) (TCons t1 t2) (TLRef i)) u2)).(\lambda (H9: (((iso (THeads (Flat Appl)
+(TCons t1 t2) (TLRef i)) u2) \to (\forall (P: Prop).P)))).(H9
+(nf2_iso_appls_lref c i H (TCons t1 t2) u2 H8) (sn3 c (THead (Flat Appl) t
+u2))))))))) H5))) H3))))))) t0))) us)))).
+
+theorem sn3_appls_cast:
+ \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat
+Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3
+c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))
+\def
+ \lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
+(u: T).((sn3 c (THeads (Flat Appl) t u)) \to (\forall (t0: T).((sn3 c (THeads
+(Flat Appl) t t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Flat Cast) u
+t0)))))))) (\lambda (u: T).(\lambda (H: (sn3 c u)).(\lambda (t: T).(\lambda
+(H0: (sn3 c t)).(sn3_cast c u H t H0))))) (\lambda (t: T).(\lambda (t0:
+TList).(TList_ind (\lambda (t1: TList).(((\forall (u: T).((sn3 c (THeads
+(Flat Appl) t1 u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) t1 t)) \to
+(sn3 c (THeads (Flat Appl) t1 (THead (Flat Cast) u t)))))))) \to (\forall (u:
+T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 u))) \to (\forall (t2:
+T).((sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 t2))) \to (sn3 c
+(THead (Flat Appl) t (THeads (Flat Appl) t1 (THead (Flat Cast) u t2))))))))))
+(\lambda (_: ((\forall (u: T).((sn3 c (THeads (Flat Appl) TNil u)) \to
+(\forall (t: T).((sn3 c (THeads (Flat Appl) TNil t)) \to (sn3 c (THeads (Flat
+Appl) TNil (THead (Flat Cast) u t))))))))).(\lambda (u: T).(\lambda (H0: (sn3
+c (THead (Flat Appl) t (THeads (Flat Appl) TNil u)))).(\lambda (t1:
+T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) TNil
+t1)))).(sn3_appl_cast c t u H0 t1 H1)))))) (\lambda (t1: T).(\lambda (t2:
+TList).(\lambda (_: ((((\forall (u: T).((sn3 c (THeads (Flat Appl) t2 u)) \to
+(\forall (t: T).((sn3 c (THeads (Flat Appl) t2 t)) \to (sn3 c (THeads (Flat
+Appl) t2 (THead (Flat Cast) u t)))))))) \to (\forall (u: T).((sn3 c (THead
+(Flat Appl) t (THeads (Flat Appl) t2 u))) \to (\forall (t0: T).((sn3 c (THead
+(Flat Appl) t (THeads (Flat Appl) t2 t0))) \to (sn3 c (THead (Flat Appl) t
+(THeads (Flat Appl) t2 (THead (Flat Cast) u t0))))))))))).(\lambda (H0:
+((\forall (u: T).((sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) \to (\forall
+(t: T).((sn3 c (THeads (Flat Appl) (TCons t1 t2) t)) \to (sn3 c (THeads (Flat
+Appl) (TCons t1 t2) (THead (Flat Cast) u t))))))))).(\lambda (u: T).(\lambda
+(H1: (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
+u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead (Flat Appl) t (THeads
+(Flat Appl) (TCons t1 t2) t3)))).(let H3 \def (sn3_gen_flat Appl c t (THeads
+(Flat Appl) (TCons t1 t2) t3) H2) in (and_ind (sn3 c t) (sn3 c (THead (Flat
+Appl) t1 (THeads (Flat Appl) t2 t3))) (sn3 c (THead (Flat Appl) t (THeads
+(Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)))) (\lambda (_: (sn3 c
+t)).(\lambda (H5: (sn3 c (THead (Flat Appl) t1 (THeads (Flat Appl) t2
+t3)))).(let H6 \def H5 in (let H7 \def (sn3_gen_flat Appl c t (THeads (Flat
+Appl) (TCons t1 t2) u) H1) in (and_ind (sn3 c t) (sn3 c (THead (Flat Appl) t1
+(THeads (Flat Appl) t2 u))) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
+(TCons t1 t2) (THead (Flat Cast) u t3)))) (\lambda (H8: (sn3 c t)).(\lambda
+(H9: (sn3 c (THead (Flat Appl) t1 (THeads (Flat Appl) t2 u)))).(let H10 \def
+H9 in (sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c (H0 u H10 t3 H6) t H8
+(\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat Appl) (TCons t1 t2)
+(THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso (THeads (Flat Appl)
+(TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl) (TCons
+t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat Appl)
+(TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11 H12) t
+Appl))))))))) H7))))) H3)))))))))) t0))) vs)).
+
+theorem sn3_lift:
+ \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t))))))))
+\def
+ \lambda (d: C).(\lambda (t: T).(\lambda (H: (sn3 d t)).(sn3_ind d (\lambda
+(t0: T).(\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d)
+\to (sn3 c (lift h i t0))))))) (\lambda (t1: T).(\lambda (_: ((\forall (t2:
+T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 d t1 t2) \to (sn3 d
+t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 d t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall
+(i: nat).((drop h i c d) \to (sn3 c (lift h i t2))))))))))).(\lambda (c:
+C).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H2: (drop h i c
+d)).(sn3_pr2_intro c (lift h i t1) (\lambda (t2: T).(\lambda (H3: (((eq T
+(lift h i t1) t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr2 c (lift h i
+t1) t2)).(let H5 \def (pr2_gen_lift c t1 t2 h i H4 d H2) in (ex2_ind T
+(\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t1 t3))
+(sn3 c t2) (\lambda (x: T).(\lambda (H6: (eq T t2 (lift h i x))).(\lambda
+(H7: (pr2 d t1 x)).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T (lift h i
+t1) t) \to (\forall (P: Prop).P))) H3 (lift h i x) H6) in (eq_ind_r T (lift h
+i x) (\lambda (t0: T).(sn3 c t0)) (H1 x (\lambda (H9: (eq T t1 x)).(\lambda
+(P: Prop).(let H10 \def (eq_ind_r T x (\lambda (t: T).((eq T (lift h i t1)
+(lift h i t)) \to (\forall (P: Prop).P))) H8 t1 H9) in (let H11 \def
+(eq_ind_r T x (\lambda (t: T).(pr2 d t1 t)) H7 t1 H9) in (H10 (refl_equal T
+(lift h i t1)) P))))) (pr3_pr2 d t1 x H7) c h i H2) t2 H6)))))
+H5))))))))))))) t H))).
+
+theorem sn3_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i)))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 d
+v)).(sn3_pr2_intro c (TLRef i) (\lambda (t2: T).(\lambda (H1: (((eq T (TLRef
+i) t2) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr2 c (TLRef i) t2)).(let
+H3 \def (pr2_gen_lref c t2 i H2) in (or_ind (eq T t2 (TLRef i)) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u: T).(getl i c (CHead d0 (Bind Abbr) u))))
+(\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S i) O u))))) (sn3 c t2)
+(\lambda (H4: (eq T t2 (TLRef i))).(let H5 \def (eq_ind T t2 (\lambda (t:
+T).((eq T (TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H4) in
+(eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c t)) (H5 (refl_equal T (TLRef i))
+(sn3 c (TLRef i))) t2 H4))) (\lambda (H4: (ex2_2 C T (\lambda (d: C).(\lambda
+(u: T).(getl i c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(eq T t2 (lift (S i) O u)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
+(u: T).(getl i c (CHead d0 (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u:
+T).(eq T t2 (lift (S i) O u)))) (sn3 c t2) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H5: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (eq T t2
+(lift (S i) O x1))).(let H7 \def (eq_ind T t2 (\lambda (t: T).((eq T (TLRef
+i) t) \to (\forall (P: Prop).P))) H1 (lift (S i) O x1) H6) in (eq_ind_r T
+(lift (S i) O x1) (\lambda (t: T).(sn3 c t)) (let H8 \def (eq_ind C (CHead d
+(Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind Abbr) x1)
+(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
+(let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) v) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) v) i H
+(CHead x0 (Bind Abbr) x1) H5)) in ((let H10 \def (f_equal C T (\lambda (e:
+C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead
+_ _ t) \Rightarrow t])) (CHead d (Bind Abbr) v) (CHead x0 (Bind Abbr) x1)
+(getl_mono c (CHead d (Bind Abbr) v) i H (CHead x0 (Bind Abbr) x1) H5)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t:
+T).(getl i c (CHead x0 (Bind Abbr) t))) H8 v H10) in (eq_ind T v (\lambda (t:
+T).(sn3 c (lift (S i) O t))) (let H13 \def (eq_ind_r C x0 (\lambda (c0:
+C).(getl i c (CHead c0 (Bind Abbr) v))) H12 d H11) in (sn3_lift d v H0 c (S
+i) O (getl_drop Abbr c d v i H13))) x1 H10)))) H9))) t2 H6)))))) H4))
+H3))))))))))).
+
+theorem sns3_lifts:
+ \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h
+i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (h: nat).(\lambda (i: nat).(\lambda
+(H: (drop h i c d)).(\lambda (ts: TList).(TList_ind (\lambda (t:
+TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
+(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
+H1 in (and_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
+(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
+(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
+H4)))) H2)))))) ts)))))).
+
+theorem sns3_lifts1:
+ \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to
+(\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
+\def
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c
+(lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda
+(ts: TList).(\lambda (H0: (sns3 e ts)).(let H1 \def (match H return (\lambda
+(p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p c0
+c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c (lifts1
+PNil ts))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil
+PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c
+(\lambda (c1: C).((eq C c1 e) \to (sns3 c (lifts1 PNil ts)))) (\lambda (H4:
+(eq C c e)).(eq_ind C e (\lambda (c: C).(sns3 c (lifts1 PNil ts))) (eq_ind_r
+TList ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts))
+c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1
+c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds)
+PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def
+(eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda
+(_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow
+True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d
+c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 PNil ts)))))) H6)) H4 H5 H1
+H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C
+e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda
+(H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to
+(sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0
+p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H2 \def (match
+H0 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_:
+(drop1 p0 c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C
+c1 e) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))))) with [(drop1_nil c0)
+\Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C
+c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda
+(e0: PList).(match e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow
+True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind
+((eq C c0 c) \to ((eq C c0 e) \to (sns3 c (lifts1 (PCons n n0 p) ts)))) H5))
+H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq
+PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda
+(H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0:
+PList).(match e0 return (\lambda (_: PList).PList) with [PNil \Rightarrow hds
+| (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let
+H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_:
+PList).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons
+h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0:
+PList).(match e0 return (\lambda (_: PList).nat) with [PNil \Rightarrow h |
+(PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind
+nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1
+c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c
+(lifts1 (PCons n n0 p) ts))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat
+n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to
+((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 (PCons n n0 p)
+ts)))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0:
+PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2
+c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))) (\lambda (H12: (eq C c1
+c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to
+((drop1 p c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))) (\lambda (H13:
+(eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2
+c0) \to (sns3 c (lifts1 (PCons n n0 p) ts))))) (\lambda (H14: (drop n n0 c
+c2)).(\lambda (H15: (drop1 p c2 e)).(eq_ind_r TList (lifts n n0 (lifts1 p
+ts)) (\lambda (t: TList).(sns3 c t)) (sns3_lifts c c2 n n0 H14 (lifts1 p ts)
+(H c2 H15 ts H1)) (lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts)))) c3
+(sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11)))
+d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))])
+in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C
+e))))))))))) hds)).
+
+theorem sn3_gen_lift:
+ \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1
+(lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (sn3 c1 (lift h d t))).(insert_eq T (lift h d t) (\lambda (t0: T).(sn3 c1
+t0)) (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))) (\lambda (y:
+T).(\lambda (H0: (sn3 c1 y)).(unintro T t (\lambda (t0: T).((eq T y (lift h d
+t0)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t0))))) (sn3_ind c1
+(\lambda (t0: T).(\forall (x: T).((eq T t0 (lift h d x)) \to (\forall (c2:
+C).((drop h d c1 c2) \to (sn3 c2 x)))))) (\lambda (t1: T).(\lambda (H1:
+((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c1 t1
+t2) \to (sn3 c1 t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c1 t1 t2) \to (\forall (x: T).((eq T t2
+(lift h d x)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2
+x)))))))))).(\lambda (x: T).(\lambda (H3: (eq T t1 (lift h d x))).(\lambda
+(c2: C).(\lambda (H4: (drop h d c1 c2)).(let H5 \def (eq_ind T t1 (\lambda
+(t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3
+c1 t t2) \to (\forall (x: T).((eq T t2 (lift h d x)) \to (\forall (c2:
+C).((drop h d c1 c2) \to (sn3 c2 x))))))))) H2 (lift h d x) H3) in (let H6
+\def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c1 t t2) \to (sn3 c1 t2))))) H1 (lift h d
+x) H3) in (sn3_sing c2 x (\lambda (t2: T).(\lambda (H7: (((eq T x t2) \to
+(\forall (P: Prop).P)))).(\lambda (H8: (pr3 c2 x t2)).(H5 (lift h d t2)
+(\lambda (H9: (eq T (lift h d x) (lift h d t2))).(\lambda (P: Prop).(let H10
+\def (eq_ind_r T t2 (\lambda (t: T).(pr3 c2 x t)) H8 x (lift_inj x t2 h d
+H9)) in (let H11 \def (eq_ind_r T t2 (\lambda (t: T).((eq T x t) \to (\forall
+(P: Prop).P))) H7 x (lift_inj x t2 h d H9)) in (H11 (refl_equal T x) P)))))
+(pr3_lift c1 c2 h d H4 x t2 H8) t2 (refl_equal T (lift h d t2)) c2
+H4)))))))))))))) y H0)))) H))))).
+
+definition sc3:
+ G \to (A \to (C \to (T \to Prop)))
+\def
+ let rec sc3 (g: G) (a: A) on a: (C \to (T \to Prop)) \def (\lambda (c:
+C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t
+(ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead
+a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is
+t)))))))))]))) in sc3.
+
+theorem sc3_arity_gen:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
+t) \to (arity g c t a)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
+(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
+t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (arity g
+c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: (sn3
+c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to (arity
+g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity g c t
+a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
+(and_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
+Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
+g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
+a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
+Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))).
+
+theorem sc3_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c
+t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c:
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3
+g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a1:
+A).((llt a1 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to
+(\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))))) \to (\forall (c:
+C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3
+g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall
+(a1: A).((llt a1 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a1
+c t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t)))))))))).(\lambda
+(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
+t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
+in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda (H3:
+(arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
+(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort g n
+n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a3 (ASort h2 n2))))) (\lambda
+(n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
+(aplus g (ASort h2 n2) k))))) (sc3 g a3 c t) (\lambda (x0: nat).(\lambda (x1:
+nat).(\lambda (x2: nat).(\lambda (H6: (eq A a3 (ASort x1 x0))).(\lambda (_:
+(eq A (aplus g (ASort n n0) x2) (aplus g (ASort x1 x0) x2))).(let H8 \def
+(eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) H6) in
+(eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity g c t
+(ASort x1 x0)) (sn3 c t) H8 H4) a3 H6))))))) H5)))))) H2)))))))))) (\lambda
+(a: A).(\lambda (_: ((((\forall (a1: A).((llt a1 a) \to (\forall (c:
+C).(\forall (t: T).((sc3 g a1 c t) \to (\forall (a2: A).((leq g a1 a2) \to
+(sc3 g a2 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to
+(\forall (a2: A).((leq g a a2) \to (sc3 g a2 c t))))))))).(\lambda (a0:
+A).(\lambda (H0: ((((\forall (a1: A).((llt a1 a0) \to (\forall (c:
+C).(\forall (t: T).((sc3 g a1 c t) \to (\forall (a2: A).((leq g a1 a2) \to
+(sc3 g a2 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t)
+\to (\forall (a2: A).((leq g a0 a2) \to (sc3 g a2 c t))))))))).(\lambda (H1:
+((\forall (a1: A).((llt a1 (AHead a a0)) \to (\forall (c: C).(\forall (t:
+T).((sc3 g a1 c t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c
+t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
+t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
+\def H2 in (and_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
+T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
+(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
+g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
+d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat
+Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head g a a0 a3 H3) in
+(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (a5: A).(eq A a3
+(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a a4))) (\lambda (_:
+A).(\lambda (a5: A).(leq g a0 a5))) (sc3 g a3 c t) (\lambda (x0: A).(\lambda
+(x1: A).(\lambda (H8: (eq A a3 (AHead x0 x1))).(\lambda (H9: (leq g a
+x0)).(\lambda (H10: (leq g a0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a4:
+A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall (d: C).(\forall
+(w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g x1
+d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t (AHead a a0) H5
+(AHead x0 x1) (leq_head g a x0 H9 a0 x1 H10)) (\lambda (d: C).(\lambda (w:
+T).(\lambda (H11: (sc3 g x0 d w)).(\lambda (is: PList).(\lambda (H12: (drop1
+is d c)).(H0 (\lambda (a4: A).(\lambda (H13: (llt a4 a0)).(\lambda (c0:
+C).(\lambda (t0: T).(\lambda (H14: (sc3 g a4 c0 t0)).(\lambda (a5:
+A).(\lambda (H15: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 (AHead a a0) H13
+(llt_head_dx a a0)) c0 t0 H14 a5 H15)))))))) d (THead (Flat Appl) w (lift1 is
+t)) (H6 d w (H1 x0 (llt_repl g a x0 H9 (AHead a a0) (llt_head_sx a a0)) d w
+H11 a (leq_sym g a x0 H9)) is H12) x1 H10))))))) a3 H8)))))) H7)))))
+H4)))))))))))) a2)) a1)).
+
+theorem sc3_lift:
+ \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e
+t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
+\to (sc3 g a c (lift h d t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e:
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t))))))))))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
+(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
+(and_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
+(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
+n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
+(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
+t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e:
+C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h:
+nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d
+t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t:
+T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e:
+C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall
+(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
+e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
+\def H1 in (and_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall (w:
+T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g a1
+d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
+(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
+(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
+(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
+(H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
+t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0:
+C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
+\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t)))))))))
+(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w:
+T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1
+is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1
+(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w
+t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t))
+(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)).
+
+theorem sc3_lift1:
+ \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds:
+PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e)
+\to (sc3 g a c (lift1 hds t)))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g
+a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c:
+C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c
+e)).(let H1 \def (match H0 return (\lambda (p: PList).(\lambda (c0:
+C).(\lambda (c1: C).(\lambda (_: (drop1 p c0 c1)).((eq PList p PNil) \to ((eq
+C c0 c) \to ((eq C c1 e) \to (sc3 g a c t)))))))) with [(drop1_nil c0)
+\Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0
+c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to
+(sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(sc3 g
+a c t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1
+c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds)
+PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def
+(eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda
+(_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow
+True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d
+c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c t))))) H6)) H4 H5 H1 H2))))]) in
+(H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c:
+C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p
+t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (sc3 g a e
+t)).(\lambda (H1: (drop1 (PCons n n0 p) c e)).(let H2 \def (match H1 return
+(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0
+c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to
+(sc3 g a c (lift n n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow
+(\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0
+c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0:
+PList).(match e0 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True
+| (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq
+C c0 c) \to ((eq C c0 e) \to (sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3
+H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq
+PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda
+(H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0:
+PList).(match e0 return (\lambda (_: PList).PList) with [PNil \Rightarrow hds
+| (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let
+H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_:
+PList).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons
+h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0:
+PList).(match e0 return (\lambda (_: PList).nat) with [PNil \Rightarrow h |
+(PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind
+nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1
+c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g
+a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat
+n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to
+((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c (lift n n0 (lift1 p
+t))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0:
+PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2
+c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1
+c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to
+((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))) (\lambda (H13:
+(eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2
+c0) \to (sc3 g a c (lift n n0 (lift1 p t)))))) (\lambda (H14: (drop n n0 c
+c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a c2 (lift1 p t) (H c2 t H0
+H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds
+(sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9)))
+H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
+(refl_equal C c) (refl_equal C e))))))))))) hds)))).
+
+axiom sc3_abbr:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i:
+nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads
+(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to
+(sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
+.
+
+theorem sc3_cast:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
+(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t:
+T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl)
+vs (THead (Flat Cast) u t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs:
+TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat
+Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to
+(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u:
+T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) |
+(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t:
+T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0))
+(sn3 c (THeads (Flat Appl) vs t)))).((match n return (\lambda (n1: nat).((sc3
+g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow
+(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads
+(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land
+(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0))
+(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) with [O
+\Rightarrow (\lambda (H1: (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl)
+vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort O
+n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g
+c (THeads (Flat Appl) vs u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl)
+vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
+(ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))
+(\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort O (next g
+n0)))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in
+(and_ind (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads
+(Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast)
+u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))
+(\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort O n0))).(\lambda
+(H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl)
+vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs
+(THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort O n0) H4 H7)
+(sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))) | (S n1) \Rightarrow
+(\lambda (H1: (sc3 g (ASort n1 n0) c (THeads (Flat Appl) vs u))).(\lambda
+(H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c
+(THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads
+(Flat Appl) vs u) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs u)) (land
+(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0))
+(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity
+g c (THeads (Flat Appl) vs u) (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads
+(Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat
+Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity
+g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c
+(THeads (Flat Appl) vs t) (ASort (S n1) n0))).(\lambda (H8: (sn3 c (THeads
+(Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat
+Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat
+Cast) u t))) (arity_appls_cast g c u t vs (ASort (S n1) n0) H4 H7)
+(sn3_appls_cast c vs u H5 t H8)))) H6)))) H3))))]) H H0))))))))) (\lambda
+(a0: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall (u:
+T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t:
+T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat
+Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: A).(\lambda (H0:
+((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a1) c
+(THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c (THeads (Flat
+Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead (Flat Cast) u
+t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda
+(H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1)))
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
+is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land
+(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
+d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
+\def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
+a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
+is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
+(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3
+g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u
+t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0
+(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
+w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
+in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
+t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
+(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity
+g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
+t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
+(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is
+(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c
+u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9:
+(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y
+\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1
+is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d
+(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1
+is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl)
+(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat
+Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w
+t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u))
+(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat
+Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w
+H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl
+is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t))
+(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl
+is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
+
+axiom sc3_bind:
+ \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1:
+A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v:
+T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts
+(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs
+(THead (Bind b) v t)))))))))))))
+.
+
+axiom sc3_appl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs:
+TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads
+(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w:
+T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead
+(Flat Appl) v (THead (Bind Abst) w t))))))))))))))
+.
+
+theorem sc3_props__sc3_sn3_abst:
+ \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g
+a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
+((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c:
+C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs:
+TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in
+(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
+(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall
+(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3
+c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
+(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to
+((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
+n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
+C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
+t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
+t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
+H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
+(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
+(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat
+Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H
+(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land
+(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall
+(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
+vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
+(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0:
+(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t))))
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
+(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to
+(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c:
+C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t))))
+(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads
+(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c
+vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))
+(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads
+(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
+T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
+(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
+d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (and_ind
+(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
+(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
+(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
+(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_:
+((\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))))).(\lambda
+(H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
+(THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs)
+\to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef i))))))))))).(let H5 \def H0 in
+(and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a1 c0 t0) \to (sn3 c0
+t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0
+(THeads (Flat Appl) vs (TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0
+vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t)
+(\lambda (H6: ((\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c
+t)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c:
+C).((arity g c (THeads (Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i))
+\to ((sns3 c vs) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef
+i))))))))))).(let H8 \def H1 in (and_ind (arity g c t (AHead a0 a1)) (\forall
+(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
+c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))) (sn3 c t)
+(\lambda (H9: (arity g c t (AHead a0 a1))).(\lambda (H10: ((\forall (d:
+C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
+\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H_y \def
+(arity_aprem g c t (AHead a0 a1) H9 O a0) in (let H11 \def (H_y (aprem_zero
+a0 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j:
+nat).(drop j O d c)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_:
+nat).(arity g d u (asucc g a0))))) (sn3 c t) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: nat).(\lambda (H12: (drop x2 O x0 c)).(\lambda (H13: (arity
+g x0 x1 (asucc g a0))).(let H_y0 \def (H10 (CHead x0 (Bind Abst) x1) (TLRef
+O) (H4 TNil O (CHead x0 (Bind Abst) x1) (arity_abst g (CHead x0 (Bind Abst)
+x1) x0 x1 O (getl_refl Abst x0 x1) a0 H13) (nf2_lref_abst (CHead x0 (Bind
+Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) I) (PCons (S x2) O PNil)) in (let
+H_y1 \def (H6 (CHead x0 (Bind Abst) x1) (THead (Flat Appl) (TLRef O) (lift (S
+x2) O t)) (H_y0 (drop1_cons (CHead x0 (Bind Abst) x1) c (S x2) O (drop_drop
+(Bind Abst) x2 x0 c H12 x1) c PNil (drop1_nil c)))) in (let H14 \def
+(sn3_gen_flat Appl (CHead x0 (Bind Abst) x1) (TLRef O) (lift (S x2) O t)
+H_y1) in (and_ind (sn3 (CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0
+(Bind Abst) x1) (lift (S x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0
+(Bind Abst) x1) (TLRef O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1)
+(lift (S x2) O t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c
+(drop_drop (Bind Abst) x2 x0 c H12 x1)))) H14))))))))) H11))))) H8)))) H5))))
+H2))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1:
+(arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2:
+(nf2 c (TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat
+Appl) vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0
+d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
+Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
+C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
+PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (and_ind (\forall (c0:
+C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
+TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
+vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
+c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
+w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c:
+C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))))).(\lambda (_: ((\forall
+(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl)
+vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c
+(THeads (Flat Appl) vs (TLRef i))))))))))).(let H9 \def H0 in (and_ind
+(\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))) (\forall
+(vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat
+Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to
+(sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead
+(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (_:
+((\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))))).(\lambda
+(H11: ((\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c
+(THeads (Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs)
+\to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(let H_y \def (H11
+(TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs)
+(lift1 is (TLRef i))) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t)))
+(eq_ind_r T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat
+Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind
+T (lift1 is (TLRef i)) (\lambda (t: T).(arity g d (THead (Flat Appl) w
+(THeads (Flat Appl) (lifts1 is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat
+Appl) vs (TLRef i))) (\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1))
+(arity_appl g d w a0 (sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat
+Appl) vs (TLRef i))) a1 (arity_lift1 g (AHead a0 a1) c is d (THeads (Flat
+Appl) vs (TLRef i)) H5 H1)) (THeads (Flat Appl) (lifts1 is vs) (lift1 is
+(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)) (TLRef (trans is i))
+(lift1_lref is i)) (eq_ind T (lift1 is (TLRef i)) (\lambda (t: T).(nf2 d t))
+(nf2_lift1 c is d (TLRef i) H5 H2) (TLRef (trans is i)) (lift1_lref is i))
+(conj (sn3 d w) (sns3 d (lifts1 is vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs
+H3))) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl)
+vs (TLRef i))) (lifts1_flat Appl is (TLRef i) vs))))) H9))))
+H6))))))))))))))))))) a)).
+
+theorem sc3_sn3:
+ \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c
+t) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
+(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
+H_x in (and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
+c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat
+Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0
+(TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda
+(H1: ((\forall (c: C).(\forall (t: T).((sc3 g a c t) \to (sn3 c
+t)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(let t \def
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
+((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))).(H1 c t H)))
+H0))))))).
+
+theorem sc3_abst:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall
+(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef
+i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))
+\def
+ \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
+(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
+a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
+(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (and_ind (\forall (c0:
+C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
+TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in
+(\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0
+vs0) \to (sc3 g a c0 t)))))))) (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))
+(\lambda (_: ((\forall (c: C).(\forall (t: T).((sc3 g a c t) \to (sn3 c
+t)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(let t \def
+(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to
+((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))).(H4 vs i c H
+H0 H1))) H2)))))))))).
-axiom csubst0_getl_ge: \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 e) \to (getl n c2 e))))))))) .
+inductive csubc (g:G): C \to (C \to Prop) \def
+| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
+| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
+(k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
+| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall
+(v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g
+a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr)
+w))))))))).
+
+definition ceqc:
+ G \to (C \to (C \to Prop))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(or (csubc g c1 c2) (csubc
+g c2 c1)))).
+
+theorem scubc_refl:
+ \forall (g: G).(\forall (c: C).(csubc g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csubc g c0 c0))
+(\lambda (n: nat).(csubc_sort g n)) (\lambda (c0: C).(\lambda (H: (csubc g c0
+c0)).(\lambda (k: K).(\lambda (t: T).(csubc_head g c0 c0 H k t))))) c)).
+
+theorem ceqc_sym:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c1 c2) \to (ceqc g
+c2 c1))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c1
+c2)).(let H0 \def H in (or_ind (csubc g c1 c2) (csubc g c2 c1) (ceqc g c2 c1)
+(\lambda (H1: (csubc g c1 c2)).(or_intror (csubc g c2 c1) (csubc g c1 c2)
+H1)) (\lambda (H1: (csubc g c2 c1)).(or_introl (csubc g c2 c1) (csubc g c1
+c2) H1)) H0))))).
+
+theorem drop_csubc_trans:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
+C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
+(c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
+(e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
+C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g
+(CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def
+(eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1))
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c
+c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
+e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h
+n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h:
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
+(e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
+(\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2
+\def (eq_ind_r C e2 (\lambda (c: C).(csubc g c e1)) H1 (CHead c k t)
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
+O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1)
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
+(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2
+e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
+(\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
+e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k
+t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t)))))
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t)
+c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
+C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
+(c: C).(csubc g c e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
+(\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall
+(e1: C).((csubc g c0 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1))
+(\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) H3) in
+(let H8 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead
+c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k x1) e1)
+\to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g
+(CHead c k t) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r
+k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1))
+(\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match H6 return
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 c1)).((eq C c0
+(CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c2: C).(drop h (S n)
+c2 e1)) (\lambda (c2: C).(csubc g (CHead c k (lift h (r k n) x1)) c2))))))))
+with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) (CHead x0 k
+x1))).(\lambda (H3: (eq C (CSort n0) e1)).((let H4 \def (eq_ind C (CSort n0)
+(\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x0 k x1) H1)
+in (False_ind ((eq C (CSort n0) e1) \to (ex2 C (\lambda (c1: C).(drop h (S n)
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) H4))
+H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq C (CHead c1
+k0 v) (CHead x0 k x1))).(\lambda (H6: (eq C (CHead c2 k0 v) e1)).((let H2
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 v)
+(CHead x0 k x1) H3) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c1 k0 v) (CHead x0 k x1) H3) in ((let H7 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 v) (CHead x0
+k x1) H3) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to
+((eq C (CHead c2 k0 v) e1) \to ((csubc g c0 c2) \to (ex2 C (\lambda (c3:
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n)
+x1)) c3)))))))) (\lambda (H8: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq
+T v x1) \to ((eq C (CHead c2 k1 v) e1) \to ((csubc g x0 c2) \to (ex2 C
+(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k
+(lift h (r k n) x1)) c3))))))) (\lambda (H9: (eq T v x1)).(eq_ind T x1
+(\lambda (t: T).((eq C (CHead c2 k t) e1) \to ((csubc g x0 c2) \to (ex2 C
+(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k
+(lift h (r k n) x1)) c3)))))) (\lambda (H10: (eq C (CHead c2 k x1)
+e1)).(eq_ind C (CHead c2 k x1) (\lambda (c0: C).((csubc g x0 c2) \to (ex2 C
+(\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c k
+(lift h (r k n) x1)) c3))))) (\lambda (H11: (csubc g x0 c2)).(let H_x \def (H
+x0 (r k n) h H5 c2 H11) in (let H5 \def H_x in (ex2_ind C (\lambda (c3:
+C).(drop h (r k n) c3 c2)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda
+(c3: C).(drop h (S n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c
+k (lift h (r k n) x1)) c3))) (\lambda (x: C).(\lambda (H12: (drop h (r k n) x
+c2)).(\lambda (H13: (csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S
+n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n)
+x1)) c3)) (CHead x k (lift h (r k n) x1)) (drop_skip k h n x c2 H12 x1)
+(csubc_head g c x H13 k (lift h (r k n) x1)))))) H5)))) e1 H10)) v (sym_eq T
+v x1 H9))) k0 (sym_eq K k0 k H8))) c1 (sym_eq C c1 x0 H7))) H4)) H2)) H6
+H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) \Rightarrow (\lambda (H6: (eq C
+(CHead c1 (Bind Abst) v) (CHead x0 k x1))).(\lambda (H7: (eq C (CHead c2
+(Bind Abbr) w) e1)).((let H2 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t)
+\Rightarrow t])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in ((let H4
+\def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow (Bind Abst) | (CHead _ k _) \Rightarrow k])) (CHead c1
+(Bind Abst) v) (CHead x0 k x1) H6) in ((let H9 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead
+c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in
+(eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abst) k) \to ((eq T v x1) \to ((eq
+C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g c0 c2) \to ((sc3 g (asucc g a)
+c0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1))
+(\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))))))))
+(\lambda (H10: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) (\lambda (k:
+K).((eq T v x1) \to ((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2)
+\to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3:
+C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n)
+x1)) c3))))))))) (\lambda (H11: (eq T v x1)).(eq_ind T x1 (\lambda (t:
+T).((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) \to ((sc3 g
+(asucc g a) x0 t) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S
+n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind
+Abst) n) x1)) c3)))))))) (\lambda (H12: (eq C (CHead c2 (Bind Abbr) w)
+e1)).(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (c0: C).((csubc g x0 c2) \to
+((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3:
+C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift
+h (r (Bind Abst) n) x1)) c3))))))) (\lambda (H13: (csubc g x0 c2)).(\lambda
+(H14: (sc3 g (asucc g a) x0 x1)).(\lambda (H15: (sc3 g a c2 w)).(let H8 \def
+(eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift
+h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k
+x1) e1) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1:
+C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))))))) H8 (Bind Abst) H10) in
+(let H16 \def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind
+Abst) H10) in (let H_x \def (H x0 (r (Bind Abst) n) h H16 c2 H13) in (let H17
+\def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abst) n) c3 c2))
+(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3
+(CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g (CHead c (Bind Abst)
+(lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: C).(\lambda (H18: (drop h
+(r (Bind Abst) n) x c2)).(\lambda (H19: (csubc g c x)).(ex_intro2 C (\lambda
+(c3: C).(drop h (S n) c3 (CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g
+(CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3)) (CHead x (Bind Abbr)
+(lift h n w)) (drop_skip_bind h n x c2 H18 Abbr w) (csubc_abst g c x H19
+(lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) x0 x1 H14 c h (r
+(Bind Abst) n) H16) (lift h n w) (sc3_lift g a c2 w H15 x h n H18))))))
+H17)))))))) e1 H12)) v (sym_eq T v x1 H11))) k H10)) c1 (sym_eq C c1 x0 H9)))
+H4)) H2)) H7 H1 H3 H5)))]) in (H9 (refl_equal C (CHead x0 k x1)) (refl_equal
+C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
+theorem csubc_drop_conf_rev:
+ \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
+(h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
+C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
+(c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
+(d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
+(e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat
+h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
+C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
+(eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
+nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1
+(CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
+C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def
+(eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C
+(\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n)))
+e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
+(\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
+(\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1
+c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
+e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h
+n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h:
+nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
+(e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
+(\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O
+(CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2
+\def (eq_ind_r C e2 (\lambda (c: C).(csubc g e1 c)) H1 (CHead c k t)
+(drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
+O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1)
+H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
+(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop
+(S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1
+e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
+(let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
+(\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
+e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda
+(H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C
+(\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c
+k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t)))))
+H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda
+(c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k
+t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
+e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda
+(e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
+n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
+C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
+x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
+(c: C).(csubc g e1 c)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
+(\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall
+(e1: C).((csubc g e1 c0) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1))
+(\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) H3) in
+(let H8 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead
+c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k x1))
+\to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1
+(CHead c k t))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k
+n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1))
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 return
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0 c1)).((eq C c0 e1)
+\to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c2: C).(drop h (S n) c2
+e1)) (\lambda (c2: C).(csubc g c2 (CHead c k (lift h (r k n) x1))))))))))
+with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0)
+e1)).(\lambda (H3: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort n0)
+(\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda (c1:
+C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k
+n) x1))))))) (\lambda (H4: (eq C (CSort n0) (CHead x0 k x1))).(let H5 \def
+(eq_ind C (CSort n0) (\lambda (e: C).(match e return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I
+(CHead x0 k x1) H4) in (False_ind (ex2 C (\lambda (c1: C).(drop h (S n) c1
+(CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))
+H5))) e1 H1 H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq
+C (CHead c1 k0 v) e1)).(\lambda (H6: (eq C (CHead c2 k0 v) (CHead x0 k
+x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c0: C).((eq C (CHead c2 k0 v)
+(CHead x0 k x1)) \to ((csubc g c1 c2) \to (ex2 C (\lambda (c3: C).(drop h (S
+n) c3 c0)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))
+(\lambda (H7: (eq C (CHead c2 k0 v) (CHead x0 k x1))).(let H2 \def (f_equal C
+T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c2 k0 v) (CHead x0 k
+x1) H7) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow
+k])) (CHead c2 k0 v) (CHead x0 k x1) H7) in ((let H8 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 k0 v) (CHead x0 k
+x1) H7) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to
+((csubc g c1 c0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k0
+v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))
+(\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to
+((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k1
+v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))
+(\lambda (H10: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to
+(ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t))) (\lambda (c3:
+C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H11: (csubc g
+c1 x0)).(let H12 \def (eq_ind T v (\lambda (t: T).(eq C (CHead c1 k0 t) e1))
+H3 x1 H10) in (let H13 \def (eq_ind K k0 (\lambda (k: K).(eq C (CHead c1 k
+x1) e1)) H12 k H9) in (let H_x \def (H x0 (r k n) h H5 c1 H11) in (let H5
+\def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda (c3:
+C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1)))
+(\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) (\lambda (x:
+C).(\lambda (H14: (drop h (r k n) x c1)).(\lambda (H15: (csubc g x
+c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) (\lambda
+(c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k (lift h (r k
+n) x1)) (drop_skip k h n x c1 H14 x1) (csubc_head g x c H15 k (lift h (r k n)
+x1)))))) H5)))))) v (sym_eq T v x1 H10))) k0 (sym_eq K k0 k H9))) c2 (sym_eq
+C c2 x0 H8))) H4)) H2))) e1 H3 H6 H1))) | (csubc_abst c1 c2 H1 v a H3 w H5)
+\Rightarrow (\lambda (H6: (eq C (CHead c1 (Bind Abst) v) e1)).(\lambda (H7:
+(eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))).(eq_ind C (CHead c1 (Bind
+Abst) v) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))
+\to ((csubc g c1 c2) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c2 w) \to
+(ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g c3
+(CHead c k (lift h (r k n) x1)))))))))) (\lambda (H9: (eq C (CHead c2 (Bind
+Abbr) w) (CHead x0 k x1))).(let H2 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t)
+\Rightarrow t])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in ((let H4
+\def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k _) \Rightarrow k])) (CHead c2
+(Bind Abbr) w) (CHead x0 k x1) H9) in ((let H10 \def (f_equal C C (\lambda
+(e: C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 |
+(CHead c _ _) \Rightarrow c])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9)
+in (eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to
+((csubc g c1 c0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c0 w) \to (ex2 C
+(\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3:
+C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H11: (eq K
+(Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k: K).((eq T w x1) \to
+((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C
+(\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3:
+C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))) (\lambda (H12: (eq T w
+x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1
+v) \to ((sc3 g a x0 t) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1
+(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r
+(Bind Abbr) n) x1))))))))) (\lambda (H13: (csubc g c1 x0)).(\lambda (H14:
+(sc3 g (asucc g a) c1 v)).(\lambda (H15: (sc3 g a x0 x1)).(let H8 \def
+(eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift
+h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k
+x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: C).(csubc g
+c1 (CHead c k (lift h (r k n) x1)))))))))) H8 (Bind Abbr) H11) in (let H16
+\def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind Abbr) H11)
+in (let H_x \def (H x0 (r (Bind Abbr) n) h H16 c1 H13) in (let H17 \def H_x
+in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda
+(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1
+(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r
+(Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h (r (Bind Abbr)
+n) x c1)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h
+(S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c
+(Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n
+v)) (drop_skip_bind h n x c1 H18 Abst v) (csubc_abst g x c H19 (lift h n v) a
+(sc3_lift g (asucc g a) c1 v H14 x h n H18) (lift h (r (Bind Abbr) n) x1)
+(sc3_lift g a x0 x1 H15 c h (r (Bind Abbr) n) H16)))))) H17)))))))) w (sym_eq
+T w x1 H12))) k H11)) c2 (sym_eq C c2 x0 H10))) H4)) H2))) e1 H6 H7 H1 H3
+H5)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t
+H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+
+theorem drop1_csubc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2
+e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
+c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H
+return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_:
+(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to
+(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2
+c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2
+(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1
+e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind
+C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda
+(c1: C).(csubc g c0 c1)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc
+g c e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1:
+C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) (ex_intro2 C
+(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) e1
+(drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2)
+H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq
+PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C
+c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
+PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False
+| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2)
+\to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C
+(\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))))
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2)
+(refl_equal C e2)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to
+(\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1
+e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda (c2: C).(\lambda (e2:
+C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1:
+(csubc g e2 e1)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda
+(c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n
+n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0
+p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def
+(eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_:
+PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False]))
+I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2
+c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow
+(\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C
+c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList
+(\lambda (e: PList).(match e return (\lambda (_: PList).PList) with [PNil
+\Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0
+p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
+return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n _)
+\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal
+PList nat (\lambda (e: PList).(match e return (\lambda (_: PList).nat) with
+[PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n
+n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList
+hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1
+hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda
+(c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0
+(\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to
+((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1
+(PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))))) (\lambda
+(H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2)
+\to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C
+(\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2
+c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C
+c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2:
+C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))))
+(\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0)
+\to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1))
+(\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: (drop n n0 c2
+c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in
+(let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda
+(c2: C).(csubc g c0 c2)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2
+e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H1: (drop1
+p x e1)).(\lambda (H16: (csubc g c0 x)).(let H_x0 \def (drop_csubc_trans g c2
+c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: C).(drop n
+n0 c2 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C (\lambda (c2: C).(drop1
+(PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x0:
+C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g c2
+x0)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda
+(c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18))))
+H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds
+(sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9)))
+H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
+(refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+
+theorem csubc_drop1_conf_rev:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1
+e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
+c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H
+return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_:
+(drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to
+(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1
+c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
+PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2
+(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1
+e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind
+C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda
+(c1: C).(csubc g c1 c0)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc
+g e1 c)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1:
+C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) (ex_intro2 C
+(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) e1
+(drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2)
+H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq
+PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C
+c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e:
+PList).(match e return (\lambda (_: PList).Prop) with [PNil \Rightarrow False
+| (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2)
+\to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C
+(\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))))
+H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2)
+(refl_equal C e2)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to
+(\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1
+e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda (c2: C).(\lambda (e2:
+C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1:
+(csubc g e1 e2)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda
+(c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n
+n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) with
+[(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0
+p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def
+(eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_:
+PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False]))
+I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1
+c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow
+(\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C
+c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList
+(\lambda (e: PList).(match e return (\lambda (_: PList).PList) with [PNil
+\Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0
+p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
+return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n _)
+\Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal
+PList nat (\lambda (e: PList).(match e return (\lambda (_: PList).nat) with
+[PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n
+n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList
+hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1
+hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda
+(c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0
+(\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to
+((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1
+(PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))))) (\lambda
+(H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2)
+\to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C
+(\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4
+c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C
+c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2:
+C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))))
+(\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0)
+\to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1))
+(\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: (drop n n0 c2
+c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in
+(let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda
+(c2: C).(csubc g c2 c0)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2
+e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H1: (drop1
+p x e1)).(\lambda (H16: (csubc g x c0)).(let H_x0 \def (csubc_drop_conf_rev g
+c2 c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2:
+C).(drop n n0 c2 x)) (\lambda (c4: C).(csubc g c4 c2)) (ex2 C (\lambda (c2:
+C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda
+(x0: C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g x0
+c2)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda
+(c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18))))
+H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds
+(sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9)))
+H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
+(refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+
+theorem drop1_ceqc_trans:
+ \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
+C).((drop1 hds c2 e2) \to (\forall (e1: C).((ceqc g e2 e1) \to (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))))))))
+\def
+ \lambda (g: G).(\lambda (hds: PList).(\lambda (c2: C).(\lambda (e2:
+C).(\lambda (H: (drop1 hds c2 e2)).(\lambda (e1: C).(\lambda (H0: (ceqc g e2
+e1)).(let H1 \def H0 in (or_ind (csubc g e2 e1) (csubc g e1 e2) (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))
+(\lambda (H2: (csubc g e2 e1)).(let H_x \def (drop1_csubc_trans g hds c2 e2 H
+e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop1 hds c1 e1))
+(\lambda (c1: C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 hds c1 e1))
+(\lambda (c1: C).(ceqc g c2 c1))) (\lambda (x: C).(\lambda (H4: (drop1 hds x
+e1)).(\lambda (H5: (csubc g c2 x)).(ex_intro2 C (\lambda (c1: C).(drop1 hds
+c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)) x H4 (or_introl (csubc g c2 x)
+(csubc g x c2) H5))))) H3)))) (\lambda (H2: (csubc g e1 e2)).(let H_x \def
+(csubc_drop1_conf_rev g hds c2 e2 H e1 H2) in (let H3 \def H_x in (ex2_ind C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) (ex2 C
+(\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))
+(\lambda (x: C).(\lambda (H4: (drop1 hds x e1)).(\lambda (H5: (csubc g x
+c2)).(ex_intro2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc
+g c2 c1)) x H4 (or_intror (csubc g c2 x) (csubc g x c2) H5))))) H3))))
+H1)))))))).
+
+axiom sc3_ceqc_trans:
+ \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c1:
+C).(\forall (t: T).((sc3 g a c1 (THeads (Flat Appl) vs t)) \to (\forall (c2:
+C).((ceqc g c2 c1) \to (sc3 g a c2 (THeads (Flat Appl) vs t)))))))))
+.
+
+theorem sc3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
+a) \to (sc3 g a c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
+A).(sc3 g a0 c0 t0)))) (\lambda (c0: C).(\lambda (n: nat).(conj (arity g c0
+(TSort n) (ASort O n)) (sn3 c0 (TSort n)) (arity_sort g c0 n) (sn3_nf2 c0
+(TSort n) (nf2_sort c0 n))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: (sc3 g a0
+d u)).(let H_y \def (sc3_abbr g a0 TNil) in (H_y i d u c0 (sc3_lift g a0 d u
+H2 c0 (S i) O (getl_drop Abbr c0 d u i H0)) H0)))))))))) (\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
+(CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc
+g a0))).(\lambda (_: (sc3 g (asucc g a0) d u)).(let H3 \def (sc3_abst g a0
+TNil) in (H3 c0 i (arity_abst g c0 d u i H0 a0 H1) (nf2_lref_abst c0 d u i
+H0) I)))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda
+(c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
+a1)).(\lambda (H2: (sc3 g a1 c0 u)).(\lambda (t0: T).(\lambda (a2:
+A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: (sc3 g
+a2 (CHead c0 (Bind b) u) t0)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in
+(H_y c0 u t0 H4 H2))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
+(a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (H1: (sc3 g
+(asucc g a1) c0 u)).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g
+(CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: (sc3 g a2 (CHead c0 (Bind
+Abst) u) t0)).(conj (arity g c0 (THead (Bind Abst) u t0) (AHead a1 a2))
+(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is:
+PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is (THead
+(Bind Abst) u t0))))))))) (arity_head g c0 u a1 H0 t0 a2 H2) (\lambda (d:
+C).(\lambda (w: T).(\lambda (H4: (sc3 g a1 d w)).(\lambda (is:
+PList).(\lambda (H5: (drop1 is d c0)).(let H6 \def (sc3_appl g a1 a2 TNil) in
+(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1:
+T).(sc3 g a2 d (THead (Flat Appl) w t1))) (H6 d w (lift1 (Ss is) t0) (let H_y
+\def (sc3_bind g Abbr (\lambda (H3: (eq B Abbr Abst)).(not_abbr_abst H3)) a1
+a2 TNil) in (H_y d w (lift1 (Ss is) t0) (let H7 \def (sc3_ceqc_trans g a2
+TNil) in (H7 (CHead d (Bind Abst) (lift1 is u)) (lift1 (Ss is) t0) (sc3_lift1
+g (CHead c0 (Bind Abst) u) a2 (Ss is) (CHead d (Bind Abst) (lift1 is u)) t0
+H3 (drop1_skip_bind Abst c0 is d u H5)) (CHead d (Bind Abbr) w) (or_intror
+(csubc g (CHead d (Bind Abbr) w) (CHead d (Bind Abst) (lift1 is u))) (csubc g
+(CHead d (Bind Abst) (lift1 is u)) (CHead d (Bind Abbr) w)) (csubc_abst g d d
+(scubc_refl g d) (lift1 is u) a1 (sc3_lift1 g c0 (asucc g a1) is d u H1 H5) w
+H4)))) H4)) H4 (lift1 is u) (sc3_lift1 g c0 (asucc g a1) is d u H1 H5))
+(lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u
+t0)))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (H1: (sc3 g a1 c0 u)).(\lambda
+(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
+(H3: (sc3 g (AHead a1 a2) c0 t0)).(let H4 \def H3 in (and_ind (arity g c0 t0
+(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall
+(is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is
+t0)))))))) (sc3 g a2 c0 (THead (Flat Appl) u t0)) (\lambda (_: (arity g c0 t0
+(AHead a1 a2))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a1 d
+w) \to (\forall (is: PList).((drop1 is d c0) \to (sc3 g a2 d (THead (Flat
+Appl) w (lift1 is t0)))))))))).(let H_y \def (H6 c0 u H1 PNil) in (H_y
+(drop1_nil c0))))) H4))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
+(a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (H1: (sc3 g (asucc
+g a0) c0 u)).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3:
+(sc3 g a0 c0 t0)).(let H_y \def (sc3_cast g a0 TNil) in (H_y c0 u H1 t0
+H3)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
+(arity g c0 t0 a1)).(\lambda (H1: (sc3 g a1 c0 t0)).(\lambda (a2: A).(\lambda
+(H2: (leq g a1 a2)).(sc3_repl g a1 c0 t0 H1 a2 H2)))))))) c t a H))))).
+
+definition pc1:
+ T \to (T \to Prop)
+\def
+ \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda
+(t: T).(pr1 t2 t)))).
+
+theorem pc1_pr0_r:
+ \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H)
+(pr1_r t2)))).
+
+theorem pc1_pr0_x:
+ \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_r t1)
+(pr1_pr0 t2 t1 H)))).
+
+theorem pc1_pr0_u:
+ \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3:
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t:
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x:
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda
+(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_u t2 t1 H x H2) H3))))
+H1)))))).
+
+theorem pc1_refl:
+ \forall (t: T).(pc1 t t)
+\def
+ \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0:
+T).(pr1 t t0)) t (pr1_r t) (pr1_r t)).
+
+theorem pc1_s:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2
+t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2
+x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2
+H1)))) H0)))).
+
+theorem pc1_head_1:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall
+(k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t:
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0))
+(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda
+(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T
+(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2
+t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k)))))
+H0)))))).
+
+theorem pc1_head_2:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall
+(k: K).(pc1 (THead k u t1) (THead k u t2))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u:
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t))
+(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x:
+T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda
+(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead
+k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
+
+theorem pc1_t:
+ \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2
+t3) \to (pc1 t1 t3)))))
+\def
+ \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3:
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t:
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x:
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1
+t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2
+x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1
+t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x
+x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1
+(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x
+H2))))) H4))))) H1)))))).
+
+theorem pc1_pr0_u2:
+ \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0
+t2) \to (pc1 t1 t2)))))
+\def
+ \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2:
+T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
+
+theorem pc1_head:
+ \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall
+(t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2
+t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead
+k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2)
+(pc1_head_2 t1 t2 H0 u2 k)))))))).
+
+definition pc3:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3
+c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
+
+theorem clear_pc3_trans:
+ \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
+(\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2))))))
+\def
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c2 t1
+t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def H in (ex2_ind
+T (\lambda (t: T).(pr3 c2 t1 t)) (\lambda (t: T).(pr3 c2 t2 t)) (pc3 c1 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c2 t1 x)).(\lambda (H3: (pr3 c2 t2
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2
+t)) x (clear_pr3_trans c2 t1 x H2 c1 H0) (clear_pr3_trans c2 t2 x H3 c1
+H0))))) H1))))))).
+
+theorem pc3_pr2_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 (pr3_pr2 c t1 t2 H) (pr3_refl c t2))))).
+
+theorem pc3_pr2_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t2
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) (pr3_pr2 c t2 t1 H))))).
+
+theorem pc3_pr3_r:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t2 H (pr3_refl c t2))))).
+
+theorem pc3_pr3_x:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c
+t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t2
+t1)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))
+t1 (pr3_refl c t1) H)))).
+
+theorem pc3_pr3_t:
+ \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall
+(t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda (H: (pr3 c t1
+t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t:
+T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
+
+theorem pc3_pr2_u:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
+
+theorem pc3_refl:
+ \forall (c: C).(\forall (t: T).(pc3 c t t))
+\def
+ \lambda (c: C).(\lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr3 c t t0))
+(\lambda (t0: T).(pr3 c t t0)) t (pr3_refl c t) (pr3_refl c t))).
+
+theorem pc3_s:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c
+t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc3 c t1
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t2 t1) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t2 t))
+(\lambda (t: T).(pr3 c t1 t)) x H2 H1)))) H0))))).
+
+theorem pc3_thin_dx:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
+(u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (u: T).(\lambda (f: F).(let H0 \def H in (ex2_ind T (\lambda
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 c (THead (Flat f) u
+t1) (THead (Flat f) u t2)) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c (THead
+(Flat f) u t1) t)) (\lambda (t: T).(pr3 c (THead (Flat f) u t2) t)) (THead
+(Flat f) u x) (pr3_thin_dx c t1 x H1 u f) (pr3_thin_dx c t2 x H2 u f)))))
+H0))))))).
+
+theorem pc3_head_1:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(let H0 \def H in (ex2_ind T (\lambda
+(t0: T).(pr3 c u1 t0)) (\lambda (t0: T).(pr3 c u2 t0)) (pc3 c (THead k u1 t)
+(THead k u2 t)) (\lambda (x: T).(\lambda (H1: (pr3 c u1 x)).(\lambda (H2:
+(pr3 c u2 x)).(ex_intro2 T (\lambda (t0: T).(pr3 c (THead k u1 t) t0))
+(\lambda (t0: T).(pr3 c (THead k u2 t) t0)) (THead k x t) (pr3_head_12 c u1 x
+H1 k t t (pr3_refl (CHead c k x) t)) (pr3_head_12 c u2 x H2 k t t (pr3_refl
+(CHead c k x) t)))))) H0))))))).
+
+theorem pc3_head_2:
+ \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u
+t2)))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pc3 (CHead c k u) t1 t2)).(let H0 \def H in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c k u) t1 t)) (\lambda (t: T).(pr3 (CHead c k u)
+t2 t)) (pc3 c (THead k u t1) (THead k u t2)) (\lambda (x: T).(\lambda (H1:
+(pr3 (CHead c k u) t1 x)).(\lambda (H2: (pr3 (CHead c k u) t2 x)).(ex_intro2
+T (\lambda (t: T).(pr3 c (THead k u t1) t)) (\lambda (t: T).(pr3 c (THead k u
+t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1)
+(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
+
+theorem pc3_pc1:
+ \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1
+t2))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (c:
+C).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t:
+T).(pr1 t2 t)) (pc3 c t1 t2) (\lambda (x: T).(\lambda (H1: (pr1 t1
+x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t))
+(\lambda (t: T).(pr3 c t2 t)) x (pr3_pr1 t1 x H1 c) (pr3_pr1 t2 x H2 c)))))
+H0))))).
+
+theorem pc3_t:
+ \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (t2: T).(\lambda (c: C).(\lambda (t1: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(let H4 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3 c t1 t3) (\lambda (x0: T).(\lambda (H5: (pr3 c t1
+x0)).(\lambda (H6: (pr3 c t2 x0)).(ex2_ind T (\lambda (t: T).(pr3 c x0 t))
+(\lambda (t: T).(pr3 c x t)) (pc3 c t1 t3) (\lambda (x1: T).(\lambda (H7:
+(pr3 c x0 x1)).(\lambda (H8: (pr3 c x x1)).(pc3_pr3_t c t1 x1 (pr3_t x0 t1 c
+H5 x1 H7) t3 (pr3_t x t3 c H3 x1 H8))))) (pr3_confluence c t2 x0 H6 x H2)))))
+H4))))) H1))))))).
+
+theorem pc3_pr2_u2:
+ \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall
+(t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2))))))
+\def
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0
+t1)).(\lambda (t2: T).(\lambda (H0: (pc3 c t0 t2)).(pc3_t t0 c t1 (pc3_pr2_x
+c t1 t0 H) t2 H0)))))).
+
+theorem pc3_head_12:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
+(CHead c k u2) t1 t2)).(pc3_t (THead k u2 t1) c (THead k u1 t1) (pc3_head_1 c
+u1 u2 H k t1) (THead k u2 t2) (pc3_head_2 c u2 t1 t2 k H0))))))))).
+
+theorem pc3_head_21:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall
+(k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3
+c (THead k u1 t1) (THead k u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc3 c u1
+u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3
+(CHead c k u1) t1 t2)).(pc3_t (THead k u1 t2) c (THead k u1 t1) (pc3_head_2 c
+u1 t1 t2 k H0) (THead k u2 t2) (pc3_head_1 c u1 u2 H k t2))))))))).
+
+theorem pc3_pr0_pr2_t:
+ \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
+(CHead c k u2) t1 t2)).(let H1 \def (match H0 return (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
+(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1
+t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
+(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
+t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
+\to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3
+(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
+(t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
+t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
+(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
+H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
+(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
+T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
+t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
+u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
+t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
+(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3
+(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
+(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
+((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
+(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
+(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
+(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1
+t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((clear (CHead
+c k u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k u1) t1 t2))) (\lambda
+(b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d
+(Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr)
+u) u2 H14)) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in ((let H16 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in
+(\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind
+T u (\lambda (t: T).(subst0 O t t3 t2)) H13 u2 H16) in (eq_ind B Abbr
+(\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t1:
+T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 t2 t1)) (pc3 (CHead c (Bind
+Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H: (subst0 O u1 t3 x)).(\lambda
+(H20: (pr0 t2 x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t1 x (pr3_pr2 (CHead c
+(Bind Abbr) u1) t1 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl
+Abbr c u1) t1 t3 H10 x H)) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) t2 x
+(pr2_free (CHead c (Bind Abbr) u1) t2 x H20)))))) (pr0_subst0_fwd u2 t3 t2 O
+H19 u1 H)) b H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (clear
+(CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d
+(Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d (Bind Abbr) u) t1 t2 (pr2_delta
+(CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t1 t3 H10 t2 H13)) (CHead
+c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f c
+(CHead d (Bind Abbr) u) u2 H14) f u1)))) k (getl_gen_O (CHead c k u2) (CHead
+d (Bind Abbr) u) H12)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead
+c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k
+u1) t1 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind
+Abbr) u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k:
+K).((((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3
+t2) \to (pc3 (CHead c k u1) t1 t2)))) \to ((getl (r k i0) c (CHead d (Bind
+Abbr) u)) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (_:
+(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
+t3 t2) \to (pc3 (CHead c (Bind b) u1) t1 t2))))).(\lambda (H0: (getl (r (Bind
+b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2
+(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d
+(Bind Abbr) u) H0 u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (_:
+(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
+t3 t2) \to (pc3 (CHead c (Flat f) u1) t1 t2))))).(\lambda (H0: (getl (r (Flat
+f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2
+(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) H0 t1 t3 H10 t2 H13) f
+u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9
+H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0
+(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2))
+(refl_equal T t1) (refl_equal T t2)))))))))).
+
+theorem pc3_pr2_pr2_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
+u1)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u2) \to ((eq T
+t0 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k
+u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))))))) with [(pr2_free c0 t1 t2
+H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1
+u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T t1 u2)
+\to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
+T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3
+t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T
+t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
+K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda
+(H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3:
+T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3
+(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0
+t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H)))))) t2 (sym_eq T t2 u1 H5))) t1
+(sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
+i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
+T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1
+u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
+t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
+K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))))))
+(\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to
+((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
+\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
+t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
+u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
+((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
+T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3
+t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
+(pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda
+(t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 t3)).(let H11
+\def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t1:
+T).(\lambda (_: (pr2 c0 t t1)).((eq C c0 (CHead c k u2)) \to ((eq T t t0) \to
+((eq T t1 t3) \to (pc3 (CHead c k u1) t0 t3)))))))) with [(pr2_free c0 t1 t4
+H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T
+t1 t0)).(\lambda (H6: (eq T t4 t3)).(eq_ind C (CHead c k u2) (\lambda (_:
+C).((eq T t1 t0) \to ((eq T t4 t3) \to ((pr0 t1 t4) \to (pc3 (CHead c k u1)
+t0 t3))))) (\lambda (H7: (eq T t1 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t4
+t3) \to ((pr0 t t4) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H8: (eq T t4
+t3)).(eq_ind T t3 (\lambda (t: T).((pr0 t0 t) \to (pc3 (CHead c k u1) t0
+t3))) (\lambda (H9: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1) t0 t3 (pr2_free
+(CHead c k u1) t0 t3 H9))) t4 (sym_eq T t4 t3 H8))) t1 (sym_eq T t1 t0 H7)))
+c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3
+t1 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda
+(H7: (eq T t1 t0)).(\lambda (H11: (eq T t t3)).(eq_ind C (CHead c k u2)
+(\lambda (c1: C).((eq T t1 t0) \to ((eq T t t3) \to ((getl i0 c1 (CHead d0
+(Bind Abbr) u0)) \to ((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c
+k u1) t0 t3))))))) (\lambda (H12: (eq T t1 t0)).(eq_ind T t0 (\lambda (t2:
+T).((eq T t t3) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
+((pr0 t2 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c k u1) t0 t3))))))
+(\lambda (H13: (eq T t t3)).(eq_ind T t3 (\lambda (t2: T).((getl i0 (CHead c
+k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2)
+\to (pc3 (CHead c k u1) t0 t3))))) (\lambda (H14: (getl i0 (CHead c k u2)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t0 t4)).(\lambda (H16:
+(subst0 i0 u0 t4 t3)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
+(CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t3) \to (pc3 (CHead c k u1)
+t0 t3)))) (\lambda (H17: (getl O (CHead c k u2) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H18: (subst0 O u0 t4 t3)).((match k return (\lambda (k:
+K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1)
+t0 t3))) with [(Bind b) \Rightarrow (\lambda (H19: (clear (CHead c (Bind b)
+u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 | (CHead
+c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H0 \def
+(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort
+_) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d0
+(Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind
+Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq
+B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t:
+T).(subst0 O t t4 t3)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pc3
+(CHead c (Bind b0) u1) t0 t3)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4
+t0)) (\lambda (t0: T).(pr0 t3 t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3)
+(\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda (H9: (pr0 t3
+x)).(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(subst0
+(S (plus i O)) u x t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) (\lambda (x0:
+T).(\lambda (H10: (subst0 O u1 t4 x0)).(\lambda (H23: (subst0 (S (plus i O))
+u x x0)).(let H24 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i
+O) (plus_n_O i))) in (let H25 \def (eq_ind nat (S (plus i O)) (\lambda (n:
+nat).(subst0 n u x x0)) H23 (S i) H24) in (pc3_pr2_u (CHead c (Bind Abbr) u1)
+x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t4
+H15 x0 H10) t3 (pc3_pr2_x (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c
+(Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u)
+H8 u1) t3 x H9 x0 H25)))))))) (subst0_subst0_back t4 x t2 O H2 u1 u i
+H10))))) (pr0_subst0_fwd u2 t4 t3 O H22 t2 H9)) b H20))))) H0)) H))) | (Flat
+f) \Rightarrow (\lambda (H8: (clear (CHead c (Flat f) u2) (CHead d0 (Bind
+Abbr) u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r
+(CHead d0 (Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O
+(getl_refl Abbr d0 u0) t0 t4 H15 t3 H18)) (CHead c (Flat f) u1) (clear_flat c
+(CHead d0 (Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2
+H8) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H17))))
+(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind
+Abbr) u0)) \to ((subst0 i1 u0 t4 t3) \to (pc3 (CHead c k u1) t0
+t3))))).(\lambda (H8: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H9: (subst0 (S i1) u0 t4 t3)).(K_ind (\lambda (k: K).((getl
+(r k i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) t0 t3)))
+(\lambda (b: B).(\lambda (H: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr)
+u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1)
+d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H u1) t0 t4
+H15 t3 H9)))) (\lambda (f: F).(\lambda (H: (getl (r (Flat f) i1) c (CHead d0
+(Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0 t3
+(pr2_delta c d0 u0 (r (Flat f) i1) H t0 t4 H15 t3 H9) f u1)))) k (getl_gen_S
+k c (CHead d0 (Bind Abbr) u0) u2 i1 H8)))))) i0 H14 H16)))) t (sym_eq T t t3
+H13))) t1 (sym_eq T t1 t0 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 H11 H3
+H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t0)
+(refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1 (sym_eq T t1 u2 H6))) c0
+(sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T
+u2) (refl_equal T u1)))))).
+
+theorem pc3_pr2_pr3_t:
+ \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
+(k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to
+(pc3 (CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(k: K).(\lambda (H: (pr3 (CHead c k u2) t1 t2)).(pr3_ind (CHead c k u2)
+(\lambda (t: T).(\lambda (t0: T).(\forall (u1: T).((pr2 c u2 u1) \to (pc3
+(CHead c k u1) t t0))))) (\lambda (t: T).(\lambda (u1: T).(\lambda (_: (pr2 c
+u2 u1)).(pc3_refl (CHead c k u1) t)))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
+(pr3 (CHead c k u2) t0 t4)).(\lambda (H2: ((\forall (u1: T).((pr2 c u2 u1)
+\to (pc3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u2
+u1)).(pc3_t t0 (CHead c k u1) t3 (pc3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2
+u1 H3)))))))))) t1 t2 H)))))).
+
+theorem pc3_pr3_pc3_t:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall
+(t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3
+(CHead c k u1) t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u2
+u1)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t1: T).(\forall
+(t2: T).(\forall (k: K).((pc3 (CHead c k t) t1 t2) \to (pc3 (CHead c k t0) t1
+t2))))))) (\lambda (t: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
+K).(\lambda (H0: (pc3 (CHead c k t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda
+(t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2
+t3)).(\lambda (H2: ((\forall (t1: T).(\forall (t4: T).(\forall (k: K).((pc3
+(CHead c k t2) t1 t4) \to (pc3 (CHead c k t3) t1 t4))))))).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pc3 (CHead c k t1) t0
+t4)).(H2 t0 t4 k (let H4 \def H3 in (ex2_ind T (\lambda (t: T).(pr3 (CHead c
+k t1) t0 t)) (\lambda (t: T).(pr3 (CHead c k t1) t4 t)) (pc3 (CHead c k t2)
+t0 t4) (\lambda (x: T).(\lambda (H5: (pr3 (CHead c k t1) t0 x)).(\lambda (H6:
+(pr3 (CHead c k t1) t4 x)).(pc3_t x (CHead c k t2) t0 (pc3_pr2_pr3_t c t1 t0
+x k H5 t2 H0) t4 (pc3_s (CHead c k t2) x t4 (pc3_pr2_pr3_t c t1 t4 x k H6 t2
+H0)))))) H4))))))))))))) u2 u1 H)))).
+
+theorem pc3_lift:
+ \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift
+h d t1) (lift h d t2)))))))))
+\def
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pc3 e t1
+t2)).(let H1 \def H0 in (ex2_ind T (\lambda (t: T).(pr3 e t1 t)) (\lambda (t:
+T).(pr3 e t2 t)) (pc3 c (lift h d t1) (lift h d t2)) (\lambda (x: T).(\lambda
+(H2: (pr3 e t1 x)).(\lambda (H3: (pr3 e t2 x)).(pc3_pr3_t c (lift h d t1)
+(lift h d x) (pr3_lift c e h d H t1 x H2) (lift h d t2) (pr3_lift c e h d H
+t2 x H3))))) H1))))))))).
+
+theorem pc3_wcpr0__pc3_wcpr0_t_aux:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall
+(u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3
+(CHead c2 k u) t1 t2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k:
+K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
+(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0:
+T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t))
+(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3:
+(pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match
+H1 return (\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (_: (pr2
+c t t1)).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pc3
+(CHead c2 k u) t3 t0)))))))) with [(pr2_free c t1 t2 H2) \Rightarrow (\lambda
+(H3: (eq C c (CHead c1 k u))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T
+t2 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2
+t0) \to ((pr0 t1 t2) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H6: (eq T
+t1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pc3
+(CHead c2 k u) t3 t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda
+(t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H8: (pr0 t3
+t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 H8))) t2
+(sym_eq T t2 t0 H7))) t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c1 k u)
+H3) H4 H5 H2)))) | (pr2_delta c d u0 i H2 t1 t2 H3 t H4) \Rightarrow (\lambda
+(H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T
+t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t1 t3) \to ((eq T t
+t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t2) \to ((subst0 i
+u0 t2 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H8: (eq T t1
+t3)).(eq_ind T t3 (\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c1 k u)
+(CHead d (Bind Abbr) u0)) \to ((pr0 t4 t2) \to ((subst0 i u0 t2 t) \to (pc3
+(CHead c2 k u) t3 t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda
+(t4: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 t3 t2)
+\to ((subst0 i u0 t2 t4) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H10:
+(getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H11: (pr0 t3
+t2)).(\lambda (H12: (subst0 i u0 t2 t0)).(ex3_2_ind C T (\lambda (e2:
+C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t3 t0) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H0: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda
+(_: (wcpr0 d x0)).(\lambda (H14: (pr0 u0 x1)).(ex2_ind T (\lambda (t0:
+T).(subst0 i x1 t2 t0)) (\lambda (t3: T).(pr0 t0 t3)) (pc3 (CHead c2 k u) t3
+t0) (\lambda (x: T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 t0
+x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H0 t3 t2
+H11 x H15) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x
+H16)))))) (pr0_subst0_fwd u0 t2 t0 i H12 x1 H14))))))) (wcpr0_getl (CHead c1
+k u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind
+Abbr) H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c
+(CHead c1 k u) H5) H6 H7 H2 H3 H4))))]) in (H4 (refl_equal C (CHead c1 k u))
+(refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))).
+
+theorem pc3_wcpr0_t:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
+T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind
+(\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1
+t2) \to (pc3 c0 t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr3 c t1 t2)).(pc3_pr3_r c t1 t2 H0))))) (\lambda (c0:
+C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1:
+T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pc3 c3 t1 t2)))))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c0 k u1) t1 t2)).(let H4 \def
+(pc3_pr2_pr3_t c0 u1 t1 t2 k H3 u2 (pr2_free c0 u1 u2 H2)) in (ex2_ind T
+(\lambda (t: T).(pr3 (CHead c0 k u2) t1 t)) (\lambda (t: T).(pr3 (CHead c0 k
+u2) t2 t)) (pc3 (CHead c3 k u2) t1 t2) (\lambda (x: T).(\lambda (H5: (pr3
+(CHead c0 k u2) t1 x)).(\lambda (H6: (pr3 (CHead c0 k u2) t2 x)).(pc3_t x
+(CHead c3 k u2) t1 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t1 x H5) t2
+(pc3_s (CHead c3 k u2) x t2 (pc3_wcpr0__pc3_wcpr0_t_aux c0 c3 H0 k u2 t2 x
+H6)))))) H4))))))))))))) c1 c2 H))).
+
+theorem pc3_wcpr0:
+ \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
+T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H0: (pc3 c1 t1 t2)).(let H1 \def H0 in (ex2_ind
+T (\lambda (t: T).(pr3 c1 t1 t)) (\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1
+t2) (\lambda (x: T).(\lambda (H2: (pr3 c1 t1 x)).(\lambda (H3: (pr3 c1 t2
+x)).(pc3_t x c2 t1 (pc3_wcpr0_t c1 c2 H t1 x H2) t2 (pc3_s c2 x t2
+(pc3_wcpr0_t c1 c2 H t2 x H3)))))) H1))))))).
-axiom csubst0_getl_lt: \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))))))) .
+inductive pc3_left (c:C): T \to (T \to Prop) \def
+| pc3_left_r: \forall (t: T).(pc3_left c t t)
+| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
+| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
+
+theorem pc3_ind_left__pc3_left_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
+(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2
+c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2:
+(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_trans:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
+(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3:
+T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda
+(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
+t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0
+t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
+(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda
+(H2: ((\forall (t3: T).((pc3_left c t4 t3) \to (pc3_left c t0
+t3))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0
+t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_sym:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
+(pc3_left c t2 t1))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t)))
+(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
+(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
+t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3
+H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4
+t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3
+(pc3_left_r c t3))))))))) t1 t2 H)))).
+
+theorem pc3_ind_left__pc3_left_pc3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to
+(pc3_left c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
+T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1
+x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
+(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x
+(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
+
+theorem pc3_ind_left__pc3_pc3_left:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
+(pc3 c t1 t2))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda
+(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3
+c t3 t4)).(pc3_pr2_u c t3 t0 H0 t4 H2))))))) (\lambda (t0: T).(\lambda (t3:
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0
+t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 (pc3_pr2_x c t3 t0 H0) t4
+H2))))))) t1 t2 H)))).
+
+theorem pc3_ind_left:
+ \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t
+t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
+T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1:
+T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to
+((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c
+t t0) \to (P t t0))))))))
+\def
+ \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall
+(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1
+t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1
+t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
+\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2
+t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t
+t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda
+(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3:
+T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3
+(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left
+c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3
+(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0
+(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
+
+theorem pc3_gen_sort:
+ \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort
+n)) \to (eq nat m n))))
+\def
+ \lambda (c: C).(\lambda (m: nat).(\lambda (n: nat).(\lambda (H: (pc3 c
+(TSort m) (TSort n))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c
+(TSort m) t)) (\lambda (t: T).(pr3 c (TSort n) t)) (eq nat m n) (\lambda (x:
+T).(\lambda (H1: (pr3 c (TSort m) x)).(\lambda (H2: (pr3 c (TSort n) x)).(let
+H3 \def (eq_ind T x (\lambda (t: T).(eq T t (TSort n))) (pr3_gen_sort c x n
+H2) (TSort m) (pr3_gen_sort c x m H1)) in (let H4 \def (f_equal T nat
+(\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort n)
+\Rightarrow n | (TLRef _) \Rightarrow m | (THead _ _ _) \Rightarrow m]))
+(TSort m) (TSort n) H3) in H4))))) H0))))).
+
+theorem pc3_gen_abst:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall
+(t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
+t1 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (H: (pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2
+t2))).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Abst)
+u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind Abst) u2 t2) t)) (land (pc3 c
+u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))))
+(\lambda (x: T).(\lambda (H1: (pr3 c (THead (Bind Abst) u1 t1) x)).(\lambda
+(H2: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H3 \def (pr3_gen_abst c u2 t2
+x H2) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t2 t3))))) (land (pc3 c u1 u2) (\forall (b: B).(\forall (u:
+T).(pc3 (CHead c (Bind b) u) t1 t2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr3 c u2
+x0)).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t2 x1))))).(let H7 \def (pr3_gen_abst c u1 t1 x H1) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t3)))))
+(land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u)
+t1 t2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T x (THead
+(Bind Abst) x2 x3))).(\lambda (H9: (pr3 c u1 x2)).(\lambda (H10: ((\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x3))))).(let H11 \def
+(eq_ind T x (\lambda (t: T).(eq T t (THead (Bind Abst) x0 x1))) H4 (THead
+(Bind Abst) x2 x3) H8) in (let H12 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | (TLRef _)
+\Rightarrow x2 | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) x2 x3)
+(THead (Bind Abst) x0 x1) H11) in ((let H13 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow x3 | (TLRef
+_) \Rightarrow x3 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) x2 x3)
+(THead (Bind Abst) x0 x1) H11) in (\lambda (H14: (eq T x2 x0)).(let H15 \def
+(eq_ind T x3 (\lambda (t: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c
+(Bind b) u) t1 t)))) H10 x1 H13) in (let H16 \def (eq_ind T x2 (\lambda (t:
+T).(pr3 c u1 t)) H9 x0 H14) in (conj (pc3 c u1 u2) (\forall (b: B).(\forall
+(u: T).(pc3 (CHead c (Bind b) u) t1 t2))) (pc3_pr3_t c u1 x0 H16 u2 H5)
+(\lambda (b: B).(\lambda (u: T).(pc3_pr3_t (CHead c (Bind b) u) t1 x1 (H15 b
+u) t2 (H6 b u))))))))) H12)))))))) H7))))))) H3))))) H0))))))).
+
+theorem pc3_gen_lift:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall
+(d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d
+c e) \to (pc3 e t1 t2))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H: (pc3 c (lift h d t1) (lift h d t2))).(\lambda (e:
+C).(\lambda (H0: (drop h d c e)).(let H1 \def H in (ex2_ind T (\lambda (t:
+T).(pr3 c (lift h d t1) t)) (\lambda (t: T).(pr3 c (lift h d t2) t)) (pc3 e
+t1 t2) (\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t1) x)).(\lambda (H3:
+(pr3 c (lift h d t2) x)).(let H4 \def (pr3_gen_lift c t2 x h d H3 e H0) in
+(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3 e
+t2 t3)) (pc3 e t1 t2) (\lambda (x0: T).(\lambda (H5: (eq T x (lift h d
+x0))).(\lambda (H6: (pr3 e t2 x0)).(let H7 \def (pr3_gen_lift c t1 x h d H2 e
+H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
+T).(pr3 e t1 t3)) (pc3 e t1 t2) (\lambda (x1: T).(\lambda (H8: (eq T x (lift
+h d x1))).(\lambda (H9: (pr3 e t1 x1)).(let H10 \def (eq_ind T x (\lambda (t:
+T).(eq T t (lift h d x0))) H5 (lift h d x1) H8) in (let H11 \def (eq_ind T x1
+(\lambda (t: T).(pr3 e t1 t)) H9 x0 (lift_inj x1 x0 h d H10)) in (pc3_pr3_t e
+t1 x0 H11 t2 H6)))))) H7))))) H4))))) H1))))))))).
+
+theorem pc3_gen_not_abst:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1:
+T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b)
+u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S
+O) O (THead (Bind Abst) u2 t2))))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
+(c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2:
+T).((pc3 c (THead (Bind b0) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead
+c (Bind b0) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))))))))))) (\lambda
+(_: (not (eq B Abbr Abst))).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Bind Abbr)
+u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in (ex2_ind T (\lambda (t:
+T).(pr3 c (THead (Bind Abbr) u1 t1) t)) (\lambda (t: T).(pr3 c (THead (Bind
+Abst) u2 t2) t)) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind
+Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3 c (THead (Bind Abbr) u1 t1)
+x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2 t2) x)).(let H4 \def
+(pr3_gen_abbr c u1 t1 x H2) in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr3 (CHead c (Bind Abbr)
+u1) t1 t3)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (pc3 (CHead
+c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))))).(ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
+(t3: T).(pr3 (CHead c (Bind Abbr) u1) t1 t3))) (pc3 (CHead c (Bind Abbr) u1)
+t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H6: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (_: (pr3 c u1
+x0)).(\lambda (_: (pr3 (CHead c (Bind Abbr) u1) t1 x1)).(let H9 \def
+(pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3
+c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u:
+T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Abbr) u1) t1
+(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2
+x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind
+Abbr) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T
+(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (THead (Bind Abbr) x0 x1) H13) in (False_ind (pc3
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
+H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Abbr) u1) t1
+(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
+t3))))) (pc3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O (THead (Bind Abst) u2
+t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 x1))))).(let H10 \def (eq_ind
+T x (\lambda (t: T).(pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O t))) H5
+(THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Abbr) u1) t1 (lift
+(S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind Abst) u2
+t2)) (pr3_lift (CHead c (Bind Abbr) u1) c (S O) O (drop_drop (Bind Abbr) O c
+c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 x1)
+(pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) H4)))))
+H1))))))))) (\lambda (H: (not (eq B Abst Abst))).(\lambda (c: C).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pc3
+c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def (match
+(H (refl_equal B Abst)) return (\lambda (_: False).(pc3 (CHead c (Bind Abst)
+u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))) with []) in H1))))))))
+(\lambda (_: (not (eq B Void Abst))).(\lambda (c: C).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (pc3 c
+(THead (Bind Void) u1 t1) (THead (Bind Abst) u2 t2))).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c (THead (Bind Void) u1 t1) t)) (\lambda (t:
+T).(pr3 c (THead (Bind Abst) u2 t2) t)) (pc3 (CHead c (Bind Void) u1) t1
+(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x: T).(\lambda (H2: (pr3
+c (THead (Bind Void) u1 t1) x)).(\lambda (H3: (pr3 c (THead (Bind Abst) u2
+t2) x)).(let H4 \def (pr3_gen_void c u1 t1 x H2) in (or_ind (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1
+t3)))))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) (pc3 (CHead c
+(Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (H5:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t1 t2))))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead
+c (Bind b0) u) t1 t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O
+(THead (Bind Abst) u2 t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6:
+(eq T x (THead (Bind Void) x0 x1))).(\lambda (_: (pr3 c u1 x0)).(\lambda (_:
+((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 x1))))).(let H9
+\def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda (t3: T).(\forall (b0: B).(\forall
+(u: T).(pr3 (CHead c (Bind b0) u) t2 t3))))) (pc3 (CHead c (Bind Void) u1) t1
+(lift (S O) O (THead (Bind Abst) u2 t2))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H10: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (_: (pr3 c u2
+x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t2 x3))))).(let H13 \def (eq_ind T x (\lambda (t: T).(eq T t (THead (Bind
+Void) x0 x1))) H6 (THead (Bind Abst) x2 x3) H10) in (let H14 \def (eq_ind T
+(THead (Bind Abst) x2 x3) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (THead (Bind Void) x0 x1) H13) in (False_ind (pc3
+(CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))
+H14)))))))) H9))))))) H5)) (\lambda (H5: (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x))).(let H6 \def (pr3_gen_abst c u2 t2 x H3) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t2
+t3))))) (pc3 (CHead c (Bind Void) u1) t1 (lift (S O) O (THead (Bind Abst) u2
+t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (pr3 c u2 x0)).(\lambda (H9: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 x1))))).(let H10 \def (eq_ind
+T x (\lambda (t: T).(pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O t))) H5
+(THead (Bind Abst) x0 x1) H7) in (pc3_pr3_t (CHead c (Bind Void) u1) t1 (lift
+(S O) O (THead (Bind Abst) x0 x1)) H10 (lift (S O) O (THead (Bind Abst) u2
+t2)) (pr3_lift (CHead c (Bind Void) u1) c (S O) O (drop_drop (Bind Void) O c
+c (drop_refl c) u1) (THead (Bind Abst) u2 t2) (THead (Bind Abst) x0 x1)
+(pr3_head_12 c u2 x0 H8 (Bind Abst) t2 x1 (H9 Abst x0)))))))))) H6))) H4)))))
+H1))))))))) b).
+
+theorem pc3_gen_lift_abst:
+ \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall
+(h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2))
+\to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda
+(t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
+t1)))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda
+(h: nat).(\lambda (d: nat).(\lambda (H: (pc3 c (lift h d t) (THead (Bind
+Abst) u2 t2))).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def H
+in (ex2_ind T (\lambda (t0: T).(pr3 c (lift h d t) t0)) (\lambda (t0: T).(pr3
+c (THead (Bind Abst) u2 t2) t0)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1:
+T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_:
+T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1)))))))
+(\lambda (x: T).(\lambda (H2: (pr3 c (lift h d t) x)).(\lambda (H3: (pr3 c
+(THead (Bind Abst) u2 t2) x)).(let H4 \def (pr3_gen_lift c t x h d H2 e H0)
+in (ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr3
+e t t3)) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind
+Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1))))
+(\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x0: T).(\lambda (H5: (eq T
+x (lift h d x0))).(\lambda (H6: (pr3 e t x0)).(let H7 \def (pr3_gen_abst c u2
+t2 x H3) in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead
+(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead
+c (Bind b) u) t2 t3))))) (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e
+t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2
+(lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H8: (eq T x (THead (Bind Abst) x1
+x2))).(\lambda (H9: (pr3 c u2 x1)).(\lambda (H10: ((\forall (b: B).(\forall
+(u: T).(pr3 (CHead c (Bind b) u) t2 x2))))).(let H11 \def (eq_ind T x
+(\lambda (t: T).(eq T t (lift h d x0))) H5 (THead (Bind Abst) x1 x2) H8) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T x1 (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T x2 (lift h (S d) z)))) (ex3_2 T T (\lambda (u1:
+T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1:
+T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d)
+t1))))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H12: (eq T x0 (THead
+(Bind Abst) x3 x4))).(\lambda (H13: (eq T x1 (lift h d x3))).(\lambda (H14:
+(eq T x2 (lift h (S d) x4))).(let H15 \def (eq_ind T x2 (\lambda (t:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 t)))) H10
+(lift h (S d) x4) H14) in (let H16 \def (eq_ind T x1 (\lambda (t: T).(pr3 c
+u2 t)) H9 (lift h d x3) H13) in (let H17 \def (eq_ind T x0 (\lambda (t0:
+T).(pr3 e t t0)) H6 (THead (Bind Abst) x3 x4) H12) in (ex3_2_intro T T
+(\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1))))
+(\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_:
+T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u)
+t2 (lift h (S d) t1)))))) x3 x4 H17 H16 H15))))))))) (lift_gen_bind Abst x1
+x2 x0 h d H11)))))))) H7))))) H4))))) H1)))))))))).
+
+theorem pc3_pr2_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pr2 c1 t1
+t)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t0 c2
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t3
+t2))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0:
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0:
+T).(\lambda (H1: (fsubst0 i u c t2 c2 t0)).(fsubst0_ind i u c t2 (\lambda
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u))
+\to (pc3 c0 t4 t3))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr)
+u))).(or_ind (pr0 t4 t3) (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2:
+T).(subst0 i u t3 w2))) (pc3 c t4 t3) (\lambda (H4: (pr0 t4 t3)).(pc3_pr2_r c
+t4 t3 (pr2_free c t4 t3 H4))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 t4
+w2)) (\lambda (w2: T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0
+t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2)) (pc3 c t4 t3) (\lambda (x:
+T).(\lambda (H5: (pr0 t4 x)).(\lambda (H6: (subst0 i u t3 x)).(pc3_pr2_u c x
+t4 (pr2_free c t4 x H5) t3 (pc3_pr2_x c x t3 (pr2_delta c e u i H3 t3 t3
+(pr0_refl t3) x H6)))))) H4)) (pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl
+u))))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e:
+C).(\lambda (_: (getl i c (CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3
+(pr2_free c0 t2 t3 H0)))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t2
+t4)).(\lambda (c0: C).(\lambda (H3: (csubst0 i u c c0)).(\lambda (e:
+C).(\lambda (H4: (getl i c (CHead e (Bind Abbr) u))).(or_ind (pr0 t4 t3) (ex2
+T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2: T).(subst0 i u t3 w2))) (pc3 c0
+t4 t3) (\lambda (H5: (pr0 t4 t3)).(pc3_pr2_r c0 t4 t3 (pr2_free c0 t4 t3
+H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t4 w2)) (\lambda (w2:
+T).(subst0 i u t3 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t4 w2)) (\lambda
+(w2: T).(subst0 i u t3 w2)) (pc3 c0 t4 t3) (\lambda (x: T).(\lambda (H6: (pr0
+t4 x)).(\lambda (H7: (subst0 i u t3 x)).(pc3_pr2_u c0 x t4 (pr2_free c0 t4 x
+H6) t3 (pc3_pr2_x c0 x t3 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c
+c0 u H3 (CHead e (Bind Abbr) u) H4) t3 t3 (pr0_refl t3) x H7)))))) H5))
+(pr0_subst0 t2 t3 H0 u t4 i H2 u (pr0_refl u))))))))) c2 t0 H1))))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H3: (fsubst0 i0 u0 c t2 c2 t4)).(fsubst0_ind i0 u0 c t2 (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr)
+u0)) \to (pc3 c0 t5 t0))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(pc3_t t2 c t5 (pc3_s c t5 t2 (pc3_pr2_r c t2 t5 (pr2_delta c e u0 i0
+H5 t2 t2 (pr0_refl t2) t5 H4))) t0 (pc3_pr2_r c t2 t0 (pr2_delta c d u i H0
+t2 t3 H1 t0 H2))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8:
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11:
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9)
+in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x3)) H11 u H14) in (let H18 \def (eq_ind_r C
+x1 (\lambda (c: C).(getl i c0 (CHead c (Bind x0) x3))) H10 d H16) in (let H19
+\def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H18
+Abbr H15) in (ex2_ind T (\lambda (t5: T).(subst0 i x3 t3 t5)) (\lambda (t5:
+T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda
+(x: T).(\lambda (H20: (subst0 i x3 t3 x)).(\lambda (H21: (subst0 (S (plus
+(minus i0 (S i)) i)) u0 t0 x)).(let H22 \def (eq_ind_r nat (S (plus (minus i0
+(S i)) i)) (\lambda (n: nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0
+H6)) in (pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0
+(pc3_pr2_x c0 x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c
+c0 u0 H4 (CHead e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H22)))))))
+(subst0_subst0_back t3 t0 u i H2 x3 u0 (minus i0 (S i)) H17)))))))) H13))
+H12))))))))) H8)) (\lambda (H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(getl i c0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (pc3 c0 t2 t0)
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10:
+(getl i c0 (CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t:
+T).(getl i c0 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r
+C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H11 d H16) in (let
+H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) u)))
+H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i H19 t2 t3 H1 t0
+H2)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl i
+c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t0) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H11:
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr
+x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C
+x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H12 d H17) in (let H20
+\def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) x4))) H10
+Abbr H16) in (ex2_ind T (\lambda (t5: T).(subst0 i x4 t3 t5)) (\lambda (t5:
+T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda
+(x: T).(\lambda (H21: (subst0 i x4 t3 x)).(\lambda (H22: (subst0 (S (plus
+(minus i0 (S i)) i)) u0 t0 x)).(let H23 \def (eq_ind_r nat (S (plus (minus i0
+(S i)) i)) (\lambda (n: nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0
+H6)) in (pc3_pr2_u c0 x t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0
+(pc3_pr2_x c0 x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c
+c0 u0 H4 (CHead e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H23)))))))
+(subst0_subst0_back t3 t0 u i H2 x4 u0 (minus i0 (S i)) H18)))))))) H14))
+H13))))))))))) H8)) H7))) (\lambda (H6: (le i0 i)).(pc3_pr2_r c0 t2 t0
+(pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr)
+u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t2
+t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i0 u0 c c0)).(\lambda (e:
+C).(\lambda (H6: (getl i0 c (CHead e (Bind Abbr) u0))).(lt_le_e i i0 (pc3 c0
+t5 t0) (\lambda (H7: (lt i i0)).(let H8 \def (csubst0_getl_lt i0 i H7 c c0 u0
+H5 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c0 (CHead d (Bind Abbr)
+u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w:
+T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i))
+u0 e1 e2))))))) (pc3 c0 t5 t0) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr)
+u))).(pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n
+i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0
+(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i H9 t2 t3 H1 t0 H2)))) (\lambda (H9:
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_:
+T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w:
+T).(subst0 (minus i0 (S i)) u0 u w))))))).(ex3_4_ind B C T T (\lambda (b:
+B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind
+x0) x3))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda
+(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr
+| (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H12 u H15)
+in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(getl i c0 (CHead c (Bind x0)
+x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0
+(CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda (t6: T).(subst0
+i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0
+t6)) (pc3 c0 t5 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x3 t3
+x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let H23
+\def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: nat).(subst0 n
+u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u2 c0 t2 t5
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
+(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_u c0 x t2
+(pr2_delta c0 d x3 i H20 t2 t3 H1 x H21) t0 (pc3_pr2_x c0 x t0 (pr2_delta c0
+e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
+H6) t0 t0 (pr0_refl t0) x H23)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0
+(minus i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C
+C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl i c0 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (pc3 c0 t5 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda
+(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr
+| (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x3 (\lambda (t: T).(getl i c0 (CHead x2 (Bind x0) t))) H11 u H15)
+in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0
+c x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0
+(CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e
+u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0)
+H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i
+H20 t2 t3 H1 t0 H2))))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex4_5 B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t5 t0)
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H12:
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: C).(match e0 return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H16
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in (\lambda (H17: (eq B Abbr
+x0)).(\lambda (H18: (eq C d x1)).(let H19 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H12 u H16) in (let H20 \def (eq_ind_r C
+x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H13 d H18) in (let H21
+\def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) x4))) H11
+Abbr H17) in (ex2_ind T (\lambda (t6: T).(subst0 i x4 t3 t6)) (\lambda (t6:
+T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t5 t0) (\lambda
+(x: T).(\lambda (H22: (subst0 i x4 t3 x)).(\lambda (H23: (subst0 (S (plus
+(minus i0 (S i)) i)) u0 t0 x)).(let H24 \def (eq_ind_r nat (S (plus (minus i0
+(S i)) i)) (\lambda (n: nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0
+H7)) in (pc3_pr2_u2 c0 t2 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0
+(le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4)
+t0 (pc3_pr2_u c0 x t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t0 (pc3_pr2_x
+c0 x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5
+(CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24))))))))
+(subst0_subst0_back t3 t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15))
+H14))))))))))) H9)) H8))) (\lambda (H7: (le i0 i)).(pc3_pr2_u2 c0 t2 t5
+(pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e
+(Bind Abbr) u0) H6) t2 t2 (pr0_refl t2) t5 H4) t0 (pc3_pr2_r c0 t2 t0
+(pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr)
+u) H0) t2 t3 H1 t0 H2))))))))))) c2 t4 H3)))))))))))))))) c1 t1 t H)))).
+
+theorem pc3_pr2_fsubst0_back:
+ \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
+c2 t t2)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t: T).(\lambda (t1: T).(\lambda (H: (pr2 c1 t
+t1)).(pr2_ind (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2 c2
+t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (pc3 c2 t0
+t3))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0:
+(pr0 t2 t3)).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t0:
+T).(\lambda (H1: (fsubst0 i u c t3 c2 t0)).(fsubst0_ind i u c t3 (\lambda
+(c0: C).(\lambda (t4: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u))
+\to (pc3 c0 t2 t4))))) (\lambda (t4: T).(\lambda (H2: (subst0 i u t3
+t4)).(\lambda (e: C).(\lambda (H3: (getl i c (CHead e (Bind Abbr)
+u))).(pc3_pr2_u c t3 t2 (pr2_free c t2 t3 H0) t4 (pc3_pr2_r c t3 t4
+(pr2_delta c e u i H3 t3 t3 (pr0_refl t3) t4 H2))))))) (\lambda (c0:
+C).(\lambda (_: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (_: (getl i c
+(CHead e (Bind Abbr) u))).(pc3_pr2_r c0 t2 t3 (pr2_free c0 t2 t3 H0))))))
+(\lambda (t4: T).(\lambda (H2: (subst0 i u t3 t4)).(\lambda (c0: C).(\lambda
+(H3: (csubst0 i u c c0)).(\lambda (e: C).(\lambda (H4: (getl i c (CHead e
+(Bind Abbr) u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H0) t4 (pc3_pr2_r c0
+t3 t4 (pr2_delta c0 e u i (csubst0_getl_ge i i (le_n i) c c0 u H3 (CHead e
+(Bind Abbr) u) H4) t3 t3 (pr0_refl t3) t4 H2))))))))) c2 t0 H1))))))))))
+(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (H1: (pr0 t2 t3)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t3
+t0)).(\lambda (i0: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H3: (fsubst0 i0 u0 c t0 c2 t4)).(fsubst0_ind i0 u0 c t0 (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i0 c (CHead e (Bind Abbr)
+u0)) \to (pc3 c0 t2 t5))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0
+t5)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(pc3_t t3 c t2 (pc3_pr3_r c t2 t3 (pr3_pr2 c t2 t3 (pr2_free c t2 t3
+H1))) t5 (pc3_pr3_r c t3 t5 (pr3_sing c t0 t3 (pr2_delta c d u i H0 t3 t3
+(pr0_refl t3) t0 H2) t5 (pr3_pr2 c t0 t5 (pr2_delta c e u0 i0 H5 t0 t0
+(pr0_refl t0) t5 H4))))))))) (\lambda (c0: C).(\lambda (H4: (csubst0 i0 u0 c
+c0)).(\lambda (e: C).(\lambda (H5: (getl i0 c (CHead e (Bind Abbr)
+u0))).(lt_le_e i i0 (pc3 c0 t2 t0) (\lambda (H6: (lt i i0)).(let H7 \def
+(csubst0_getl_lt i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr) u) H0) in (or4_ind
+(getl i c0 (CHead d (Bind Abbr) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))))) (pc3 c0 t2 t0) (\lambda (H8:
+(getl i c0 (CHead d (Bind Abbr) u))).(pc3_pr2_r c0 t2 t0 (pr2_delta c0 d u i
+H8 t2 t3 H1 t0 H2))) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda
+(u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b)
+u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))
+(pc3 c0 t2 t0) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H10: (getl i c0 (CHead x1 (Bind x0) x3))).(\lambda (H11:
+(subst0 (minus i0 (S i)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9)
+in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abbr
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x3)) H11 u H14) in (let H18 \def (eq_ind_r C
+x1 (\lambda (c: C).(getl i c0 (CHead c (Bind x0) x3))) H10 d H16) in (let H19
+\def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead d (Bind b) x3))) H18
+Abbr H15) in (ex2_ind T (\lambda (t5: T).(subst0 i x3 t3 t5)) (\lambda (t5:
+T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda
+(x: T).(\lambda (H20: (subst0 i x3 t3 x)).(\lambda (H21: (subst0 (S (plus
+(minus i0 (S i)) i)) u0 t0 x)).(let H22 \def (eq_ind_r nat (S (plus (minus i0
+(S i)) i)) (\lambda (n: nat).(subst0 n u0 t0 x)) H21 i0 (lt_plus_minus_r i i0
+H6)) in (pc3_pr2_u c0 x t2 (pr2_delta c0 d x3 i H19 t2 t3 H1 x H20) t0
+(pc3_pr2_x c0 x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c
+c0 u0 H4 (CHead e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H22)))))))
+(subst0_subst0_back t3 t0 u i H2 x3 u0 (minus i0 (S i)) H17)))))))) H13))
+H12))))))))) H8)) (\lambda (H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(getl i c0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl i c0
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2))))) (pc3 c0 t2 t0)
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10:
+(getl i c0 (CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t:
+T).(getl i c0 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r
+C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H11 d H16) in (let
+H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) u)))
+H17 Abbr H15) in (pc3_pr2_r c0 t2 t0 (pr2_delta c0 x2 u i H19 t2 t3 H1 t0
+H2)))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0))))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl i
+c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t0) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H10: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H11:
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr
+x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C
+x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H12 d H17) in (let H20
+\def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) x4))) H10
+Abbr H16) in (ex2_ind T (\lambda (t5: T).(subst0 i x4 t3 t5)) (\lambda (t5:
+T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0 t5)) (pc3 c0 t2 t0) (\lambda
+(x: T).(\lambda (H21: (subst0 i x4 t3 x)).(\lambda (H22: (subst0 (S (plus
+(minus i0 (S i)) i)) u0 t0 x)).(let H23 \def (eq_ind_r nat (S (plus (minus i0
+(S i)) i)) (\lambda (n: nat).(subst0 n u0 t0 x)) H22 i0 (lt_plus_minus_r i i0
+H6)) in (pc3_pr2_u c0 x t2 (pr2_delta c0 x2 x4 i H20 t2 t3 H1 x H21) t0
+(pc3_pr2_x c0 x t0 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c
+c0 u0 H4 (CHead e (Bind Abbr) u0) H5) t0 t0 (pr0_refl t0) x H23)))))))
+(subst0_subst0_back t3 t0 u i H2 x4 u0 (minus i0 (S i)) H18)))))))) H14))
+H13))))))))))) H8)) H7))) (\lambda (H6: (le i0 i)).(pc3_pr2_r c0 t2 t0
+(pr2_delta c0 d u i (csubst0_getl_ge i0 i H6 c c0 u0 H4 (CHead d (Bind Abbr)
+u) H0) t2 t3 H1 t0 H2)))))))) (\lambda (t5: T).(\lambda (H4: (subst0 i0 u0 t0
+t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i0 u0 c c0)).(\lambda (e:
+C).(\lambda (H6: (getl i0 c (CHead e (Bind Abbr) u0))).(lt_le_e i i0 (pc3 c0
+t2 t5) (\lambda (H7: (lt i i0)).(let H8 \def (csubst0_getl_lt i0 i H7 c c0 u0
+H5 (CHead d (Bind Abbr) u) H0) in (or4_ind (getl i c0 (CHead d (Bind Abbr)
+u)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(w: T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda
+(_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e2 (Bind b) w)))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w:
+T).(subst0 (minus i0 (S i)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (S i))
+u0 e1 e2))))))) (pc3 c0 t2 t5) (\lambda (H9: (getl i c0 (CHead d (Bind Abbr)
+u))).(pc3_pr2_u c0 t3 t2 (pr2_free c0 t2 t3 H1) t5 (pc3_pr3_r c0 t3 t5
+(pr3_sing c0 t0 t3 (pr2_delta c0 d u i H9 t3 t3 (pr0_refl t3) t0 H2) t5
+(pr3_pr2 c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c c0
+u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4)))))) (\lambda
+(H9: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u:
+T).(\lambda (w: T).(subst0 (minus i0 (S i)) u0 u w))))))).(ex3_4_ind B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl i c0 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i0 (S i)) u0 u1 w))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H11: (getl i c0 (CHead x1 (Bind
+x0) x3))).(\lambda (H12: (subst0 (minus i0 (S i)) u0 x2 x3)).(let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) H10) in ((let H14 \def (f_equal C B (\lambda
+(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr
+| (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H10) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i0 (S i)) u0 t x3)) H12 u H15)
+in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(getl i c0 (CHead c (Bind x0)
+x3))) H11 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0
+(CHead d (Bind b) x3))) H19 Abbr H16) in (ex2_ind T (\lambda (t6: T).(subst0
+i x3 t3 t6)) (\lambda (t6: T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0
+t6)) (pc3 c0 t2 t5) (\lambda (x: T).(\lambda (H21: (subst0 i x3 t3
+x)).(\lambda (H22: (subst0 (S (plus (minus i0 (S i)) i)) u0 t0 x)).(let H23
+\def (eq_ind_r nat (S (plus (minus i0 (S i)) i)) (\lambda (n: nat).(subst0 n
+u0 t0 x)) H22 i0 (lt_plus_minus_r i i0 H7)) in (pc3_pr2_u c0 x t2 (pr2_delta
+c0 d x3 i H20 t2 t3 H1 x H21) t5 (pc3_pr2_u2 c0 t0 x (pr2_delta c0 e u0 i0
+(csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0
+t0 (pr0_refl t0) x H23) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0
+(csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0
+t0 (pr0_refl t0) t5 H4)))))))) (subst0_subst0_back t3 t0 u i H2 x3 u0 (minus
+i0 (S i)) H18)))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex3_4 B C C T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (u: T).(getl i c0 (CHead e2 (Bind b) u))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0
+(minus i0 (S i)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
+C).(\lambda (u1: T).(getl i c0 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (S
+i)) u0 e1 e2))))) (pc3 c0 t2 t5) (\lambda (x0: B).(\lambda (x1: C).(\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0)
+x3))).(\lambda (H12: (csubst0 (minus i0 (S i)) u0 x1 x2)).(let H13 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H14 \def (f_equal C B (\lambda
+(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr
+| (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x3) H10) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in
+(\lambda (H16: (eq B Abbr x0)).(\lambda (H17: (eq C d x1)).(let H18 \def
+(eq_ind_r T x3 (\lambda (t: T).(getl i c0 (CHead x2 (Bind x0) t))) H11 u H15)
+in (let H19 \def (eq_ind_r C x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0
+c x2)) H12 d H17) in (let H20 \def (eq_ind_r B x0 (\lambda (b: B).(getl i c0
+(CHead x2 (Bind b) u))) H18 Abbr H16) in (pc3_pr2_u c0 t0 t2 (pr2_delta c0 x2
+u i H20 t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0
+(csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0
+t0 (pr0_refl t0) t5 H4))))))))) H14)) H13))))))))) H9)) (\lambda (H9: (ex4_5
+B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))))).(ex4_5_ind B C
+C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1)))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w:
+T).(getl i c0 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i0 (S i))
+u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
+T).(\lambda (_: T).(csubst0 (minus i0 (S i)) u0 e1 e2)))))) (pc3 c0 t2 t5)
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H10: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H11: (getl i c0 (CHead x2 (Bind x0) x4))).(\lambda (H12:
+(subst0 (minus i0 (S i)) u0 x3 x4)).(\lambda (H13: (csubst0 (minus i0 (S i))
+u0 x1 x2)).(let H14 \def (f_equal C C (\lambda (e0: C).(match e0 return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow
+c])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H15 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H10) in ((let H16
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H10) in (\lambda (H17: (eq B Abbr
+x0)).(\lambda (H18: (eq C d x1)).(let H19 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i0 (S i)) u0 t x4)) H12 u H16) in (let H20 \def (eq_ind_r C
+x1 (\lambda (c: C).(csubst0 (minus i0 (S i)) u0 c x2)) H13 d H18) in (let H21
+\def (eq_ind_r B x0 (\lambda (b: B).(getl i c0 (CHead x2 (Bind b) x4))) H11
+Abbr H17) in (ex2_ind T (\lambda (t6: T).(subst0 i x4 t3 t6)) (\lambda (t6:
+T).(subst0 (S (plus (minus i0 (S i)) i)) u0 t0 t6)) (pc3 c0 t2 t5) (\lambda
+(x: T).(\lambda (H22: (subst0 i x4 t3 x)).(\lambda (H23: (subst0 (S (plus
+(minus i0 (S i)) i)) u0 t0 x)).(let H24 \def (eq_ind_r nat (S (plus (minus i0
+(S i)) i)) (\lambda (n: nat).(subst0 n u0 t0 x)) H23 i0 (lt_plus_minus_r i i0
+H7)) in (pc3_pr2_u c0 x t2 (pr2_delta c0 x2 x4 i H21 t2 t3 H1 x H22) t5
+(pc3_pr2_u2 c0 t0 x (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c
+c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) x H24) t5
+(pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0 (csubst0_getl_ge i0 i0 (le_n i0) c
+c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0 t0 (pr0_refl t0) t5 H4))))))))
+(subst0_subst0_back t3 t0 u i H2 x4 u0 (minus i0 (S i)) H19)))))))) H15))
+H14))))))))))) H9)) H8))) (\lambda (H7: (le i0 i)).(pc3_pr2_u c0 t0 t2
+(pr2_delta c0 d u i (csubst0_getl_ge i0 i H7 c c0 u0 H5 (CHead d (Bind Abbr)
+u) H0) t2 t3 H1 t0 H2) t5 (pc3_pr2_r c0 t0 t5 (pr2_delta c0 e u0 i0
+(csubst0_getl_ge i0 i0 (le_n i0) c c0 u0 H5 (CHead e (Bind Abbr) u0) H6) t0
+t0 (pr0_refl t0) t5 H4))))))))))) c2 t4 H3)))))))))))))))) c1 t t1 H)))).
+
+theorem pc3_fsubst0:
+ \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1
+t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3
+c2 t2 t)))))))))))
+\def
+ \lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (pc3 c1 t1
+t)).(pc3_ind_left c1 (\lambda (t0: T).(\lambda (t2: T).(\forall (i:
+nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c1 t0 c2
+t3) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t3
+t2)))))))))) (\lambda (t0: T).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2:
+C).(\lambda (t2: T).(\lambda (H0: (fsubst0 i u c1 t0 c2 t2)).(fsubst0_ind i u
+c1 t0 (\lambda (c: C).(\lambda (t3: T).(\forall (e: C).((getl i c1 (CHead e
+(Bind Abbr) u)) \to (pc3 c t3 t0))))) (\lambda (t3: T).(\lambda (H1: (subst0
+i u t0 t3)).(\lambda (e: C).(\lambda (H2: (getl i c1 (CHead e (Bind Abbr)
+u))).(pc3_pr2_x c1 t3 t0 (pr2_delta c1 e u i H2 t0 t0 (pr0_refl t0) t3
+H1)))))) (\lambda (c0: C).(\lambda (_: (csubst0 i u c1 c0)).(\lambda (e:
+C).(\lambda (_: (getl i c1 (CHead e (Bind Abbr) u))).(pc3_refl c0 t0)))))
+(\lambda (t3: T).(\lambda (H1: (subst0 i u t0 t3)).(\lambda (c0: C).(\lambda
+(H2: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H3: (getl i c1 (CHead e
+(Bind Abbr) u))).(pc3_pr2_x c0 t3 t0 (pr2_delta c0 e u i (csubst0_getl_ge i i
+(le_n i) c1 c0 u H2 (CHead e (Bind Abbr) u) H3) t0 t0 (pr0_refl t0) t3
+H1)))))))) c2 t2 H0))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (H0:
+(pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda (H1: (pc3 c1 t2 t3)).(\lambda (H2:
+((\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t4:
+T).((fsubst0 i u c1 t2 c2 t4) \to (\forall (e: C).((getl i c1 (CHead e (Bind
+Abbr) u)) \to (pc3 c2 t4 t3)))))))))).(\lambda (i: nat).(\lambda (u:
+T).(\lambda (c2: C).(\lambda (t4: T).(\lambda (H3: (fsubst0 i u c1 t0 c2
+t4)).(fsubst0_ind i u c1 t0 (\lambda (c: C).(\lambda (t5: T).(\forall (e:
+C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c t5 t3))))) (\lambda (t5:
+T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda (e: C).(\lambda (H5: (getl i c1
+(CHead e (Bind Abbr) u))).(pc3_t t2 c1 t5 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c1
+t5 (fsubst0_snd i u c1 t0 t5 H4) e H5) t3 H1))))) (\lambda (c0: C).(\lambda
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e
+(Bind Abbr) u))).(pc3_t t2 c0 t0 (pc3_pr2_fsubst0 c1 t0 t2 H0 i u c0 t0
+(fsubst0_fst i u c1 t0 c0 H4) e H5) t3 (H2 i u c0 t2 (fsubst0_fst i u c1 t2
+c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t0 t5)).(\lambda
+(c0: C).(\lambda (H5: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H6:
+(getl i c1 (CHead e (Bind Abbr) u))).(pc3_t t2 c0 t5 (pc3_pr2_fsubst0 c1 t0
+t2 H0 i u c0 t5 (fsubst0_both i u c1 t0 t5 H4 c0 H5) e H6) t3 (H2 i u c0 t2
+(fsubst0_fst i u c1 t2 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) (\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H0: (pr2 c1 t0 t2)).(\lambda (t3: T).(\lambda
+(H1: (pc3 c1 t0 t3)).(\lambda (H2: ((\forall (i: nat).(\forall (u:
+T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t0 c2 t2) \to (\forall
+(e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t2
+t3)))))))))).(\lambda (i: nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H3: (fsubst0 i u c1 t2 c2 t4)).(fsubst0_ind i u c1 t2 (\lambda
+(c: C).(\lambda (t5: T).(\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u))
+\to (pc3 c t5 t3))))) (\lambda (t5: T).(\lambda (H4: (subst0 i u t2
+t5)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e (Bind Abbr)
+u))).(pc3_t t0 c1 t5 (pc3_s c1 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c1
+t5 (fsubst0_snd i u c1 t2 t5 H4) e H5)) t3 H1))))) (\lambda (c0: C).(\lambda
+(H4: (csubst0 i u c1 c0)).(\lambda (e: C).(\lambda (H5: (getl i c1 (CHead e
+(Bind Abbr) u))).(pc3_t t0 c0 t2 (pc3_s c0 t2 t0 (pc3_pr2_fsubst0_back c1 t0
+t2 H0 i u c0 t2 (fsubst0_fst i u c1 t2 c0 H4) e H5)) t3 (H2 i u c0 t0
+(fsubst0_fst i u c1 t0 c0 H4) e H5)))))) (\lambda (t5: T).(\lambda (H4:
+(subst0 i u t2 t5)).(\lambda (c0: C).(\lambda (H5: (csubst0 i u c1
+c0)).(\lambda (e: C).(\lambda (H6: (getl i c1 (CHead e (Bind Abbr)
+u))).(pc3_t t0 c0 t5 (pc3_s c0 t5 t0 (pc3_pr2_fsubst0_back c1 t0 t2 H0 i u c0
+t5 (fsubst0_both i u c1 t2 t5 H4 c0 H5) e H6)) t3 (H2 i u c0 t0 (fsubst0_fst
+i u c1 t0 c0 H5) e H6)))))))) c2 t4 H3)))))))))))) t1 t H)))).
+
+theorem pc3_gen_cabbr:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
+(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
+\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall
+(x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2))))))))))))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
+t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H0: (getl d
+c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H1: (csubst1 d u c
+a0)).(\lambda (a: C).(\lambda (H2: (drop (S O) d a0 a)).(\lambda (x1:
+T).(\lambda (H3: (subst1 d u t1 (lift (S O) d x1))).(\lambda (x2: T).(\lambda
+(H4: (subst1 d u t2 (lift (S O) d x2))).(let H5 \def H in (ex2_ind T (\lambda
+(t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) (pc3 a x1 x2) (\lambda (x:
+T).(\lambda (H6: (pr3 c t1 x)).(\lambda (H7: (pr3 c t2 x)).(ex2_ind T
+(\lambda (x3: T).(subst1 d u x (lift (S O) d x3))) (\lambda (x3: T).(pr3 a x2
+x3)) (pc3 a x1 x2) (\lambda (x0: T).(\lambda (H8: (subst1 d u x (lift (S O) d
+x0))).(\lambda (H9: (pr3 a x2 x0)).(ex2_ind T (\lambda (x3: T).(subst1 d u x
+(lift (S O) d x3))) (\lambda (x3: T).(pr3 a x1 x3)) (pc3 a x1 x2) (\lambda
+(x3: T).(\lambda (H10: (subst1 d u x (lift (S O) d x3))).(\lambda (H11: (pr3
+a x1 x3)).(let H12 \def (eq_ind T x3 (\lambda (t: T).(pr3 a x1 t)) H11 x0
+(subst1_confluence_lift x x3 u d H10 x0 H8)) in (pc3_pr3_t a x1 x0 H12 x2
+H9))))) (pr3_gen_cabbr c t1 x H6 e u d H0 a0 H1 a H2 x1 H3)))))
+(pr3_gen_cabbr c t2 x H7 e u d H0 a0 H1 a H2 x2 H4))))) H5))))))))))))))))).
-axiom csubst0_getl_ge_back: \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 e) \to (getl n c1 e))))))))) .
-
-inductive csubst1 (i:nat) (v:T) (c1:C): C \to Prop \def
-| csubst1_refl: csubst1 i v c1 c1
-| csubst1_sing: \forall (c2: C).((csubst0 i v c1 c2) \to (csubst1 i v c1 c2)).
-
-axiom csubst1_head: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (s k i) v (CHead c1 k u1) (CHead c2 k u2)))))))))) .
-
-axiom csubst1_bind: \forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 (S i) v (CHead c1 (Bind b) u1) (CHead c2 (Bind b) u2)))))))))) .
-
-axiom csubst1_flat: \forall (f: F).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall (u2: T).((subst1 i v u1 u2) \to (\forall (c1: C).(\forall (c2: C).((csubst1 i v c1 c2) \to (csubst1 i v (CHead c1 (Flat f) u1) (CHead c2 (Flat f) u2)))))))))) .
-
-axiom csubst1_gen_head: \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall (v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2)))))))))) .
-
-axiom csubst1_getl_ge: \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 e) \to (getl n c2 e))))))))) .
-
-axiom csubst1_getl_lt: \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e1: C).((getl n c1 e1) \to (ex2 C (\lambda (e2: C).(csubst1 (minus i n) v e1 e2)) (\lambda (e2: C).(getl n c2 e2))))))))))) .
-
-axiom csubst1_getl_ge_back: \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c2 e) \to (getl n c1 e))))))))) .
-
-axiom getl_csubst1: \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 a)))))))) .
-
-inductive fsubst0 (i:nat) (v:T) (c1:C) (t1:T): C \to (T \to Prop) \def
-| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 t2))
-| fsubst0_fst: \forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t1))
-| fsubst0_both: \forall (t2: T).((subst0 i v t1 t2) \to (\forall (c2: C).((csubst0 i v c1 c2) \to (fsubst0 i v c1 t1 c2 t2)))).
-
-axiom fsubst0_gen_base: \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall (v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2))))))))) .
-
-record G : Set \def {
- next: (nat \to nat);
- next_lt: (\forall (n: nat).(lt n (next n)))
-}.
-
-definition next_plus: G \to (nat \to (nat \to nat)) \def let rec next_plus (g: G) (n: nat) (i: nat): nat \def (match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus.
-
-axiom next_plus_assoc: \forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq nat (next_plus g (next_plus g n h1) h2) (next_plus g n (plus h1 h2)))))) .
-
-axiom next_plus_next: \forall (g: G).(\forall (n: nat).(\forall (h: nat).(eq nat (next_plus g (next g n) h) (next g (next_plus g n h))))) .
-
-axiom next_plus_lt: \forall (g: G).(\forall (h: nat).(\forall (n: nat).(lt n (next_plus g (next g n) h)))) .
-
-inductive tau0 (g:G): C \to (T \to (T \to Prop)) \def
-| tau0_sort: \forall (c: C).(\forall (n: nat).(tau0 g c (TSort n) (TSort (next g n))))
-| tau0_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau0 g d v w) \to (tau0 g c (TLRef i) (lift (S i) O w))))))))
-| tau0_abst: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) v)) \to (\forall (w: T).((tau0 g d v w) \to (tau0 g c (TLRef i) (lift (S i) O v))))))))
-| tau0_bind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((tau0 g (CHead c (Bind b) v) t1 t2) \to (tau0 g c (THead (Bind b) v t1) (THead (Bind b) v t2)))))))
-| tau0_appl: \forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((tau0 g c t1 t2) \to (tau0 g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t2))))))
-| tau0_cast: \forall (c: C).(\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to (\forall (t1: T).(\forall (t2: T).((tau0 g c t1 t2) \to (tau0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t2)))))))).
-
-axiom tau0_lift: \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (tau0 g c (lift h d t1) (lift h d t2)))))))))) .
-
-axiom tau0_correct: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2))))))) .
-
-inductive tau1 (g:G) (c:C) (t1:T): T \to Prop \def
-| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
-| tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c t t2) \to (tau1 g c t1 t2)))).
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-axiom tau1_trans: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau1 g c t t2) \to (tau1 g c t1 t2))))))) .
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-axiom tau1_bind: \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((tau1 g (CHead c (Bind b) v) t1 t2) \to (tau1 g c (THead (Bind b) v t1) (THead (Bind b) v t2)))))))) .
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-axiom tau1_appl: \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((tau1 g c t1 t2) \to (tau1 g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t2))))))) .
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-axiom tau1_lift: \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau1 g e t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (tau1 g c (lift h d t1) (lift h d t2)))))))))) .
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-axiom tau1_correct: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2))))))) .
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-axiom tau1_abbr: \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau1 g d v w) \to (tau1 g c (TLRef i) (lift (S i) O w))))))))) .
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-axiom tau1_cast2: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((tau1 g c t1 t2) \to (\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t2))))))))))) .
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-axiom tau1_cnt: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c t1 t) \to (ex2 T (\lambda (t2: T).(tau1 g c t1 t2)) (\lambda (t2: T).(cnt t2))))))) .
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-inductive A: Set \def
-| ASort: nat \to (nat \to A)
-| AHead: A \to (A \to A).
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-definition asucc: G \to (A \to A) \def let rec asucc (g: G) (l: A): A \def (match l with [(ASort n0 n) \Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) \Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g a2))]) in asucc.
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-definition aplus: G \to (A \to (nat \to A)) \def let rec aplus (g: G) (a: A) (n: nat): A \def (match n with [O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus.
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-inductive leq (g:G): A \to (A \to Prop) \def
-| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).(\forall (k: nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort h1 n1) (ASort h2 n2)))))))
-| leq_head: \forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leq g a3 a4) \to (leq g (AHead a1 a3) (AHead a2 a4))))))).
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-axiom leq_gen_sort: \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)))))))))) .
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-axiom leq_gen_head: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g (AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))))) .
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-axiom asucc_gen_sort: \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A (ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a (ASort h0 n0))))))))) .
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-axiom asucc_gen_head: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A (AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))))) .
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-axiom aplus_reg_r: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall (h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A (aplus g a1 (plus h h1)) (aplus g a2 (plus h h2))))))))) .
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-axiom aplus_assoc: \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A (aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) .
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-axiom aplus_asucc: \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) h) (asucc g (aplus g a h))))) .
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-axiom aplus_sort_O_S_simpl: \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O n) (S k)) (aplus g (ASort O (next g n)) k)))) .
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-axiom aplus_sort_S_S_simpl: \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A (aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) .
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-axiom asucc_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g (asucc g a1) (asucc g a2))))) .
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-axiom asucc_inj: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc g a2)) \to (leq g a1 a2)))) .
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-axiom aplus_asort_O_simpl: \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O n) h) (ASort O (next_plus g n h))))) .
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-axiom aplus_asort_le_simpl: \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) .
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-axiom aplus_asort_simpl: \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) .
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-axiom aplus_ahead_simpl: \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) .
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-axiom aplus_asucc_false: \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) h) a) \to (\forall (P: Prop).P)))) .
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-axiom aplus_inj: \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) .
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-axiom ahead_inj_snd: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) .
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-axiom leq_refl: \forall (g: G).(\forall (a: A).(leq g a a)) .
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-axiom leq_eq: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 a2)))) .
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-axiom leq_asucc: \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g a0))))) .
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-axiom leq_sym: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g a2 a1)))) .
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-axiom leq_trans: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) .
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-axiom leq_ahead_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) \to (\forall (P: Prop).P)))) .
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-axiom leq_ahead_asucc_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) (asucc g a1)) \to (\forall (P: Prop).P)))) .
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-axiom leq_asucc_false: \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: Prop).P))) .
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-definition lweight: A \to nat \def let rec lweight (a: A): nat \def (match a with [(ASort _ _) \Rightarrow O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in lweight.
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-definition llt: A \to (A \to Prop) \def \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
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-axiom lweight_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat (lweight a1) (lweight a2))))) .
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-axiom llt_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((llt a1 a3) \to (llt a2 a3)))))) .
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-axiom llt_trans: \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 a3) \to (llt a1 a3))))) .
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-axiom llt_head_sx: \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) .
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-axiom llt_head_dx: \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) .
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-axiom llt_wf__q_ind: \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P: ((A \to Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P a))))) P n))) \to (\forall (a: A).(P a))) .
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-axiom llt_wf_ind: \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) .
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-inductive aprem: nat \to (A \to (A \to Prop)) \def
-| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
-| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
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-axiom aprem_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) .
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-axiom aprem_asucc: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i a1 a2) \to (aprem i (asucc g a1) a2))))) .
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-definition gz: G \def Build_G S lt_n_Sn.
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-inductive leqz: A \to (A \to Prop) \def
-| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort h2 n2))))))
-| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
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-axiom aplus_gz_le: \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A (aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) .
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-axiom aplus_gz_ge: \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A (aplus gz (ASort h n) k) (ASort (minus h k) n))))) .
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-axiom next_plus_gz: \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) .
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-axiom leqz_leq: \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) .
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-axiom leq_leqz: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) .
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-inductive arity (g:G): C \to (T \to (A \to Prop)) \def
-| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O n)))
-| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to (arity g c (TLRef i) a)))))))
-| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u (asucc g a)) \to (arity g c (TLRef i) a)))))))
-| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c (THead (Bind b) u t) a2)))))))))
-| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
-| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to (arity g c (THead (Flat Appl) u t) a2)))))))
-| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat Cast) u t) a))))))
-| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
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-axiom arity_gen_sort: \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c (TSort n) a) \to (leq g a (ASort O n)))))) .
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-axiom arity_gen_lref: \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c (TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))))) .
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-axiom arity_gen_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))))))))) .
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-axiom arity_gen_abst: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c (Bind Abst) u) t a2))))))))) .
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-axiom arity_gen_appl: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2))))))))) .
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-axiom arity_gen_cast: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) (arity g c t a))))))) .
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-axiom arity_gen_appls: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))))) .
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-axiom node_inh: \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort k n))))))) .
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-axiom arity_gen_cvoid_subst0: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to (\forall (P: Prop).P)))))))))))) .
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-axiom arity_gen_cvoid: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))))))))) .
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-axiom arity_gen_lift: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t a))))))))) .
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-axiom arity_lift: \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (arity g c1 (lift h d t) a))))))))) .
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-axiom arity_lift1: \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 hds t) a)))))))) .
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-axiom arity_mono: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2))))))) .
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-axiom arity_cimp_conf: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a))))))) .
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-axiom arity_aprem: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))))) .
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-axiom arity_appls_cast: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) a)))))))) .
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-axiom arity_appls_abbr: \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) vs (TLRef i)) a))))))))) .
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-axiom arity_appls_bind: \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) a2))))))))))) .
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-axiom arity_fsubst0: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t1 c2 t2) \to (arity g c2 t2 a)))))))))))) .
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-axiom arity_subst0: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 a))))))))))) .
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-inductive pr0: T \to (T \to Prop) \def
-| pr0_refl: \forall (t: T).(pr0 t t)
-| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) (THead k u2 t2))))))))
-| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2))))))))
-| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))))))))))
-| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to (pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
-| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O t1)) t2))))))
-| pr0_epsilon: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Flat Cast) u t1) t2)))).
-
-axiom pr0_gen_sort: \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n)))) .
-
-axiom pr0_gen_lref: \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) .
-
-axiom pr0_gen_abst: \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) .
-
-axiom pr0_gen_appl: \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) .
-
-axiom pr0_gen_cast: \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))) .
-
-axiom pr0_lift: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t1) (lift h d t2)))))) .
-
-axiom pr0_gen_abbr: \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) .
-
-axiom pr0_gen_void: \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) .
-
-axiom pr0_gen_lift: \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 (lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr0 t1 t2))))))) .
-
-axiom pr0_subst0_back: \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) .
-
-axiom pr0_subst0_fwd: \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) .
-
-axiom pr0_subst0: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)))))))))))) .
-
-axiom pr0_confluence__pr0_cong_upsilon_refl: \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))) .
-
-axiom pr0_confluence__pr0_cong_upsilon_cong: \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall (t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))))))))))))))))) .
-
-axiom pr0_confluence__pr0_cong_upsilon_delta: (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))))))))))))))))) .
-
-axiom pr0_confluence__pr0_cong_upsilon_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t))))))))))))))))) .
-
-axiom pr0_confluence__pr0_cong_delta: \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to (\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)))))))))))))) .
-
-axiom pr0_confluence__pr0_upsilon_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to (\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) t))))))))))))))))))) .
-
-axiom pr0_confluence__pr0_delta_delta: \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to (\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to (\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)))))))))))))))) .
-
-axiom pr0_confluence__pr0_delta_epsilon: \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to (\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))))))))) .
-
-axiom pr0_confluence: \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) .
-
-axiom pr0_delta1: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) .
-
-axiom pr0_subst1_back: \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) .
-
-axiom pr0_subst1_fwd: \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) .
-
-axiom pr0_subst1: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall (w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))))))))))) .
-
-axiom nf0_dec: \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))) .
-
-inductive pr1: T \to (T \to Prop) \def
-| pr1_r: \forall (t: T).(pr1 t t)
-| pr1_u: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pr1 t2 t3) \to (pr1 t1 t3))))).
-
-axiom pr1_pr0: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) .
-
-axiom pr1_t: \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 t3) \to (pr1 t1 t3))))) .
-
-axiom pr1_head_1: \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall (k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) .
-
-axiom pr1_head_2: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall (k: K).(pr1 (THead k u t1) (THead k u t2)))))) .
-
-axiom pr1_strip: \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) .
-
-axiom pr1_confluence: \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) .
-
-inductive wcpr0: C \to (C \to Prop) \def
-| wcpr0_refl: \forall (c: C).(wcpr0 c c)
-| wcpr0_comp: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (k: K).(wcpr0 (CHead c1 k u1) (CHead c2 k u2)))))))).
-
-axiom wcpr0_gen_sort: \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort n)))) .
-
-axiom wcpr0_gen_head: \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 (CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) .
-
-axiom wcpr0_drop: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) .
-
-axiom wcpr0_drop_back: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) .
-
-axiom wcpr0_getl: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) .
-
-axiom wcpr0_getl_back: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) .
-
-inductive pr2: C \to (T \to (T \to Prop)) \def
-| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr2 c t1 t2))))
-| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 t)))))))))).
-
-axiom pr2_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to (eq T x (TSort n))))) .
-
-axiom pr2_gen_lref: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))) .
-
-axiom pr2_gen_abst: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))) .
-
-axiom pr2_gen_cast: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x)))))) .
-
-axiom pr2_gen_csort: \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) \to (pr0 t1 t2)))) .
-
-axiom pr2_gen_ctail: \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen c) u t t2))))))))) .
-
-axiom pr2_thin_dx: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) .
-
-axiom pr2_head_1: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall (k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) .
-
-axiom pr2_head_2: \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u t2))))))) .
-
-axiom clear_pr2_trans: \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to (\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) .
-
-axiom pr2_cflat: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) .
-
-axiom pr2_ctail: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) .
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-axiom pr2_gen_cbind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))) .
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-axiom pr2_gen_cflat: \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) .
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-axiom pr2_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) .
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-axiom pr2_gen_appl: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) .
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-axiom pr2_gen_abbr: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) .
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-axiom pr2_gen_void: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) .
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-axiom pr2_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) .
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-axiom pr2_confluence__pr2_free_free: \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))) .
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-axiom pr2_confluence__pr2_free_delta: \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))))))))) .
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-axiom pr2_confluence__pr2_delta_delta: \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))))))))))))))) .
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-axiom pr2_confluence: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))) .
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-axiom pr2_delta1: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) .
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-axiom pr2_subst1: \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) .
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-axiom pr2_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)))))))))))))))) .
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-inductive pr3 (c:C): T \to (T \to Prop) \def
-| pr3_refl: \forall (t: T).(pr3 c t t)
-| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
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-axiom pr3_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to (eq T x (TSort n))))) .
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-axiom pr3_gen_abst: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) .
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-axiom pr3_gen_cast: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c t1 x)))))) .
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-axiom clear_pr3_trans: \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to (\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2)))))) .
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-axiom pr3_pr2: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c t1 t2)))) .
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-axiom pr3_t: \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall (t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3)))))) .
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-axiom pr3_thin_dx: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) .
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-axiom pr3_head_1: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t))))))) .
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-axiom pr3_head_2: \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u t2))))))) .
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-axiom pr3_head_21: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 c (THead k u1 t1) (THead k u2 t2))))))))) .
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-axiom pr3_head_12: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 c (THead k u1 t1) (THead k u2 t2))))))))) .
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-axiom pr3_pr1: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 t2)))) .
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-axiom pr3_cflat: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2)))))) .
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-axiom pr3_pr0_pr2_t: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_pr2_pr2_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_pr2_pr3_t: \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_pr3_pr3_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift h d t1) (lift h d t2))))))))) .
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-axiom pr3_wcpr0_t: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) .
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-axiom pr3_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 t2)))))))))) .
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-axiom pr3_gen_lref: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to (or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T x (lift (S n) O v)))))))))) .
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-axiom pr3_gen_void: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))))))) .
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-axiom pr3_gen_abbr: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) .
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-axiom pr3_gen_appl: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) .
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-axiom pr3_strip: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)))))))) .
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-axiom pr3_confluence: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall (t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)))))))) .
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-axiom pr3_subst1: \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) .
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-axiom pr3_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))) .
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-axiom pr3_iso_appls_cast: \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs t) u2)))))))) .
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-inductive csuba (g:G): C \to (C \to Prop) \def
-| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
-| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
-| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))))).
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-axiom csuba_gen_abbr: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g (CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) .
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-axiom csuba_gen_void: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g (CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) .
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-axiom csuba_gen_abst: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g (CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) .
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-axiom csuba_gen_flat: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall (f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) .
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-axiom csuba_gen_bind: \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) .
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-axiom csuba_refl: \forall (g: G).(\forall (c: C).(csuba g c c)) .
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-axiom csuba_clear_conf: \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c2 e2)))))))) .
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-axiom csuba_drop_abbr: \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) .
-
-axiom csuba_drop_abst: \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))))) .
-
-axiom csuba_getl_abbr: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) .
-
-axiom csuba_getl_abst: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))))) .
-
-axiom csuba_arity: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a))))))) .
-
-axiom csuba_arity_rev: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a))))))) .
-
-axiom arity_appls_appl: \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (u: T).((arity g c u (asucc g a1)) \to (\forall (t: T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) a2) \to (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) u t))) a2))))))))))) .
-
-axiom arity_sred_wcpr0_pr0: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) \to (arity g c2 t2 a))))))))) .
-
-axiom arity_sred_wcpr0_pr1: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall (c1: C).(\forall (a: A).((arity g c1 t1 a) \to (\forall (c2: C).((wcpr0 c1 c2) \to (arity g c2 t2 a))))))))) .
-
-axiom arity_sred_pr2: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) .
-
-axiom arity_sred_pr3: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (g: G).(\forall (a: A).((arity g c t1 a) \to (arity g c t2 a))))))) .
-
-definition nf2: C \to (T \to Prop) \def \lambda (c: C).(\lambda (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (eq T t1 t2)))).
-
-axiom nf2_gen_base__aux: \forall (k: K).(\forall (t: T).(\forall (u: T).((eq T (THead k u t) t) \to (\forall (P: Prop).P)))) .
-
-axiom nf2_gen_lref: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P)))))) .
-
-axiom nf2_gen_abst: \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t))))) .
-
-axiom nf2_gen_cast: \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u t)) \to (\forall (P: Prop).P)))) .
-
-axiom nf2_gen_flat: \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t)))))) .
-
-axiom nf2_sort: \forall (c: C).(\forall (n: nat).(nf2 c (TSort n))) .
-
-axiom nf2_abst: \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v: T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind Abst) u t)))))))) .
-
-axiom nf2_pr3_unfold: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c t1) \to (eq T t1 t2))))) .
-
-axiom nf2_pr3_confluence: \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) \to (\forall (t: T).((pr3 c t t1) \to ((pr3 c t t2) \to (eq T t1 t2)))))))) .
-
-axiom nf2_appl_lref: \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c (TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i))))))) .
-
-axiom nf2_lref_abst: \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e (Bind Abst) u)) \to (nf2 c (TLRef i)))))) .
-
-axiom nf2_lift: \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t)))))))) .
-
-axiom nf2_lift1: \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t))))))) .
-
-axiom nf2_iso_appls_lref: \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso (THeads (Flat Appl) vs (TLRef i)) u)))))) .
-
-axiom nf2_dec: \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2))))) .
-
-inductive sn3 (c:C): T \to Prop \def
-| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
-
-definition sns3: C \to (TList \to Prop) \def let rec sns3 (c: C) (ts: TList): Prop \def (match ts with [TNil \Rightarrow True | (TCons t ts0) \Rightarrow (land (sn3 c t) (sns3 c ts0))]) in sns3.
-
-axiom sn3_gen_flat: \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c (THead (Flat f) u t)) \to (land (sn3 c u) (sn3 c t)))))) .
-
-axiom sn3_nf2: \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t))) .
-
-axiom sn3_pr3_trans: \forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (sn3 c t2))))) .
-
-axiom sn3_pr2_intro: \forall (c: C).(\forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr2 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1))) .
-
-axiom sn3_cast: \forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t: T).((sn3 c t) \to (sn3 c (THead (Flat Cast) u t)))))) .
-
-axiom nf2_sn3: \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c t u)) (\lambda (u: T).(nf2 c u))))) .
-
-axiom sn3_appl_lref: \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (v: T).((sn3 c v) \to (sn3 c (THead (Flat Appl) v (TLRef i))))))) .
-
-axiom sn3_appl_cast: \forall (c: C).(\forall (v: T).(\forall (u: T).((sn3 c (THead (Flat Appl) v u)) \to (\forall (t: T).((sn3 c (THead (Flat Appl) v t)) \to (sn3 c (THead (Flat Appl) v (THead (Flat Cast) u t)))))))) .
-
-axiom sn3_appl_appl: \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 u1))))))))) .
-
-axiom sn3_appl_appls: \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 u1)))))))))) .
-
-axiom sn3_appls_lref: \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (us: TList).((sns3 c us) \to (sn3 c (THeads (Flat Appl) us (TLRef i))))))) .
-
-axiom sn3_appls_cast: \forall (c: C).(\forall (vs: TList).(\forall (u: T).((sn3 c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sn3 c (THeads (Flat Appl) vs t)) \to (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))) .
-
-axiom sn3_lift: \forall (d: C).(\forall (t: T).((sn3 d t) \to (\forall (c: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) \to (sn3 c (lift h i t)))))))) .
-
-axiom sn3_abbr: \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 d v) \to (sn3 c (TLRef i))))))) .
-
-axiom sns3_lifts: \forall (c: C).(\forall (d: C).(\forall (h: nat).(\forall (i: nat).((drop h i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts)))))))) .
-
-axiom sns3_lifts1: \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts))))))) .
-
-axiom sn3_gen_lift: \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t))))))) .
-
-definition sc3: G \to (A \to (C \to (T \to Prop))) \def let rec sc3 (g: G) (a: A): (C \to (T \to Prop)) \def (\lambda (c: C).(\lambda (t: T).(match a with [(ASort h n) \Rightarrow (land (arity g c t (ASort h n)) (sn3 c t)) | (AHead a1 a2) \Rightarrow (land (arity g c t (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is t)))))))))]))) in sc3.
-
-axiom sc3_arity_gen: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c t) \to (arity g c t a))))) .
-
-axiom sc3_repl: \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))) .
-
-axiom sc3_lift: \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a c (lift h d t)))))))))) .
-
-axiom sc3_lift1: \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) \to (sc3 g a c (lift1 hds t))))))))) .
-
-axiom sc3_abbr: \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))) .
-
-axiom sc3_cast: \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) .
-
-axiom sc3_bind: \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead (Bind b) v t))))))))))))) .
-
-axiom sc3_appl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))) .
-
-axiom sc3_props__sc3_sn3_abst: \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))) .
-
-axiom sc3_sn3: \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c t) \to (sn3 c t))))) .
-
-axiom sc3_abst: \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall (i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) .
-
-inductive csubc (g:G): C \to (C \to Prop) \def
-| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
-| csubc_head: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall (k: K).(\forall (v: T).(csubc g (CHead c1 k v) (CHead c2 k v))))))
-| csubc_abst: \forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (\forall (v: T).(\forall (a: A).((sc3 g (asucc g a) c1 v) \to (\forall (w: T).((sc3 g a c2 w) \to (csubc g (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) w))))))))).
-
-definition ceqc: G \to (C \to (C \to Prop)) \def \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(or (csubc g c1 c2) (csubc g c2 c1)))).
-
-axiom scubc_refl: \forall (g: G).(\forall (c: C).(csubc g c c)) .
-
-axiom ceqc_sym: \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c1 c2) \to (ceqc g c2 c1)))) .
-
-axiom drop_csubc_trans: \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) .
-
-axiom csubc_drop_conf_rev: \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) .
-
-axiom drop1_csubc_trans: \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) .
-
-axiom csubc_drop1_conf_rev: \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) .
-
-axiom drop1_ceqc_trans: \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: C).((drop1 hds c2 e2) \to (\forall (e1: C).((ceqc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1))))))))) .
-
-axiom sc3_ceqc_trans: \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c1: C).(\forall (t: T).((sc3 g a c1 (THeads (Flat Appl) vs t)) \to (\forall (c2: C).((ceqc g c2 c1) \to (sc3 g a c2 (THeads (Flat Appl) vs t))))))))) .
-
-axiom sc3_arity: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (sc3 g a c t))))) .
-
-definition pc1: T \to (T \to Prop) \def \lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)))).
-
-axiom pc1_pr0_r: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2))) .
-
-axiom pc1_pr0_x: \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2))) .
-
-axiom pc1_pr0_u: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 t3) \to (pc1 t1 t3))))) .
-
-axiom pc1_refl: \forall (t: T).(pc1 t t) .
-
-axiom pc1_s: \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1))) .
-
-axiom pc1_head_1: \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall (k: K).(pc1 (THead k u1 t) (THead k u2 t)))))) .
-
-axiom pc1_head_2: \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall (k: K).(pc1 (THead k u t1) (THead k u t2)))))) .
-
-axiom pc1_t: \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 t3) \to (pc1 t1 t3))))) .
-
-axiom pc1_pr0_u2: \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 t2) \to (pc1 t1 t2))))) .
-
-axiom pc1_head: \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (k: K).(pc1 (THead k u1 t1) (THead k u2 t2)))))))) .
-
-definition pc3: C \to (T \to (T \to Prop)) \def \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t))))).
-
-axiom clear_pc3_trans: \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to (\forall (c1: C).((clear c1 c2) \to (pc3 c1 t1 t2)))))) .
-
-axiom pc3_pr2_r: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pc3 c t1 t2)))) .
-
-axiom pc3_pr2_x: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t2 t1) \to (pc3 c t1 t2)))) .
-
-axiom pc3_pr3_r: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3 c t1 t2)))) .
-
-axiom pc3_pr3_x: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t2 t1) \to (pc3 c t1 t2)))) .
-
-axiom pc3_pr3_t: \forall (c: C).(\forall (t1: T).(\forall (t0: T).((pr3 c t1 t0) \to (\forall (t2: T).((pr3 c t2 t0) \to (pc3 c t1 t2)))))) .
-
-axiom pc3_pr2_u: \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) .
-
-axiom pc3_refl: \forall (c: C).(\forall (t: T).(pc3 c t t)) .
-
-axiom pc3_s: \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pc3 c t1 t2) \to (pc3 c t2 t1)))) .
-
-axiom pc3_thin_dx: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall (u: T).(\forall (f: F).(pc3 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) .
-
-axiom pc3_head_1: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall (k: K).(\forall (t: T).(pc3 c (THead k u1 t) (THead k u2 t))))))) .
-
-axiom pc3_head_2: \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u) t1 t2) \to (pc3 c (THead k u t1) (THead k u t2))))))) .
-
-axiom pc3_pc1: \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1 t2)))) .
-
-axiom pc3_t: \forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall (t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3)))))) .
-
-axiom pc3_pr2_u2: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall (t2: T).((pc3 c t0 t2) \to (pc3 c t1 t2)))))) .
-
-axiom pc3_head_12: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u2) t1 t2) \to (pc3 c (THead k u1 t1) (THead k u2 t2))))))))) .
-
-axiom pc3_head_21: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pc3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pc3 (CHead c k u1) t1 t2) \to (pc3 c (THead k u1 t1) (THead k u2 t2))))))))) .
-
-axiom pc3_pr0_pr2_t: \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))) .
-
-axiom pc3_pr2_pr2_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))) .
-
-axiom pc3_pr2_pr3_t: \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to (pc3 (CHead c k u1) t1 t2)))))))) .
-
-axiom pc3_pr3_pc3_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u2 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pc3 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))) .
-
-axiom pc3_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (\forall (t1: T).(\forall (t2: T).((pc3 e t1 t2) \to (pc3 c (lift h d t1) (lift h d t2))))))))) .
-
-axiom pc3_wcpr0__pc3_wcpr0_t_aux: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 (CHead c2 k u) t1 t2)))))))) .
-
-axiom pc3_wcpr0_t: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) .
-
-axiom pc3_wcpr0: \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: T).(\forall (t2: T).((pc3 c1 t1 t2) \to (pc3 c2 t1 t2)))))) .
-
-inductive pc3_left (c:C): T \to (T \to Prop) \def
-| pc3_left_r: \forall (t: T).(pc3_left c t t)
-| pc3_left_ur: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))
-| pc3_left_ux: \forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3_left c t1 t3) \to (pc3_left c t2 t3))))).
-
-axiom pc3_ind_left__pc3_left_pr3: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (pc3_left c t1 t2)))) .
-
-axiom pc3_ind_left__pc3_left_trans: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to (\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3)))))) .
-
-axiom pc3_ind_left__pc3_left_sym: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to (pc3_left c t2 t1)))) .
-
-axiom pc3_ind_left__pc3_left_pc3: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (pc3_left c t1 t2)))) .
-
-axiom pc3_ind_left__pc3_pc3_left: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to (pc3 c t1 t2)))) .
-
-axiom pc3_ind_left: \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c t t0) \to (P t t0)))))))) .
-
-axiom pc3_gen_sort: \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort n)) \to (eq nat m n)))) .
-
-axiom pc3_gen_abst: \forall (c: C).(\forall (u1: T).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).((pc3 c (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)) \to (land (pc3 c u1 u2) (\forall (b: B).(\forall (u: T).(pc3 (CHead c (Bind b) u) t1 t2))))))))) .
-
-axiom pc3_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (h: nat).(\forall (d: nat).((pc3 c (lift h d t1) (lift h d t2)) \to (\forall (e: C).((drop h d c e) \to (pc3 e t1 t2)))))))) .
-
-axiom pc3_gen_not_abst: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (u1: T).(\forall (u2: T).((pc3 c (THead (Bind b) u1 t1) (THead (Bind Abst) u2 t2)) \to (pc3 (CHead c (Bind b) u1) t1 (lift (S O) O (THead (Bind Abst) u2 t2)))))))))) .
-
-axiom pc3_gen_lift_abst: \forall (c: C).(\forall (t: T).(\forall (t2: T).(\forall (u2: T).(\forall (h: nat).(\forall (d: nat).((pc3 c (lift h d t) (THead (Bind Abst) u2 t2)) \to (\forall (e: C).((drop h d c e) \to (ex3_2 T T (\lambda (u1: T).(\lambda (t1: T).(pr3 e t (THead (Bind Abst) u1 t1)))) (\lambda (u1: T).(\lambda (_: T).(pr3 c u2 (lift h d u1)))) (\lambda (_: T).(\lambda (t1: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t2 (lift h (S d) t1))))))))))))))) .
-
-axiom pc3_pr2_fsubst0: \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t2 t))))))))))) .
-
-axiom pc3_pr2_fsubst0_back: \forall (c1: C).(\forall (t: T).(\forall (t1: T).((pr2 c1 t t1) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t t2))))))))))) .
-
-axiom pc3_fsubst0: \forall (c1: C).(\forall (t1: T).(\forall (t: T).((pc3 c1 t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (pc3 c2 t2 t))))))))))) .
-
-axiom pc3_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (\forall (x2: T).((subst1 d u t2 (lift (S O) d x2)) \to (pc3 a x1 x2)))))))))))))))) .
-
-inductive ty3 (g:G): C \to (T \to (T \to Prop)) \def
-| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) \to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to (ty3 g c u t2))))))))
-| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next g m))))
-| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to (ty3 g c (TLRef n) (lift (S n) O t))))))))
-| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u: T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to (ty3 g c (TLRef n) (lift (S n) O u))))))))
-| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) u) t1 t2) \to (\forall (t0: T).((ty3 g (CHead c (Bind b) u) t2 t0) \to (ty3 g c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))))
-| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to (\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3 g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u t)))))))))
-| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) \to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1) t2)))))).
-
-axiom ty3_gen_sort: \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c (TSort n) x) \to (pc3 c (TSort (next g n)) x))))) .
-
-axiom ty3_gen_lref: \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c (TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))))))) .
-
-axiom ty3_gen_bind: \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind b) u) t2 t0))))))))))) .
-
-axiom ty3_gen_appl: \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x: T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))))))))) .
-
-axiom ty3_gen_cast: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (land (pc3 c t2 x) (ty3 g c t1 t2))))))) .
-
-axiom ty3_lift: \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e) \to (ty3 g c (lift h d t1) (lift h d t2)))))))))) .
-
-axiom ty3_correct: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t))))))) .
-
-axiom ty3_unique: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2))))))) .
-
-axiom ty3_fsubst0: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind Abbr) u)) \to (ty3 g c2 t2 t)))))))))))) .
-
-axiom ty3_csubst0: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g c2 t1 t2))))))))))) .
-
-axiom ty3_subst0: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1 t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2 t))))))))))) .
-
-axiom ty3_gen_cabbr: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))))) .
-
-axiom ty3_gen_cvoid: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))) .
+inductive ty3 (g:G): C \to (T \to (T \to Prop)) \def
+| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t)
+\to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((pc3 c t1 t2) \to
+(ty3 g c u t2))))))))
+| ty3_sort: \forall (c: C).(\forall (m: nat).(ty3 g c (TSort m) (TSort (next
+g m))))
+| ty3_abbr: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u:
+T).((getl n c (CHead d (Bind Abbr) u)) \to (\forall (t: T).((ty3 g d u t) \to
+(ty3 g c (TLRef n) (lift (S n) O t))))))))
+| ty3_abst: \forall (n: nat).(\forall (c: C).(\forall (d: C).(\forall (u:
+T).((getl n c (CHead d (Bind Abst) u)) \to (\forall (t: T).((ty3 g d u t) \to
+(ty3 g c (TLRef n) (lift (S n) O u))))))))
+| ty3_bind: \forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u t) \to
+(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b)
+u) t1 t2) \to (\forall (t0: T).((ty3 g (CHead c (Bind b) u) t2 t0) \to (ty3 g
+c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))))))
+| ty3_appl: \forall (c: C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to
+(\forall (v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to (ty3
+g c (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u
+t)))))))))
+| ty3_cast: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2)
+\to (\forall (t0: T).((ty3 g c t2 t0) \to (ty3 g c (THead (Flat Cast) t2 t1)
+t2)))))).
+
+theorem ty3_gen_sort:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
+(TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (ty3 g c (TSort n) x)).(insert_eq T (TSort n) (\lambda (t: T).(ty3 g c t
+x)) (pc3 c (TSort (next g n)) x) (\lambda (y: T).(\lambda (H0: (ty3 g c y
+x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t0))))) (\lambda (c0: C).(\lambda
+(t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
+(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (u: T).(\lambda (t1:
+T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TSort n)) \to (pc3
+c0 (TSort (next g n)) t1)))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq
+T u (TSort n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TSort n) H6)
+in (let H8 \def (eq_ind T u (\lambda (t: T).((eq T t (TSort n)) \to (pc3 c0
+(TSort (next g n)) t1))) H4 (TSort n) H7) in (let H9 \def (eq_ind T u
+(\lambda (t: T).(ty3 g c0 t t1)) H3 (TSort n) H7) in (pc3_t t1 c0 (TSort
+(next g n)) (H8 (refl_equal T (TSort n))) t2 H5))))))))))))))) (\lambda (c0:
+C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TSort n))).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with
+[(TSort n) \Rightarrow n | (TLRef _) \Rightarrow m | (THead _ _ _)
+\Rightarrow m])) (TSort m) (TSort n) H1) in (eq_ind_r nat n (\lambda (n0:
+nat).(pc3 c0 (TSort (next g n)) (TSort (next g n0)))) (pc3_refl c0 (TSort
+(next g n))) m H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abbr)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u
+(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0)
+(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H4) in
+(False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) H5)))))))))))
+(\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d (TSort (next g n))
+t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 \def (eq_ind T
+(TLRef n0) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n))
+(lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
+(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TSort n)) \to
+(pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq
+T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort (next g n))
+t2)))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2
+t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort
+(next g n)) t0)))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TSort n))).(let
+H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in
+(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2))
+H8)))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort
+(next g n)) u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v
+(THead (Bind Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0
+(TSort (next g n)) (THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead
+(Flat Appl) w v) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v)
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat
+Appl) w (THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
+t1 (TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda
+(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort
+(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort
+n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match
+ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (pc3 c0 (TSort (next g n)) t2) H6))))))))))) c y x H0))) H))))).
+
+theorem ty3_gen_lref:
+ \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
+(TLRef n) x) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(pc3 c (lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda
+(H: (ty3 g c (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(ty3 g c t
+x)) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c
+(lift (S n) O t) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(pc3 c (lift (S n) O u) x)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t)))))) (\lambda (y: T).(\lambda (H0: (ty3 g c
+y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t1:
+T).(pc3 c0 (lift (S n) O t1) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t1: T).(ty3 g e u t1))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t1: T).(ty3 g e u t1))))))))))
+(\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2
+t)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u)
+t)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))))))).(\lambda (u: T).(\lambda (t1: T).(\lambda (H3: (ty3 g c0 u
+t1)).(\lambda (H4: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t1)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u)
+t1)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u (TLRef
+n))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (TLRef n) H6) in (let H8
+\def (eq_ind T u (\lambda (t: T).((eq T t (TLRef n)) \to (or (ex3_3 C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t1)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t0: T).(ty3 g e
+u t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3
+c0 (lift (S n) O u) t1)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t0: T).(ty3 g e u t0)))))))) H4 (TLRef n) H7) in (let H9 \def
+(eq_ind T u (\lambda (t: T).(ty3 g c0 t t1)) H3 (TLRef n) H7) in (let H10
+\def (H8 (refl_equal T (TLRef n))) in (or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t1)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0)
+t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))))) (\lambda (H11: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t1)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))) (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H12: (pc3 c0
+(lift (S n) O x2) t1)).(\lambda (H13: (getl n c0 (CHead x0 (Bind Abbr)
+x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_introl (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0
+(lift (S n) O x2) H12 t2 H5) H13 H14)))))))) H11)) (\lambda (H11: (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u)
+t1)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) t1)))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (H12: (pc3 c0 (lift (S n) O x1) t1)).(\lambda (H13: (getl n c0
+(CHead x0 (Bind Abst) x1))).(\lambda (H14: (ty3 g x0 x1 x2)).(or_intror
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift
+(S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0:
+T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0)
+t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 (lift (S n) O x1) H12 t2 H5) H13
+H14)))))))) H11)) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (H1: (eq T (TSort m) (TLRef n))).(let H2 \def (eq_ind T (TSort
+m) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (TSort (next g
+m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0
+(lift (S n) O u) (TSort (next g m)))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) H2))))) (\lambda (n0:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H1: (getl n0
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d u
+t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 d (lift (S n) O u)
+t)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T
+nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow n0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n0]))
+(TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n:
+nat).(getl n c0 (CHead d (Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n
+(\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O t))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O
+u0) (lift (S n1) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_introl (ex3_3 C T T (\lambda
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n)
+O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (lift (S n) O t))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+(lift (S n) O t))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O t)) H6 H2)) n0
+H5)))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t:
+T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S
+n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n d
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(pc3 d (lift (S n) O u) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n d (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H4: (eq T (TLRef n0)
+(TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e return
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | (TLRef n) \Rightarrow n
+| (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def
+(eq_ind nat n0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst) u))) H1 n
+H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O
+u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O
+t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0
+t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0
+(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
+n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t)))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
+(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to
+(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 (CHead
+c0 (Bind b) u) (lift (S n) O t) t2)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) u)
+(lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t) t0)))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e
+u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3
+(CHead c0 (Bind b) u) (lift (S n) O u0) t0)))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (TLRef n))).(let H8 \def
+(eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind (or (ex3_3
+C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t3: T).(pc3 c0 (lift (S n) O
+t3) (THead (Bind b) u t2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t3: T).(ty3 g e u0 t3))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind b) u t2)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t3: T).(ty3 g e u0
+t3)))))) H8)))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) u)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) u)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (_: (((eq T v (TLRef n)) \to (or (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead (Bind
+Abst) u t))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u0) (THead (Bind Abst) u t))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6
+\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in
+(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u
+t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u
+t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g
+e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda
+(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T
+(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat
+Cast) t2 t1) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda
+(_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t2))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0
+(lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c0 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t)))))) H6))))))))))) c y x H0))) H))))).
+
+theorem ty3_gen_bind:
+ \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t1:
+T).(\forall (x: T).((ty3 g c (THead (Bind b) u t1) x) \to (ex4_3 T T T
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2)
+x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u)
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c
+(Bind b) u) t2 t0)))))))))))
+\def
+ \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (H: (ty3 g c (THead (Bind b) u t1) x)).(insert_eq
+T (THead (Bind b) u t1) (\lambda (t: T).(ty3 g c t x)) (ex4_3 T T T (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))))
+(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind b) u) t1 t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind b) u)
+t2 t0))))) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g (\lambda
+(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u t1)) \to
+(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t2) t0)))) (\lambda (_: T).(\lambda (t3: T).(\lambda (_: T).(ty3 g
+c0 u t3)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
+(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t4: T).(ty3
+g (CHead c0 (Bind b) u) t2 t4))))))))) (\lambda (c0: C).(\lambda (t2:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
+(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t)))) (\lambda (_:
+T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b)
+u) t2 t0)))))))).(\lambda (u0: T).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 u0
+t0)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T T T
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t2) t0)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0
+(Bind b) u) t2 t0)))))))).(\lambda (H5: (pc3 c0 t0 t2)).(\lambda (H6: (eq T
+u0 (THead (Bind b) u t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u0
+(THead (Bind b) u t1) H6) in (let H8 \def (eq_ind T u0 (\lambda (t: T).((eq T
+t (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t0)))) (\lambda (_:
+T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c0 (Bind b)
+u) t2 t1))))))) H4 (THead (Bind b) u t1) H7) in (let H9 \def (eq_ind T u0
+(\lambda (t: T).(ty3 g c0 t t0)) H3 (THead (Bind b) u t1) H7) in (let H10
+\def (H8 (refl_equal T (THead (Bind b) u t1))) in (ex4_3_ind T T T (\lambda
+(t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t0))))
+(\lambda (_: T).(\lambda (t4: T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda
+(t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1
+t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0
+(Bind b) u) t3 t5)))) (ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda
+(_: T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t4:
+T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3:
+T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t3 t5)))))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H11: (pc3 c0
+(THead (Bind b) u x0) t0)).(\lambda (H12: (ty3 g c0 u x1)).(\lambda (H13:
+(ty3 g (CHead c0 (Bind b) u) t1 x0)).(\lambda (H14: (ty3 g (CHead c0 (Bind b)
+u) x0 x2)).(ex4_3_intro T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_:
+T).(pc3 c0 (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t4:
+T).(\lambda (_: T).(ty3 g c0 u t4)))) (\lambda (t3: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3)))) (\lambda (t3:
+T).(\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t3 t5)))) x0
+x1 x2 (pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13 H14))))))))
+H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T
+(TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda
+(ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind b) u t1) H1) in (False_ind (ex4_3 T T T (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next
+g m)))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0
+(Bind b) u) t2 t0))))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d: C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr)
+u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0
+(THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t)))) (\lambda (_:
+T).(\lambda (t: T).(\lambda (_: T).(ty3 g d u t)))) (\lambda (t2: T).(\lambda
+(_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead d (Bind b) u) t2
+t0)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def
+(eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _
+_) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind (ex4_3 T T
+T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t2) (lift (S n) O t))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_:
+T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda
+(t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3))))) H5))))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n
+c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0
+t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))))
+(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g d u t)))) (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead d (Bind b) u)
+t2 t0)))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind
+(ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t2) (lift (S n) O u0))))) (\lambda (_: T).(\lambda (t0:
+T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3)))))
+H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1:
+(ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex4_3 T
+T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t2) t)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0
+(Bind b) u) t2 t0)))))))).(\lambda (b0: B).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H3: (ty3 g (CHead c0 (Bind b0) u0) t0 t2)).(\lambda (H4: (((eq T
+t0 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t3: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3) t2))))
+(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0)
+u0) u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead
+(CHead c0 (Bind b0) u0) (Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (t0: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t2
+t0)))))))).(\lambda (t3: T).(\lambda (H5: (ty3 g (CHead c0 (Bind b0) u0) t2
+t3)).(\lambda (H6: (((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0)
+(THead (Bind b) u t2) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
+T).(ty3 g (CHead c0 (Bind b0) u0) u t)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1
+t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead (CHead
+c0 (Bind b0) u0) (Bind b) u) t2 t0)))))))).(\lambda (H7: (eq T (THead (Bind
+b0) u0 t0) (THead (Bind b) u t1))).(let H8 \def (f_equal T B (\lambda (e:
+T).(match e return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef
+_) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead
+(Bind b0) u0 t0) (THead (Bind b) u t1) H7) in ((let H9 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Bind b0) u0 t0) (THead (Bind b) u t1) H7) in ((let H10 \def (f_equal
+T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind b0) u0 t0) (THead (Bind b) u t1) H7) in (\lambda (H11: (eq T u0
+u)).(\lambda (H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t:
+T).((eq T t (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t3: T).(\lambda
+(_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t3)
+t2)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g (CHead c0
+(Bind b0) u0) u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3
+g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead (CHead c0 (Bind b0) u0)
+(Bind b) u) t2 t1))))))) H4 t1 H10) in (let H14 \def (eq_ind T t0 (\lambda
+(t: T).(ty3 g (CHead c0 (Bind b0) u0) t t2)) H3 t1 H10) in (let H15 \def
+(eq_ind B b0 (\lambda (b0: B).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T
+T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b0)
+u0) (THead (Bind b) u t2) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
+T).(ty3 g (CHead c0 (Bind b0) u0) u t)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1
+t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead (CHead
+c0 (Bind b0) u0) (Bind b) u) t2 t0))))))) H6 b H12) in (let H16 \def (eq_ind
+B b0 (\lambda (b: B).(ty3 g (CHead c0 (Bind b) u0) t2 t3)) H5 b H12) in (let
+H17 \def (eq_ind B b0 (\lambda (b0: B).((eq T t1 (THead (Bind b) u t1)) \to
+(ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0
+(Bind b0) u0) (THead (Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b0) u0) u t)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0)
+(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3
+g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t2 t0))))))) H13 b H12) in (let
+H18 \def (eq_ind B b0 (\lambda (b: B).(ty3 g (CHead c0 (Bind b) u0) t1 t2))
+H14 b H12) in (eq_ind_r B b (\lambda (b1: B).(ex4_3 T T T (\lambda (t4:
+T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind
+b1) u0 t2))))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c0 u
+t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
+(Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3
+g (CHead c0 (Bind b) u) t4 t6)))))) (let H19 \def (eq_ind T u0 (\lambda (t:
+T).((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t) (THead (Bind b)
+u t2) t3)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g (CHead
+c0 (Bind b) t) u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3
+g (CHead (CHead c0 (Bind b) t) (Bind b) u) t1 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead (CHead c0 (Bind b) t) (Bind
+b) u) t2 t1))))))) H15 u H11) in (let H20 \def (eq_ind T u0 (\lambda (t:
+T).(ty3 g (CHead c0 (Bind b) t) t2 t3)) H16 u H11) in (let H21 \def (eq_ind T
+u0 (\lambda (t: T).((eq T t1 (THead (Bind b) u t1)) \to (ex4_3 T T T (\lambda
+(t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t) (THead
+(Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) t) u t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_:
+T).(ty3 g (CHead (CHead c0 (Bind b) t) (Bind b) u) t1 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead (CHead c0 (Bind b) t) (Bind
+b) u) t2 t1))))))) H17 u H11) in (let H22 \def (eq_ind T u0 (\lambda (t:
+T).(ty3 g (CHead c0 (Bind b) t) t1 t2)) H18 u H11) in (let H23 \def (eq_ind T
+u0 (\lambda (t0: T).((eq T t0 (THead (Bind b) u t1)) \to (ex4_3 T T T
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t2) t)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c0
+(Bind b) u) t2 t1))))))) H2 u H11) in (let H24 \def (eq_ind T u0 (\lambda
+(t0: T).(ty3 g c0 t0 t)) H1 u H11) in (eq_ind_r T u (\lambda (t4: T).(ex4_3 T
+T T (\lambda (t5: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b)
+u t5) (THead (Bind b) t4 t2))))) (\lambda (_: T).(\lambda (t6: T).(\lambda
+(_: T).(ty3 g c0 u t6)))) (\lambda (t5: T).(\lambda (_: T).(\lambda (_:
+T).(ty3 g (CHead c0 (Bind b) u) t1 t5)))) (\lambda (t5: T).(\lambda (_:
+T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b) u) t5 t7)))))) (ex4_3_intro T T
+T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t4) (THead (Bind b) u t2))))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_:
+T).(ty3 g c0 u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
+(CHead c0 (Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
+(t6: T).(ty3 g (CHead c0 (Bind b) u) t4 t6)))) t2 t t3 (pc3_refl c0 (THead
+(Bind b) u t2)) H24 H22 H20) u0 H11))))))) b0 H12)))))))))) H9))
+H8)))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda
+(_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u t1)) \to (ex4_3
+T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind
+b) u t2) u0)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u
+t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind
+b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g
+(CHead c0 (Bind b) u) t2 t0)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda
+(_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (_: (((eq T v (THead
+(Bind b) u t1)) \to (ex4_3 T T T (\lambda (t2: T).(\lambda (_: T).(\lambda
+(_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 t))))) (\lambda
+(_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b)
+u) t2 t0)))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Bind b)
+u t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee:
+T).(match ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex4_3 T T T (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead
+(Flat Appl) w (THead (Bind Abst) u0 t)))))) (\lambda (_: T).(\lambda (t0:
+T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind b) u) t2 t3)))))
+H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (ty3 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u t1)) \to
+(ex4_3 T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t3) t2)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g
+c0 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
+(Bind b) u) t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3
+g (CHead c0 (Bind b) u) t2 t0)))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0
+t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex4_3 T T T
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u
+t2) t3)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
+t1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c0
+(Bind b) u) t2 t0)))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t0)
+(THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0)
+(\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind
+(ex4_3 T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead
+(Bind b) u t4) t2)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g
+c0 u t)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0
+(Bind b) u) t1 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t5: T).(ty3
+g (CHead c0 (Bind b) u) t4 t5))))) H6))))))))))) c y x H0))) H))))))).
+
+theorem ty3_gen_appl:
+ \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
+T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex3_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
+T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(insert_eq T (THead
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c t x)) (ex3_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
+(\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))) (\lambda (y: T).(\lambda
+(H0: (ty3 g c y x)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda
+(t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t1)) t0))) (\lambda
+(u: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u t1)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c0 w u)))))))) (\lambda (c0: C).(\lambda (t2:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t0: T).(pc3
+c0 (THead (Flat Appl) w (THead (Bind Abst) u t0)) t))) (\lambda (u:
+T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (u: T).(\lambda (t1:
+T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (THead (Flat Appl)
+w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) t1))) (\lambda (u: T).(\lambda (t: T).(ty3 g
+c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w
+u))))))).(\lambda (H5: (pc3 c0 t1 t2)).(\lambda (H6: (eq T u (THead (Flat
+Appl) w v))).(let H7 \def (f_equal T T (\lambda (e: T).e) u (THead (Flat
+Appl) w v) H6) in (let H8 \def (eq_ind T u (\lambda (t: T).((eq T t (THead
+(Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t0: T).(pc3 c0
+(THead (Flat Appl) w (THead (Bind Abst) u t0)) t1))) (\lambda (u: T).(\lambda
+(t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) (\lambda (u: T).(\lambda (_:
+T).(ty3 g c0 w u)))))) H4 (THead (Flat Appl) w v) H7) in (let H9 \def (eq_ind
+T u (\lambda (t: T).(ty3 g c0 t t1)) H3 (THead (Flat Appl) w v) H7) in (let
+H10 \def (H8 (refl_equal T (THead (Flat Appl) w v))) in (ex3_2_ind T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) t1))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))
+(ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0
+v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w
+u0)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) x0 x1)) t1)).(\lambda (H12: (ty3 g c0 v (THead
+(Bind Abst) x0 x1))).(\lambda (H13: (ty3 g c0 w x0)).(ex3_2_intro T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))) x0
+x1 (pc3_t t1 c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) H11 t2 H5)
+H12 H13)))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
+(H1: (eq T (TSort m) (THead (Flat Appl) w v))).(let H2 \def (eq_ind T (TSort
+m) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) w v) H1) in (False_ind (ex3_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+(TSort (next g m))))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead
+(Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H2)))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
+d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T
+(\lambda (u: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) t))) (\lambda (u: T).(\lambda (t: T).(ty3 g d v (THead (Bind
+Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g d w u))))))).(\lambda
+(H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef
+n) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0:
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0))
+(lift (S n) O t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2
+T T (\lambda (u: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) t))) (\lambda (u: T).(\lambda (t: T).(ty3 g d v (THead (Bind
+Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g d w u))))))).(\lambda
+(H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef
+n) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0:
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0))
+(lift (S n) O u)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))
+H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
+(ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T
+T (\lambda (u: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) t))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind
+Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda
+(b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind
+b) u) t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T
+(\lambda (u0: T).(\lambda (t: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat
+Appl) w (THead (Bind Abst) u0 t)) t2))) (\lambda (u0: T).(\lambda (t: T).(ty3
+g (CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t2 t0)).(\lambda (_: (((eq T t2
+(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3
+(CHead c0 (Bind b) u) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) t0)))
+(\lambda (u0: T).(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u) v (THead (Bind
+Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u)
+w u0))))))).(\lambda (H7: (eq T (THead (Bind b) u t1) (THead (Flat Appl) w
+v))).(let H8 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Appl) w v) H7) in (False_ind (ex3_2 T T (\lambda (u0:
+T).(\lambda (t3: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t3))
+(THead (Bind b) u t2)))) (\lambda (u0: T).(\lambda (t3: T).(ty3 g c0 v (THead
+(Bind Abst) u0 t3)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))
+H8)))))))))))))))) (\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda
+(H1: (ty3 g c0 w0 u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t)) u))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w
+u))))))).(\lambda (v0: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead
+(Bind Abst) u t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to
+(ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T (THead (Flat
+Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef
+_) \Rightarrow w0 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) w0 v0)
+(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef
+_) \Rightarrow v0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) w0 v0)
+(THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let H9 \def
+(eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to (ex3_2 T
+T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (t: T).(ty3
+g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w
+u)))))) H4 v H7) in (let H10 \def (eq_ind T v0 (\lambda (t0: T).(ty3 g c0 t0
+(THead (Bind Abst) u t))) H3 v H7) in (let H11 \def (eq_ind T w0 (\lambda (t:
+T).((eq T t (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda
+(t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) u))) (\lambda
+(u: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u t0)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c0 w u)))))) H2 w H8) in (let H12 \def (eq_ind T w0
+(\lambda (t: T).(ty3 g c0 t u)) H1 w H8) in (eq_ind_r T w (\lambda (t0:
+T).(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w
+(THead (Bind Abst) u0 t1)) (THead (Flat Appl) t0 (THead (Bind Abst) u t)))))
+(\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1))))
+(\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T
+(\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t0)) (THead (Flat Appl) w (THead (Bind Abst) u t))))) (\lambda (u0:
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
+T).(\lambda (_: T).(ty3 g c0 w u0))) u t (pc3_refl c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t))) H10 H12) w0 H8))))))) H6)))))))))))) (\lambda (c0:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda
+(_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u:
+T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+t2))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl)
+w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g
+c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w
+u))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (THead (Flat Appl) w
+v))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match
+ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
+True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda
+(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t))
+t2))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t))))
+(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) c y x H0)))
+H)))))).
+
+theorem ty3_gen_cast:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall
+(x: T).((ty3 g c (THead (Flat Cast) t2 t1) x) \to (land (pc3 c t2 x) (ty3 g c
+t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(x: T).(\lambda (H: (ty3 g c (THead (Flat Cast) t2 t1) x)).(insert_eq T
+(THead (Flat Cast) t2 t1) (\lambda (t: T).(ty3 g c t x)) (land (pc3 c t2 x)
+(ty3 g c t1 t2)) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(ty3_ind g
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast)
+t2 t1)) \to (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t0 t)).(\lambda
+(_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t) (ty3 g c0
+t1 t2))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
+t3)).(\lambda (H4: (((eq T u (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2
+t3) (ty3 g c0 t1 t2))))).(\lambda (H5: (pc3 c0 t3 t0)).(\lambda (H6: (eq T u
+(THead (Flat Cast) t2 t1))).(let H7 \def (f_equal T T (\lambda (e: T).e) u
+(THead (Flat Cast) t2 t1) H6) in (let H8 \def (eq_ind T u (\lambda (t:
+T).((eq T t (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t3) (ty3 g c0 t1
+t2)))) H4 (THead (Flat Cast) t2 t1) H7) in (let H9 \def (eq_ind T u (\lambda
+(t: T).(ty3 g c0 t t3)) H3 (THead (Flat Cast) t2 t1) H7) in (let H10 \def (H8
+(refl_equal T (THead (Flat Cast) t2 t1))) in (and_ind (pc3 c0 t2 t3) (ty3 g
+c0 t1 t2) (land (pc3 c0 t2 t0) (ty3 g c0 t1 t2)) (\lambda (H11: (pc3 c0 t2
+t3)).(\lambda (H12: (ty3 g c0 t1 t2)).(conj (pc3 c0 t2 t0) (ty3 g c0 t1 t2)
+(pc3_t t3 c0 t2 H11 t0 H5) H12))) H10)))))))))))))))) (\lambda (c0:
+C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2
+t1))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast)
+t2 t1) H1) in (False_ind (land (pc3 c0 t2 (TSort (next g m))) (ty3 g c0 t1
+t2)) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to
+(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef n) (THead
+(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match
+ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2
+t1) H4) in (False_ind (land (pc3 c0 t2 (lift (S n) O t)) (ty3 g c0 t1 t2))
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to
+(land (pc3 d t2 t) (ty3 g d t1 t2))))).(\lambda (H4: (eq T (TLRef n) (THead
+(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match
+ee return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2
+t1) H4) in (False_ind (land (pc3 c0 t2 (lift (S n) O u)) (ty3 g c0 t1 t2))
+H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
+(ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (land
+(pc3 c0 t2 t) (ty3 g c0 t1 t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda
+(t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq
+T t0 (THead (Flat Cast) t2 t1)) \to (land (pc3 (CHead c0 (Bind b) u) t2 t3)
+(ty3 g (CHead c0 (Bind b) u) t1 t2))))).(\lambda (t4: T).(\lambda (_: (ty3 g
+(CHead c0 (Bind b) u) t3 t4)).(\lambda (_: (((eq T t3 (THead (Flat Cast) t2
+t1)) \to (land (pc3 (CHead c0 (Bind b) u) t2 t4) (ty3 g (CHead c0 (Bind b) u)
+t1 t2))))).(\lambda (H7: (eq T (THead (Bind b) u t0) (THead (Flat Cast) t2
+t1))).(let H8 \def (eq_ind T (THead (Bind b) u t0) (\lambda (ee: T).(match ee
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Cast) t2 t1) H7) in (False_ind (land (pc3 c0 t2 (THead (Bind b)
+u t3)) (ty3 g c0 t1 t2)) H8)))))))))))))))) (\lambda (c0: C).(\lambda (w:
+T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (THead
+(Flat Cast) t2 t1)) \to (land (pc3 c0 t2 u) (ty3 g c0 t1 t2))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2
+(THead (Bind Abst) u t)) (ty3 g c0 t1 t2))))).(\lambda (H5: (eq T (THead
+(Flat Appl) w v) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead
+(Flat Appl) w v) (\lambda (ee: T).(match ee return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t2 t1)
+H5) in (False_ind (land (pc3 c0 t2 (THead (Flat Appl) w (THead (Bind Abst) u
+t))) (ty3 g c0 t1 t2)) H6)))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0
+(THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t3) (ty3 g c0 t1
+t2))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: (((eq
+T t3 (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t4) (ty3 g c0 t1
+t2))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2
+t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t
+_) \Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5)
+in ((let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5)
+in (\lambda (H8: (eq T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq
+T t (THead (Flat Cast) t2 t1)) \to (land (pc3 c0 t2 t4) (ty3 g c0 t1 t2))))
+H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3
+t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat
+Cast) t2 t1)) \to (land (pc3 c0 t2 t) (ty3 g c0 t1 t2)))) H2 t2 H8) in (let
+H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r
+T t2 (\lambda (t: T).(land (pc3 c0 t2 t) (ty3 g c0 t1 t2))) (let H13 \def
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (land
+(pc3 c0 t2 t2) (ty3 g c0 t1 t2)))) H11 t1 H7) in (let H14 \def (eq_ind T t0
+(\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (conj (pc3 c0 t2 t2) (ty3 g c0
+t1 t2) (pc3_refl c0 t2) H14))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
+
+theorem ty3_lift:
+ \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
+t1 t2) \to (\forall (c: C).(\forall (d: nat).(\forall (h: nat).((drop h d c
+e) \to (ty3 g c (lift h d t1) (lift h d t2))))))))))
+\def
+ \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g e t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to
+(ty3 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda (t0:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda (H1: ((\forall (c0:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h
+d t0) (lift h d t)))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3
+g c u t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
+t3)))))))).(\lambda (H4: (pc3 c t3 t0)).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H5: (drop h d c0 c)).(ty3_conv g c0 (lift h
+d t0) (lift h d t) (H1 c0 d h H5) (lift h d u) (lift h d t3) (H3 c0 d h H5)
+(pc3_lift c0 c h d H5 t3 t0 H4)))))))))))))))) (\lambda (c: C).(\lambda (m:
+nat).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (_: (drop
+h d c0 c)).(eq_ind_r T (TSort m) (\lambda (t: T).(ty3 g c0 t (lift h d (TSort
+(next g m))))) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(ty3 g c0
+(TSort m) t)) (ty3_sort g c0 m) (lift h d (TSort (next g m))) (lift_sort
+(next g m) h d)) (lift h d (TSort m)) (lift_sort m h d)))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c: C).(\forall (d0: nat).(\forall (h:
+nat).((drop h d0 c d) \to (ty3 g c (lift h d0 u) (lift h d0
+t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda (H3:
+(drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0
+(lift (S n) O t))) (\lambda (H4: (lt n d0)).(let H5 \def (drop_getl_trans_le
+n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) u) H0)
+in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n O c0 e0)))
+(\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1))) (\lambda (_:
+C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) u)))) (ty3 g c0 (lift h d0
+(TLRef n)) (lift h d0 (lift (S n) O t))) (\lambda (x0: C).(\lambda (x1:
+C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h (minus d0 n) x0
+x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) u))).(let H9 \def (eq_ind
+nat (minus d0 n) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S n)))
+(minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S n))
+H9 Abbr d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
+Abbr) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h (minus d0 (S
+n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O t)))
+(\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
+d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x d)).(eq_ind_r T
+(TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t))))
+(eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3 g c0 (TLRef
+n) (lift h n0 (lift (S n) O t)))) (eq_ind_r T (lift (S n) O (lift h (minus d0
+(S n)) t)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat d0 (\lambda
+(_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S n)) t))))
+(ty3_abbr g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0 (CHead x
+(Bind Abbr) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus d0 (S n))
+t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
+n) O t)) (lift_d t h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O t)))) (eq_ind nat
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
+t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
+(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus
+h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
+(plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda
+(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c: C).(\forall
+(d0: nat).(\forall (h: nat).((drop h d0 c d) \to (ty3 g c (lift h d0 u) (lift
+h d0 t)))))))).(\lambda (c0: C).(\lambda (d0: nat).(\lambda (h: nat).(\lambda
+(H3: (drop h d0 c0 c)).(lt_le_e n d0 (ty3 g c0 (lift h d0 (TLRef n)) (lift h
+d0 (lift (S n) O u))) (\lambda (H4: (lt n d0)).(let H5 \def
+(drop_getl_trans_le n d0 (le_S_n n d0 (le_S (S n) d0 H4)) c0 c h H3 (CHead d
+(Bind Abst) u) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop n
+O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 n) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) u)))) (ty3 g
+c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S n) O u))) (\lambda (x0:
+C).(\lambda (x1: C).(\lambda (H6: (drop n O c0 x0)).(\lambda (H7: (drop h
+(minus d0 n) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) u))).(let
+H9 \def (eq_ind nat (minus d0 n) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
+(minus d0 (S n))) (minus_x_Sy d0 n H4)) in (let H10 \def (drop_clear_S x1 x0
+h (minus d0 (S n)) H9 Abst d u H8) in (ex2_ind C (\lambda (c1: C).(clear x0
+(CHead c1 (Bind Abst) (lift h (minus d0 (S n)) u)))) (\lambda (c1: C).(drop h
+(minus d0 (S n)) c1 d)) (ty3 g c0 (lift h d0 (TLRef n)) (lift h d0 (lift (S
+n) O u))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
+h (minus d0 (S n)) u)))).(\lambda (H12: (drop h (minus d0 (S n)) x
+d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S
+n) O u)))) (eq_ind nat (plus (S n) (minus d0 (S n))) (\lambda (n0: nat).(ty3
+g c0 (TLRef n) (lift h n0 (lift (S n) O u)))) (eq_ind_r T (lift (S n) O (lift
+h (minus d0 (S n)) u)) (\lambda (t0: T).(ty3 g c0 (TLRef n) t0)) (eq_ind nat
+d0 (\lambda (_: nat).(ty3 g c0 (TLRef n) (lift (S n) O (lift h (minus d0 (S
+n)) u)))) (ty3_abst g n c0 x (lift h (minus d0 (S n)) u) (getl_intro n c0
+(CHead x (Bind Abst) (lift h (minus d0 (S n)) u)) x0 H6 H11) (lift h (minus
+d0 (S n)) t) (H2 x (minus d0 (S n)) h H12)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H4)) (lift h (plus (S n) (minus d0 (S n))) (lift (S
+n) O u)) (lift_d u h (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n))))) d0
+(le_plus_minus_r (S n) d0 H4)) (lift h d0 (TLRef n)) (lift_lref_lt n h d0
+H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 n)).(eq_ind_r T (TLRef (plus n
+h)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d0 (lift (S n) O u)))) (eq_ind nat
+(S n) (\lambda (_: nat).(ty3 g c0 (TLRef (plus n h)) (lift h d0 (lift (S n) O
+u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef
+(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
+(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u
+(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus
+h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
+h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
+n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
+(plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
+t)))))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3
+g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
+(lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda (_: (ty3 g
+(CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 (CHead c (Bind b) u)) \to (ty3 g c0
+(lift h d t3) (lift h d t4)))))))).(\lambda (c0: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H6: (drop h d c0 c)).(eq_ind_r T (THead
+(Bind b) (lift h d u) (lift h (s (Bind b) d) t0)) (\lambda (t5: T).(ty3 g c0
+t5 (lift h d (THead (Bind b) u t3)))) (eq_ind_r T (THead (Bind b) (lift h d
+u) (lift h (s (Bind b) d) t3)) (\lambda (t5: T).(ty3 g c0 (THead (Bind b)
+(lift h d u) (lift h (s (Bind b) d) t0)) t5)) (ty3_bind g c0 (lift h d u)
+(lift h d t) (H1 c0 d h H6) b (lift h (S d) t0) (lift h (S d) t3) (H3 (CHead
+c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0 c H6 b u)) (lift h
+(S d) t4) (H5 (CHead c0 (Bind b) (lift h d u)) (S d) h (drop_skip_bind h d c0
+c H6 b u))) (lift h d (THead (Bind b) u t3)) (lift_head (Bind b) u t3 h d))
+(lift h d (THead (Bind b) u t0)) (lift_head (Bind b) u t0 h
+d))))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c0: C).(\forall (d: nat).(\forall
+(h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d w) (lift h d
+u)))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c0: C).(\forall (d: nat).(\forall
+(h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d v) (lift h d (THead (Bind
+Abst) u t))))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (H4: (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d
+w) (lift h (s (Flat Appl) d) v)) (\lambda (t0: T).(ty3 g c0 t0 (lift h d
+(THead (Flat Appl) w (THead (Bind Abst) u t))))) (eq_ind_r T (THead (Flat
+Appl) (lift h d w) (lift h (s (Flat Appl) d) (THead (Bind Abst) u t)))
+(\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) (lift h d w) (lift h (s (Flat
+Appl) d) v)) t0)) (eq_ind_r T (THead (Bind Abst) (lift h (s (Flat Appl) d) u)
+(lift h (s (Bind Abst) (s (Flat Appl) d)) t)) (\lambda (t0: T).(ty3 g c0
+(THead (Flat Appl) (lift h d w) (lift h (s (Flat Appl) d) v)) (THead (Flat
+Appl) (lift h d w) t0))) (ty3_appl g c0 (lift h d w) (lift h d u) (H1 c0 d h
+H4) (lift h d v) (lift h (S d) t) (eq_ind T (lift h d (THead (Bind Abst) u
+t)) (\lambda (t0: T).(ty3 g c0 (lift h d v) t0)) (H3 c0 d h H4) (THead (Bind
+Abst) (lift h d u) (lift h (S d) t)) (lift_bind Abst u t h d))) (lift h (s
+(Flat Appl) d) (THead (Bind Abst) u t)) (lift_head (Bind Abst) u t h (s (Flat
+Appl) d))) (lift h d (THead (Flat Appl) w (THead (Bind Abst) u t)))
+(lift_head (Flat Appl) w (THead (Bind Abst) u t) h d)) (lift h d (THead (Flat
+Appl) w v)) (lift_head (Flat Appl) w v h d))))))))))))))) (\lambda (c:
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
+(H1: ((\forall (c0: C).(\forall (d: nat).(\forall (h: nat).((drop h d c0 c)
+\to (ty3 g c0 (lift h d t0) (lift h d t3)))))))).(\lambda (t4: T).(\lambda
+(_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d t3) (lift h d
+t4)))))))).(\lambda (c0: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H4:
+(drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d t3) (lift h (s
+(Flat Cast) d) t0)) (\lambda (t: T).(ty3 g c0 t (lift h d t3))) (ty3_cast g
+c0 (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 c0 d h H4) (lift h d t4)
+(H3 c0 d h H4)) (lift h d (THead (Flat Cast) t3 t0)) (lift_head (Flat Cast)
+t3 t0 h d)))))))))))))) e t1 t2 H))))).
+
+theorem ty3_correct:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (ex T (\lambda (t: T).(ty3 g c t2 t)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda
+(t0: T).(ex T (\lambda (t3: T).(ty3 g c0 t0 t3)))))) (\lambda (c0:
+C).(\lambda (t0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t0 t)).(\lambda
+(_: (ex T (\lambda (t0: T).(ty3 g c0 t t0)))).(\lambda (u: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g c0 u t3)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0
+t3 t)))).(\lambda (_: (pc3 c0 t3 t0)).(ex_intro T (\lambda (t4: T).(ty3 g c0
+t0 t4)) t H0))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro T
+(\lambda (t: T).(ty3 g c0 (TSort (next g m)) t)) (TSort (next g (next g m)))
+(ty3_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex T (\lambda
+(t0: T).(ty3 g d t t0)))).(let H3 \def H2 in (ex_ind T (\lambda (t0: T).(ty3
+g d t t0)) (ex T (\lambda (t0: T).(ty3 g c0 (lift (S n) O t) t0))) (\lambda
+(x: T).(\lambda (H4: (ty3 g d t x)).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(lift (S n) O t) t0)) (lift (S n) O x) (ty3_lift g d t x H4 c0 O (S n)
+(getl_drop Abbr c0 d u n H0))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind
+Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (_: (ex T
+(\lambda (t0: T).(ty3 g d t t0)))).(ex_intro T (\lambda (t0: T).(ty3 g c0
+(lift (S n) O u) t0)) (lift (S n) O t) (ty3_lift g d u t H1 c0 O (S n)
+(getl_drop Abst c0 d u n H0))))))))))) (\lambda (c0: C).(\lambda (u:
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (_: (ex T (\lambda
+(t0: T).(ty3 g c0 t t0)))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (ex T
+(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u) t3 t)))).(\lambda (t4:
+T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H5: (ex T
+(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u) t4 t)))).(let H6 \def H5 in
+(ex_ind T (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) u) t4 t5)) (ex T
+(\lambda (t5: T).(ty3 g c0 (THead (Bind b) u t3) t5))) (\lambda (x:
+T).(\lambda (H7: (ty3 g (CHead c0 (Bind b) u) t4 x)).(ex_intro T (\lambda
+(t5: T).(ty3 g c0 (THead (Bind b) u t3) t5)) (THead (Bind b) u t4) (ty3_bind
+g c0 u t H0 b t3 t4 H4 x H7)))) H6))))))))))))))) (\lambda (c0: C).(\lambda
+(w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c0 w u)).(\lambda (H1: (ex T
+(\lambda (t: T).(ty3 g c0 u t)))).(\lambda (v: T).(\lambda (t: T).(\lambda
+(_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex T (\lambda (t0:
+T).(ty3 g c0 (THead (Bind Abst) u t) t0)))).(let H4 \def H1 in (ex_ind T
+(\lambda (t0: T).(ty3 g c0 u t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) t0))) (\lambda (x: T).(\lambda (_:
+(ty3 g c0 u x)).(let H6 \def H3 in (ex_ind T (\lambda (t0: T).(ty3 g c0
+(THead (Bind Abst) u t) t0)) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) t0))) (\lambda (x0: T).(\lambda (H7: (ty3 g
+c0 (THead (Bind Abst) u t) x0)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda
+(_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u t3) x0)))) (\lambda (_:
+T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c0 u t0)))) (\lambda (t3:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u) t t3))))
+(\lambda (t3: T).(\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind
+Abst) u) t3 t4)))) (ex T (\lambda (t0: T).(ty3 g c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) t0))) (\lambda (x1: T).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (_: (pc3 c0 (THead (Bind Abst) u x1) x0)).(\lambda (H9: (ty3
+g c0 u x2)).(\lambda (H10: (ty3 g (CHead c0 (Bind Abst) u) t x1)).(\lambda
+(H11: (ty3 g (CHead c0 (Bind Abst) u) x1 x3)).(ex_intro T (\lambda (t0:
+T).(ty3 g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t0)) (THead (Flat
+Appl) w (THead (Bind Abst) u x1)) (ty3_appl g c0 w u H0 (THead (Bind Abst) u
+t) x1 (ty3_bind g c0 u x2 H9 Abst t x1 H10 x3 H11)))))))))) (ty3_gen_bind g
+Abst c0 u t x0 H7)))) H6)))) H4))))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t0 t3)).(\lambda (H1: (ex T
+(\lambda (t: T).(ty3 g c0 t3 t)))).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3
+t4)).(\lambda (_: (ex T (\lambda (t: T).(ty3 g c0 t4 t)))).H1)))))))) c t1 t2
+H))))).
+
+theorem ty3_unique:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((ty3 g c u t2) \to (pc3 c t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (t2: T).((ty3 g c0 t t2) \to (pc3 c0 t0 t2)))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: ((\forall (t3: T).((ty3 g c0 t2 t3) \to (pc3 c0 t t3))))).(\lambda (u0:
+T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H3: ((\forall
+(t2: T).((ty3 g c0 u0 t2) \to (pc3 c0 t0 t2))))).(\lambda (H4: (pc3 c0 t0
+t2)).(\lambda (t3: T).(\lambda (H5: (ty3 g c0 u0 t3)).(pc3_t t0 c0 t2 (pc3_s
+c0 t2 t0 H4) t3 (H3 t3 H5)))))))))))))) (\lambda (c0: C).(\lambda (m:
+nat).(\lambda (t2: T).(\lambda (H0: (ty3 g c0 (TSort m) t2)).(ty3_gen_sort g
+c0 t2 m H0))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t:
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((ty3 g d u0
+t2) \to (pc3 d t t2))))).(\lambda (t2: T).(\lambda (H3: (ty3 g c0 (TLRef n)
+t2)).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0:
+T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u1: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u1)))))
+(\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0))))) (pc3 c0
+(lift (S n) O t) t2) (\lambda (H4: (ex3_3 C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t2)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0)
+t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abbr) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g
+e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O x2) t2)).(\lambda
+(H6: (getl n c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1
+x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n
+c0 c)) H0 (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n
+H0 (CHead x0 (Bind Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+((let H10 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead
+d (Bind Abbr) u0) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind
+Abbr) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d
+x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t: T).(getl n c0 (CHead x0 (Bind
+Abbr) t))) H8 u0 H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(ty3 g
+x0 t x2)) H7 u0 H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c: C).(getl n
+c0 (CHead c (Bind Abbr) u0))) H12 d H11) in (let H15 \def (eq_ind_r C x0
+(\lambda (c: C).(ty3 g c u0 x2)) H13 d H11) in (pc3_t (lift (S n) O x2) c0
+(lift (S n) O t) (pc3_lift c0 d (S n) O (getl_drop Abbr c0 d u0 n H14) t x2
+(H2 x2 H15)) t2 H5))))))) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u)
+t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O t) t2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c0 (lift (S n) O
+x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (_:
+(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda
+(c: C).(getl n c0 c)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abbr) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C
+(CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
+Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead x0 (Bind Abst)
+x1) H6)) in (False_ind (pc3 c0 (lift (S n) O t) t2) H9))))))))) H4))
+(ty3_gen_lref g c0 t2 n H3)))))))))))) (\lambda (n: nat).(\lambda (c0:
+C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind
+Abst) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_:
+((\forall (t2: T).((ty3 g d u0 t2) \to (pc3 d t t2))))).(\lambda (t2:
+T).(\lambda (H3: (ty3 g c0 (TLRef n) t2)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda
+(e: C).(\lambda (u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr)
+u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))))
+(ex3_3 C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(pc3 c0 (lift
+(S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(getl n
+c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(t0: T).(ty3 g e u1 t0))))) (pc3 c0 (lift (S n) O u0) t2) (\lambda (H4:
+(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S
+n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda
+(u1: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u1))))) (\lambda (e:
+C).(\lambda (u1: T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O
+u0) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3
+c0 (lift (S n) O x2) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind
+Abst) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead x0 (Bind Abbr) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in
+(let H9 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (pc3 c0
+(lift (S n) O u0) t2) H9))))))))) H4)) (\lambda (H4: (ex3_3 C T T (\lambda
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t2))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind
+Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u1: T).(\lambda (_:
+T).(pc3 c0 (lift (S n) O u1) t2)))) (\lambda (e: C).(\lambda (u1: T).(\lambda
+(_: T).(getl n c0 (CHead e (Bind Abst) u1))))) (\lambda (e: C).(\lambda (u1:
+T).(\lambda (t0: T).(ty3 g e u1 t0)))) (pc3 c0 (lift (S n) O u0) t2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H5: (pc3 c0 (lift (S n) O
+x1) t2)).(\lambda (H6: (getl n c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
+(ty3 g x0 x1 x2)).(let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda
+(c: C).(getl n c0 c)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d
+(Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (f_equal
+C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u0)
+(CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+x0 (Bind Abst) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match
+e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abst) u0) (CHead x0 (Bind Abst) x1)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead x0 (Bind Abst) x1) H6)) in
+(\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t:
+T).(getl n c0 (CHead x0 (Bind Abst) t))) H8 u0 H10) in (let H13 \def
+(eq_ind_r T x1 (\lambda (t: T).(ty3 g x0 t x2)) H7 u0 H10) in (let H14 \def
+(eq_ind_r T x1 (\lambda (t: T).(pc3 c0 (lift (S n) O t) t2)) H5 u0 H10) in
+(let H15 \def (eq_ind_r C x0 (\lambda (c: C).(getl n c0 (CHead c (Bind Abst)
+u0))) H12 d H11) in (let H16 \def (eq_ind_r C x0 (\lambda (c: C).(ty3 g c u0
+x2)) H13 d H11) in H14))))))) H9))))))))) H4)) (ty3_gen_lref g c0 t2 n
+H3)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (_:
+(ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 u0 t2) \to (pc3 c0
+t t2))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3
+g (CHead c0 (Bind b) u0) t0 t2)).(\lambda (H3: ((\forall (t3: T).((ty3 g
+(CHead c0 (Bind b) u0) t0 t3) \to (pc3 (CHead c0 (Bind b) u0) t2
+t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2
+t3)).(\lambda (_: ((\forall (t4: T).((ty3 g (CHead c0 (Bind b) u0) t2 t4) \to
+(pc3 (CHead c0 (Bind b) u0) t3 t4))))).(\lambda (t4: T).(\lambda (H6: (ty3 g
+c0 (THead (Bind b) u0 t0) t4)).(ex4_3_ind T T T (\lambda (t5: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u0 t5) t4)))) (\lambda (_:
+T).(\lambda (t6: T).(\lambda (_: T).(ty3 g c0 u0 t6)))) (\lambda (t5:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u0) t0 t5))))
+(\lambda (t5: T).(\lambda (_: T).(\lambda (t7: T).(ty3 g (CHead c0 (Bind b)
+u0) t5 t7)))) (pc3 c0 (THead (Bind b) u0 t2) t4) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (H7: (pc3 c0 (THead (Bind b) u0 x0)
+t4)).(\lambda (_: (ty3 g c0 u0 x1)).(\lambda (H9: (ty3 g (CHead c0 (Bind b)
+u0) t0 x0)).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) x0 x2)).(pc3_t (THead
+(Bind b) u0 x0) c0 (THead (Bind b) u0 t2) (pc3_head_2 c0 u0 t2 x0 (Bind b)
+(H3 x0 H9)) t4 H7)))))))) (ty3_gen_bind g b c0 u0 t0 t4 H6)))))))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w
+u0)).(\lambda (_: ((\forall (t2: T).((ty3 g c0 w t2) \to (pc3 c0 u0
+t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
+Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((ty3 g c0 v t2) \to (pc3 c0
+(THead (Bind Abst) u0 t) t2))))).(\lambda (t2: T).(\lambda (H4: (ty3 g c0
+(THead (Flat Appl) w v) t2)).(ex3_2_ind T T (\lambda (u1: T).(\lambda (t0:
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u1 t0)) t2))) (\lambda
+(u1: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u1 t0)))) (\lambda
+(u1: T).(\lambda (_: T).(ty3 g c0 w u1))) (pc3 c0 (THead (Flat Appl) w (THead
+(Bind Abst) u0 t)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (pc3
+c0 (THead (Flat Appl) w (THead (Bind Abst) x0 x1)) t2)).(\lambda (H6: (ty3 g
+c0 v (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c0 w x0)).(pc3_t (THead
+(Flat Appl) w (THead (Bind Abst) x0 x1)) c0 (THead (Flat Appl) w (THead (Bind
+Abst) u0 t)) (pc3_thin_dx c0 (THead (Bind Abst) u0 t) (THead (Bind Abst) x0
+x1) (H3 (THead (Bind Abst) x0 x1) H6) w Appl) t2 H5)))))) (ty3_gen_appl g c0
+w v t2 H4))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: ((\forall (t3: T).((ty3 g c0
+t0 t3) \to (pc3 c0 t2 t3))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2
+t3)).(\lambda (_: ((\forall (t4: T).((ty3 g c0 t2 t4) \to (pc3 c0 t3
+t4))))).(\lambda (t4: T).(\lambda (H4: (ty3 g c0 (THead (Flat Cast) t2 t0)
+t4)).(and_ind (pc3 c0 t2 t4) (ty3 g c0 t0 t2) (pc3 c0 t2 t4) (\lambda (H5:
+(pc3 c0 t2 t4)).(\lambda (_: (ty3 g c0 t0 t2)).H5)) (ty3_gen_cast g c0 t0 t2
+t4 H4)))))))))))) c u t1 H))))).
+
+theorem ty3_fsubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
+t1 t) \to (\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u c1 t1 c2 t2) \to (\forall (e: C).((getl i c1 (CHead e (Bind
+Abbr) u)) \to (ty3 g c2 t2 t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
+(t2: T).(\forall (i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3:
+T).((fsubst0 i u c t0 c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u)) \to (ty3 g c2 t3 t2))))))))))) (\lambda (c: C).(\lambda (t2:
+T).(\lambda (t0: T).(\lambda (H0: (ty3 g c t2 t0)).(\lambda (H1: ((\forall
+(i: nat).(\forall (u: T).(\forall (c2: C).(\forall (t3: T).((fsubst0 i u c t2
+c2 t3) \to (\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to (ty3 g c2
+t3 t0)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u
+t3)).(\lambda (H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2:
+C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (\forall (e: C).((getl i c
+(CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2 t3)))))))))).(\lambda (H4: (pc3 c
+t3 t2)).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t4:
+T).(\lambda (H5: (fsubst0 i u0 c u c2 t4)).(fsubst0_ind i u0 c u (\lambda
+(c0: C).(\lambda (t5: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0))
+\to (ty3 g c0 t5 t2))))) (\lambda (t5: T).(\lambda (H6: (subst0 i u0 u
+t5)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr)
+u0))).(ty3_conv g c t2 t0 H0 t5 t3 (H3 i u0 c t5 (fsubst0_snd i u0 c u t5 H6)
+e H7) H4))))) (\lambda (c3: C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda
+(e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) u0))).(ty3_conv g c3 t2
+t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H6) e H7) u t3 (H3 i u0 c3 u
+(fsubst0_fst i u0 c u c3 H6) e H7) (pc3_fsubst0 c t3 t2 H4 i u0 c3 t3
+(fsubst0_fst i u0 c t3 c3 H6) e H7)))))) (\lambda (t5: T).(\lambda (H6:
+(subst0 i u0 u t5)).(\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c
+c3)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind Abbr)
+u0))).(ty3_conv g c3 t2 t0 (H1 i u0 c3 t2 (fsubst0_fst i u0 c t2 c3 H7) e H8)
+t5 t3 (H3 i u0 c3 t5 (fsubst0_both i u0 c u t5 H6 c3 H7) e H8) (pc3_fsubst0 c
+t3 t2 H4 i u0 c3 t3 (fsubst0_fst i u0 c t3 c3 H7) e H8)))))))) c2 t4
+H5)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (i:
+nat).(\lambda (u: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H0: (fsubst0
+i u c (TSort m) c2 t2)).(fsubst0_ind i u c (TSort m) (\lambda (c0:
+C).(\lambda (t0: T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u)) \to
+(ty3 g c0 t0 (TSort (next g m))))))) (\lambda (t3: T).(\lambda (H1: (subst0 i
+u (TSort m) t3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind Abbr)
+u))).(subst0_gen_sort u t3 i m H1 (ty3 g c t3 (TSort (next g m))))))))
+(\lambda (c3: C).(\lambda (_: (csubst0 i u c c3)).(\lambda (e: C).(\lambda
+(_: (getl i c (CHead e (Bind Abbr) u))).(ty3_sort g c3 m))))) (\lambda (t3:
+T).(\lambda (H1: (subst0 i u (TSort m) t3)).(\lambda (c3: C).(\lambda (_:
+(csubst0 i u c c3)).(\lambda (e: C).(\lambda (_: (getl i c (CHead e (Bind
+Abbr) u))).(subst0_gen_sort u t3 i m H1 (ty3 g c3 t3 (TSort (next g
+m)))))))))) c2 t2 H0)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
+(t0: T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i:
+nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2
+t2) \to (\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
+t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda
+(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c
+(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead
+e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3:
+T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5:
+(getl i c (CHead e (Bind Abbr) u0))).(and_ind (eq nat n i) (eq T t3 (lift (S
+n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda
+(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4:
+T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n:
+nat).(getl n c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C
+(CHead d (Bind Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind
+Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0)
+H8)) in (let H10 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c]))
+(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
+Abbr) u) n H0 (CHead e (Bind Abbr) u0) H8)) in ((let H11 \def (f_equal C T
+(\lambda (e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead
+e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind
+Abbr) u0) H8)) in (\lambda (H12: (eq C d e)).(let H13 \def (eq_ind_r C e
+(\lambda (c0: C).(getl n c (CHead c0 (Bind Abbr) u0))) H9 d H12) in (let H14
+\def (eq_ind_r T u0 (\lambda (t: T).(getl n c (CHead d (Bind Abbr) t))) H13 u
+H11) in (eq_ind T u (\lambda (t4: T).(ty3 g c (lift (S n) O t4) (lift (S n) O
+t0))) (ty3_lift g d u t0 H1 c O (S n) (getl_drop Abbr c d u n H14)) u0
+H11))))) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n H4)))))) (\lambda (c3:
+C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H5: (getl i c
+(CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3 (TLRef n) (lift (S n) O
+t0)) (\lambda (H6: (lt n i)).(let H7 \def (csubst0_getl_lt i n H6 c c3 u0 H4
+(CHead d (Bind Abbr) u) H0) in (or4_ind (getl n c3 (CHead d (Bind Abbr) u))
+(ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_:
+T).(eq C (CHead d (Bind Abbr) u) (CHead e0 (Bind b) u1)))))) (\lambda (b:
+B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0
+(Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (ex3_4 B C C T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+u0 e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead e1
+(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(_: T).(\lambda (w: T).(getl n c3 (CHead e2 (Bind b) w))))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0
+(minus i (S n)) u0 u1 w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))))
+(ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (H8: (getl n c3 (CHead d
+(Bind Abbr) u))).(ty3_abbr g n c3 d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T
+(\lambda (b: B).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e0 (Bind b) u0)))))) (\lambda (b: B).(\lambda
+(e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w))))))
+(\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
+(minus i (S n)) u0 u w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abbr) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n))
+u0 u1 w))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0: B).(\lambda
+(x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq C (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind
+x0) x3))).(\lambda (H11: (subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x2) H9) in ((let H13 \def (f_equal C B (\lambda
+(e0: C).(match e0 return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr
+| (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead
+x1 (Bind x0) x2) H9) in ((let H14 \def (f_equal C T (\lambda (e0: C).(match
+e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x2) H9) in
+(\lambda (H15: (eq B Abbr x0)).(\lambda (H16: (eq C d x1)).(let H17 \def
+(eq_ind_r T x2 (\lambda (t: T).(subst0 (minus i (S n)) u0 t x3)) H11 u H14)
+in (let H18 \def (eq_ind_r C x1 (\lambda (c: C).(getl n c3 (CHead c (Bind x0)
+x3))) H10 d H16) in (let H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3
+(CHead d (Bind b) x3))) H18 Abbr H15) in (let H20 \def (eq_ind nat (minus i
+n) (\lambda (n: nat).(getl n (CHead d (Bind Abbr) x3) (CHead e (Bind Abbr)
+u0))) (getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n
+i) c c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead d (Bind Abbr) x3) n H19
+(le_S_n n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in
+(ty3_abbr g n c3 d x3 H19 t0 (H2 (minus i (S n)) u0 d x3 (fsubst0_snd (minus
+i (S n)) u0 d u x3 H17) e (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0)
+x3 (minus i (S n)) H20)))))))))) H13)) H12))))))))) H8)) (\lambda (H8: (ex3_4
+B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq
+C (CHead d (Bind Abbr) u) (CHead e1 (Bind b) u0)))))) (\lambda (b:
+B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c3 (CHead e2
+(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))).(ex3_4_ind B C C T (\lambda
+(b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u1)))))) (\lambda (b: B).(\lambda (_: C).(\lambda
+(e2: C).(\lambda (u1: T).(getl n c3 (CHead e2 (Bind b) u1)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n))
+u0 e1 e2))))) (ty3 g c3 (TLRef n) (lift (S n) O t0)) (\lambda (x0:
+B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H9: (eq C
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3))).(\lambda (H10: (getl n c3
+(CHead x2 (Bind x0) x3))).(\lambda (H11: (csubst0 (minus i (S n)) u0 x1
+x2)).(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c]))
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H13 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abbr
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t:
+T).(getl n c3 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r
+C x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c x2)) H11 d H16) in (let
+H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u)))
+H17 Abbr H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n:
+nat).(getl n (CHead x2 (Bind Abbr) u) (CHead e (Bind Abbr) u0)))
+(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
+c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abbr) u) n H19 (le_S_n
+n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr
+g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n))
+u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n
+(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr)
+x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13))
+H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abbr) u) (CHead e1 (Bind b) u0))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abbr) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O t0))
+(\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda
+(x4: T).(\lambda (H9: (eq C (CHead d (Bind Abbr) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11:
+(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0
+x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c]))
+(CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d (Bind Abbr) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abbr) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abbr
+x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i (S n)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C
+x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c x2)) H12 d H17) in (let H20
+\def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) H10
+Abbr H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n: nat).(getl n
+(CHead x2 (Bind Abbr) x4) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e
+(Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind
+Abbr) u0) H5) (CHead x2 (Bind Abbr) x4) n H20 (le_S_n n i (le_S (S n) i H6)))
+(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abbr g n c3 x2 x4 H20 t0 (H2
+(minus i (S n)) u0 x2 x4 (fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19)
+e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n (minus i (S
+n))) d x2 u0 H19 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abbr) x2 (CHead e
+(Bind Abbr) u0) x4 (minus i (S n)) H21))))))))))) H14)) H13))))))))))) H8))
+H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u (csubst0_getl_ge i n H6 c
+c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda
+(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0
+c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
+u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift (S
+n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O
+u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n)
+O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n: nat).(getl n c (CHead e
+(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n:
+nat).(csubst0 n u0 c c3)) H5 n H7) in (let H11 \def (eq_ind C (CHead d (Bind
+Abbr) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
+(let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d
+(Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u)
+n H0 (CHead e (Bind Abbr) u0) H9)) in ((let H13 \def (f_equal C T (\lambda
+(e0: C).(match e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr)
+u0) (getl_mono c (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9))
+in (\lambda (H14: (eq C d e)).(let H15 \def (eq_ind_r C e (\lambda (c0:
+C).(getl n c (CHead c0 (Bind Abbr) u0))) H11 d H14) in (let H16 \def
+(eq_ind_r T u0 (\lambda (t: T).(getl n c (CHead d (Bind Abbr) t))) H15 u H13)
+in (let H17 \def (eq_ind_r T u0 (\lambda (t: T).(csubst0 n t c c3)) H10 u
+H13) in (eq_ind T u (\lambda (t4: T).(ty3 g c3 (lift (S n) O t4) (lift (S n)
+O t0))) (ty3_lift g d u t0 H1 c3 O (S n) (getl_drop Abbr c3 d u n
+(csubst0_getl_ge n n (le_n n) c c3 u H17 (CHead d (Bind Abbr) u) H16))) u0
+H13)))))) H12))))) t3 H8))) (subst0_gen_lref u0 t3 i n H4)))))))) c2 t2
+H3)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0:
+T).(\lambda (H1: (ty3 g d u t0)).(\lambda (H2: ((\forall (i: nat).(\forall
+(u0: T).(\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 d u c2 t2) \to
+(\forall (e: C).((getl i d (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t2
+t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda
+(t2: T).(\lambda (H3: (fsubst0 i u0 c (TLRef n) c2 t2)).(fsubst0_ind i u0 c
+(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead
+e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O u)))))) (\lambda (t3:
+T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5:
+(getl i c (CHead e (Bind Abbr) u0))).(and_ind (eq nat n i) (eq T t3 (lift (S
+n) O u0)) (ty3 g c t3 (lift (S n) O u)) (\lambda (H6: (eq nat n i)).(\lambda
+(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4:
+T).(ty3 g c t4 (lift (S n) O u))) (let H8 \def (eq_ind_r nat i (\lambda (n:
+nat).(getl n c (CHead e (Bind Abbr) u0))) H5 n H6) in (let H9 \def (eq_ind C
+(CHead d (Bind Abst) u) (\lambda (c0: C).(getl n c c0)) H0 (CHead e (Bind
+Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0)
+H8)) in (let H10 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (ee:
+C).(match ee return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind
+Abst) u) n H0 (CHead e (Bind Abbr) u0) H8)) in (False_ind (ty3 g c (lift (S
+n) O u0) (lift (S n) O u)) H10)))) t3 H7))) (subst0_gen_lref u0 t3 i n
+H4)))))) (\lambda (c3: C).(\lambda (H4: (csubst0 i u0 c c3)).(\lambda (e:
+C).(\lambda (H5: (getl i c (CHead e (Bind Abbr) u0))).(lt_le_e n i (ty3 g c3
+(TLRef n) (lift (S n) O u)) (\lambda (H6: (lt n i)).(let H7 \def
+(csubst0_getl_lt i n H6 c c3 u0 H4 (CHead d (Bind Abst) u) H0) in (or4_ind
+(getl n c3 (CHead d (Bind Abst) u)) (ex3_4 B C T T (\lambda (b: B).(\lambda
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead
+e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_:
+T).(\lambda (w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_:
+B).(\lambda (_: C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n))
+u0 u1 w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2)))))) (ex4_5 B C C T T
+(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
+(_: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda
+(b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl
+n c3 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
+C).(\lambda (u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))))
+(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
+(_: T).(csubst0 (minus i (S n)) u0 e1 e2))))))) (ty3 g c3 (TLRef n) (lift (S
+n) O u)) (\lambda (H8: (getl n c3 (CHead d (Bind Abst) u))).(ty3_abst g n c3
+d u H8 t0 H1)) (\lambda (H8: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0:
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0
+(Bind b) u0)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda
+(w: T).(getl n c3 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_:
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u
+w))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1:
+T).(\lambda (_: T).(eq C (CHead d (Bind Abst) u) (CHead e0 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c3
+(CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
+T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w))))) (ty3 g c3 (TLRef n)
+(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x2))).(\lambda (H10: (getl n c3 (CHead x1 (Bind x0) x3))).(\lambda (H11:
+(subst0 (minus i (S n)) u0 x2 x3)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9)
+in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x2) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x2) H9) in (\lambda (H15: (eq B Abst
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x2 (\lambda (t:
+T).(subst0 (minus i (S n)) u0 t x3)) H11 u H14) in (let H18 \def (eq_ind_r C
+x1 (\lambda (c: C).(getl n c3 (CHead c (Bind x0) x3))) H10 d H16) in (let H19
+\def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead d (Bind b) x3))) H18
+Abst H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n: nat).(getl n
+(CHead d (Bind Abst) x3) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e
+(Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind
+Abbr) u0) H5) (CHead d (Bind Abst) x3) n H19 (le_S_n n i (le_S (S n) i H6)))
+(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_conv g c3 (lift (S n) O u)
+(lift (S n) O t0) (ty3_lift g d u t0 H1 c3 O (S n) (getl_drop Abst c3 d x3 n
+H19)) (TLRef n) (lift (S n) O x3) (ty3_abst g n c3 d x3 H19 t0 (H2 (minus i
+(S n)) u0 d x3 (fsubst0_snd (minus i (S n)) u0 d u x3 H17) e (getl_gen_S
+(Bind Abst) d (CHead e (Bind Abbr) u0) x3 (minus i (S n)) H20))) (pc3_lift c3
+d (S n) O (getl_drop Abst c3 d x3 n H19) x3 u (pc3_pr2_x d x3 u (pr2_delta d
+e u0 (r (Bind Abst) (minus i (S n))) (getl_gen_S (Bind Abst) d (CHead e (Bind
+Abbr) u0) x3 (minus i (S n)) H20) u u (pr0_refl u) x3 H17))))))))))) H13))
+H12))))))))) H8)) (\lambda (H8: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead d (Bind Abst) u) (CHead e1
+(Bind b) u0)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
+(u: T).(getl n c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1
+e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
+C).(\lambda (u1: T).(eq C (CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))
+(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u1: T).(getl n c3
+(CHead e2 (Bind b) u1)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
+C).(\lambda (_: T).(csubst0 (minus i (S n)) u0 e1 e2))))) (ty3 g c3 (TLRef n)
+(lift (S n) O u)) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda
+(x3: T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x3))).(\lambda (H11:
+(csubst0 (minus i (S n)) u0 x1 x2)).(let H12 \def (f_equal C C (\lambda (e0:
+C).(match e0 return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead
+c _ _) \Rightarrow c])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9)
+in ((let H13 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H14
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H15: (eq B Abst
+x0)).(\lambda (H16: (eq C d x1)).(let H17 \def (eq_ind_r T x3 (\lambda (t:
+T).(getl n c3 (CHead x2 (Bind x0) t))) H10 u H14) in (let H18 \def (eq_ind_r
+C x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c x2)) H11 d H16) in (let
+H19 \def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) u)))
+H17 Abst H15) in (let H20 \def (eq_ind nat (minus i n) (\lambda (n:
+nat).(getl n (CHead x2 (Bind Abst) u) (CHead e (Bind Abbr) u0)))
+(getl_conf_le i (CHead e (Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c
+c3 u0 H4 (CHead e (Bind Abbr) u0) H5) (CHead x2 (Bind Abst) u) n H19 (le_S_n
+n i (le_S (S n) i H6))) (S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_abst
+g n c3 x2 u H19 t0 (H2 (minus i (S n)) u0 x2 u (fsubst0_fst (minus i (S n))
+u0 d u x2 H18) e (csubst0_getl_ge_back (minus i (S n)) (minus i (S n)) (le_n
+(minus i (S n))) d x2 u0 H18 (CHead e (Bind Abbr) u0) (getl_gen_S (Bind Abst)
+x2 (CHead e (Bind Abbr) u0) u (minus i (S n)) H20))))))))))) H13))
+H12))))))))) H8)) (\lambda (H8: (ex4_5 B C C T T (\lambda (b: B).(\lambda
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead d (Bind
+Abst) u) (CHead e1 (Bind b) u0))))))) (\lambda (b: B).(\lambda (_:
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b:
+B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C
+(CHead d (Bind Abst) u) (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c3 (CHead e2
+(Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
+(u1: T).(\lambda (w: T).(subst0 (minus i (S n)) u0 u1 w)))))) (\lambda (_:
+B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
+(minus i (S n)) u0 e1 e2)))))) (ty3 g c3 (TLRef n) (lift (S n) O u)) (\lambda
+(x0: B).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H9: (eq C (CHead d (Bind Abst) u) (CHead x1 (Bind x0)
+x3))).(\lambda (H10: (getl n c3 (CHead x2 (Bind x0) x4))).(\lambda (H11:
+(subst0 (minus i (S n)) u0 x3 x4)).(\lambda (H12: (csubst0 (minus i (S n)) u0
+x1 x2)).(let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c]))
+(CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H14 \def
+(f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead d (Bind Abst) u) (CHead x1 (Bind x0) x3) H9) in ((let H15
+\def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind
+Abst) u) (CHead x1 (Bind x0) x3) H9) in (\lambda (H16: (eq B Abst
+x0)).(\lambda (H17: (eq C d x1)).(let H18 \def (eq_ind_r T x3 (\lambda (t:
+T).(subst0 (minus i (S n)) u0 t x4)) H11 u H15) in (let H19 \def (eq_ind_r C
+x1 (\lambda (c: C).(csubst0 (minus i (S n)) u0 c x2)) H12 d H17) in (let H20
+\def (eq_ind_r B x0 (\lambda (b: B).(getl n c3 (CHead x2 (Bind b) x4))) H10
+Abst H16) in (let H21 \def (eq_ind nat (minus i n) (\lambda (n: nat).(getl n
+(CHead x2 (Bind Abst) x4) (CHead e (Bind Abbr) u0))) (getl_conf_le i (CHead e
+(Bind Abbr) u0) c3 (csubst0_getl_ge i i (le_n i) c c3 u0 H4 (CHead e (Bind
+Abbr) u0) H5) (CHead x2 (Bind Abst) x4) n H20 (le_S_n n i (le_S (S n) i H6)))
+(S (minus i (S n))) (minus_x_Sy i n H6)) in (ty3_conv g c3 (lift (S n) O u)
+(lift (S n) O t0) (ty3_lift g x2 u t0 (H2 (minus i (S n)) u0 x2 u
+(fsubst0_fst (minus i (S n)) u0 d u x2 H19) e (csubst0_getl_ge_back (minus i
+(S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind
+Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S
+n)) H21))) c3 O (S n) (getl_drop Abst c3 x2 x4 n H20)) (TLRef n) (lift (S n)
+O x4) (ty3_abst g n c3 x2 x4 H20 t0 (H2 (minus i (S n)) u0 x2 x4
+(fsubst0_both (minus i (S n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back
+(minus i (S n)) (minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e
+(Bind Abbr) u0) (getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus
+i (S n)) H21)))) (pc3_lift c3 x2 (S n) O (getl_drop Abst c3 x2 x4 n H20) x4 u
+(pc3_fsubst0 d u u (pc3_refl d u) (minus i (S n)) u0 x2 x4 (fsubst0_both
+(minus i (S n)) u0 d u x4 H18 x2 H19) e (csubst0_getl_ge_back (minus i (S n))
+(minus i (S n)) (le_n (minus i (S n))) d x2 u0 H19 (CHead e (Bind Abbr) u0)
+(getl_gen_S (Bind Abst) x2 (CHead e (Bind Abbr) u0) x4 (minus i (S n))
+H21)))))))))))) H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i
+n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind
+Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef
+n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e:
+C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u0))).(and_ind (eq nat n i)
+(eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift (S n) O u)) (\lambda (H7: (eq
+nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n)
+O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n) O u))) (let H9 \def (eq_ind_r
+nat i (\lambda (n: nat).(getl n c (CHead e (Bind Abbr) u0))) H6 n H7) in (let
+H10 \def (eq_ind_r nat i (\lambda (n: nat).(csubst0 n u0 c c3)) H5 n H7) in
+(let H11 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c0: C).(getl n c
+c0)) H0 (CHead e (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abst) u) n H0
+(CHead e (Bind Abbr) u0) H9)) in (let H12 \def (eq_ind C (CHead d (Bind Abst)
+u) (\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False]) | (Flat _) \Rightarrow False])])) I (CHead e (Bind Abbr) u0)
+(getl_mono c (CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
+(False_ind (ty3 g c3 (lift (S n) O u0) (lift (S n) O u)) H12))))) t3 H8)))
+(subst0_gen_lref u0 t3 i n H4)))))))) c2 t2 H3)))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (t0: T).(\lambda (H0: (ty3 g c u t0)).(\lambda
+(H1: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c u c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t2 t0)))))))))).(\lambda (b: B).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda
+(H3: ((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t4:
+T).((fsubst0 i u0 (CHead c (Bind b) u) t2 c2 t4) \to (\forall (e: C).((getl i
+(CHead c (Bind b) u) (CHead e (Bind Abbr) u0)) \to (ty3 g c2 t4
+t3)))))))))).(\lambda (t4: T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3
+t4)).(\lambda (_: ((\forall (i: nat).(\forall (u0: T).(\forall (c2:
+C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind b) u) t3 c2 t2) \to
+(\forall (e: C).((getl i (CHead c (Bind b) u) (CHead e (Bind Abbr) u0)) \to
+(ty3 g c2 t2 t4)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2:
+C).(\lambda (t5: T).(\lambda (H6: (fsubst0 i u0 c (THead (Bind b) u t2) c2
+t5)).(fsubst0_ind i u0 c (THead (Bind b) u t2) (\lambda (c0: C).(\lambda (t6:
+T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t6
+(THead (Bind b) u t3)))))) (\lambda (t6: T).(\lambda (H7: (subst0 i u0 (THead
+(Bind b) u t2) t6)).(\lambda (e: C).(\lambda (H8: (getl i c (CHead e (Bind
+Abbr) u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind b) u2
+t2))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t7: T).(eq T t6
+(THead (Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2
+t7)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c t6 (THead
+(Bind b) u t3)) (\lambda (H9: (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind
+b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
+T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))
+(ty3 g c t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H10: (eq T t6
+(THead (Bind b) x t2))).(\lambda (H11: (subst0 i u0 u x)).(eq_ind_r T (THead
+(Bind b) x t2) (\lambda (t7: T).(ty3 g c t7 (THead (Bind b) u t3))) (ex_ind T
+(\lambda (t7: T).(ty3 g (CHead c (Bind b) u) t4 t7)) (ty3 g c (THead (Bind b)
+x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H12: (ty3 g (CHead c
+(Bind b) u) t4 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c (Bind b) x) t3
+t7)) (ty3 g c (THead (Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x1:
+T).(\lambda (H13: (ty3 g (CHead c (Bind b) x) t3 x1)).(ty3_conv g c (THead
+(Bind b) u t3) (THead (Bind b) u t4) (ty3_bind g c u t0 H0 b t3 t4 H4 x0 H12)
+(THead (Bind b) x t2) (THead (Bind b) x t3) (ty3_bind g c x t0 (H1 i u0 c x
+(fsubst0_snd i u0 c u x H11) e H8) b t2 t3 (H3 (S i) u0 (CHead c (Bind b) x)
+t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c (Bind b) x)
+(csubst0_snd_bind b i u0 u x H11 c)) e (getl_head (Bind b) i c (CHead e (Bind
+Abbr) u0) H8 u)) x1 H13) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b)
+u t3) (pc3_refl c (THead (Bind b) u t3)) i u0 c (THead (Bind b) x t3)
+(fsubst0_snd i u0 c (THead (Bind b) u t3) (THead (Bind b) x t3) (subst0_fst
+u0 x u i H11 t3 (Bind b))) e H8)))) (ty3_correct g (CHead c (Bind b) x) t2 t3
+(H3 (S i) u0 (CHead c (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b)
+u) t2 (CHead c (Bind b) x) (csubst0_snd_bind b i u0 u x H11 c)) e (getl_head
+(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)))))) (ty3_correct g (CHead c
+(Bind b) u) t3 t4 H4)) t6 H10)))) H9)) (\lambda (H9: (ex2 T (\lambda (t2:
+T).(eq T t6 (THead (Bind b) u t2))) (\lambda (t3: T).(subst0 (s (Bind b) i)
+u0 t2 t3)))).(ex2_ind T (\lambda (t7: T).(eq T t6 (THead (Bind b) u t7)))
+(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)) (ty3 g c t6 (THead (Bind
+b) u t3)) (\lambda (x: T).(\lambda (H10: (eq T t6 (THead (Bind b) u
+x))).(\lambda (H11: (subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead (Bind
+b) u x) (\lambda (t7: T).(ty3 g c t7 (THead (Bind b) u t3))) (ex_ind T
+(\lambda (t7: T).(ty3 g (CHead c (Bind b) u) t3 t7)) (ty3 g c (THead (Bind b)
+u x) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H12: (ty3 g (CHead c
+(Bind b) u) t3 x0)).(ty3_bind g c u t0 H0 b x t3 (H3 (S i) u0 (CHead c (Bind
+b) u) x (fsubst0_snd (S i) u0 (CHead c (Bind b) u) t2 x H11) e (getl_head
+(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x0 H12))) (ty3_correct g (CHead
+c (Bind b) u) x t3 (H3 (S i) u0 (CHead c (Bind b) u) x (fsubst0_snd (S i) u0
+(CHead c (Bind b) u) t2 x H11) e (getl_head (Bind b) i c (CHead e (Bind Abbr)
+u0) H8 u)))) t6 H10)))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t6 (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t2 t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7:
+T).(subst0 (s (Bind b) i) u0 t2 t7))) (ty3 g c t6 (THead (Bind b) u t3))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t6 (THead (Bind b) x0
+x1))).(\lambda (H11: (subst0 i u0 u x0)).(\lambda (H12: (subst0 (s (Bind b)
+i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t7: T).(ty3 g c t7
+(THead (Bind b) u t3))) (ex_ind T (\lambda (t7: T).(ty3 g (CHead c (Bind b)
+u) t4 t7)) (ty3 g c (THead (Bind b) x0 x1) (THead (Bind b) u t3)) (\lambda
+(x: T).(\lambda (H13: (ty3 g (CHead c (Bind b) u) t4 x)).(ex_ind T (\lambda
+(t7: T).(ty3 g (CHead c (Bind b) x0) t3 t7)) (ty3 g c (THead (Bind b) x0 x1)
+(THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H14: (ty3 g (CHead c (Bind
+b) x0) t3 x2)).(ty3_conv g c (THead (Bind b) u t3) (THead (Bind b) u t4)
+(ty3_bind g c u t0 H0 b t3 t4 H4 x H13) (THead (Bind b) x0 x1) (THead (Bind
+b) x0 t3) (ty3_bind g c x0 t0 (H1 i u0 c x0 (fsubst0_snd i u0 c u x0 H11) e
+H8) b x1 t3 (H3 (S i) u0 (CHead c (Bind b) x0) x1 (fsubst0_both (S i) u0
+(CHead c (Bind b) u) t2 x1 H12 (CHead c (Bind b) x0) (csubst0_snd_bind b i u0
+u x0 H11 c)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x2
+H14) (pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c
+(THead (Bind b) u t3)) i u0 c (THead (Bind b) x0 t3) (fsubst0_snd i u0 c
+(THead (Bind b) u t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H11 t3
+(Bind b))) e H8)))) (ty3_correct g (CHead c (Bind b) x0) x1 t3 (H3 (S i) u0
+(CHead c (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1
+H12 (CHead c (Bind b) x0) (csubst0_snd_bind b i u0 u x0 H11 c)) e (getl_head
+(Bind b) i c (CHead e (Bind Abbr) u0) H8 u)))))) (ty3_correct g (CHead c
+(Bind b) u) t3 t4 H4)) t6 H10)))))) H9)) (subst0_gen_head (Bind b) u0 u t2 t6
+i H7)))))) (\lambda (c3: C).(\lambda (H7: (csubst0 i u0 c c3)).(\lambda (e:
+C).(\lambda (H8: (getl i c (CHead e (Bind Abbr) u0))).(ex_ind T (\lambda (t6:
+T).(ty3 g (CHead c3 (Bind b) u) t3 t6)) (ty3 g c3 (THead (Bind b) u t2)
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H9: (ty3 g (CHead c3 (Bind
+b) u) t3 x)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H7) e
+H8) b t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0
+(CHead c (Bind b) u) t2 (CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0
+H7 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H8 u)) x H9)))
+(ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) u)
+t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3 (Bind b) u)
+(csubst0_fst_bind b i c c3 u0 H7 u)) e (getl_head (Bind b) i c (CHead e (Bind
+Abbr) u0) H8 u)))))))) (\lambda (t6: T).(\lambda (H7: (subst0 i u0 (THead
+(Bind b) u t2) t6)).(\lambda (c3: C).(\lambda (H8: (csubst0 i u0 c
+c3)).(\lambda (e: C).(\lambda (H9: (getl i c (CHead e (Bind Abbr)
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind b) u2 t2)))
+(\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T (\lambda (t7: T).(eq T t6 (THead
+(Bind b) u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7))) (ex3_2 T
+T (\lambda (u2: T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)))) (ty3 g c3 t6 (THead
+(Bind b) u t3)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t6 (THead (Bind
+b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
+T).(eq T t6 (THead (Bind b) u2 t2))) (\lambda (u2: T).(subst0 i u0 u u2))
+(ty3 g c3 t6 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6
+(THead (Bind b) x t2))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead
+(Bind b) x t2) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind
+T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (ty3 g
+(CHead c3 (Bind b) u) t3 x0)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3
+(Bind b) u) x0 t7)) (ty3 g c3 (THead (Bind b) x t2) (THead (Bind b) u t3))
+(\lambda (x1: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) x0 x1)).(ex_ind
+T (\lambda (t7: T).(ty3 g (CHead c3 (Bind b) x) t3 t7)) (ty3 g c3 (THead
+(Bind b) x t2) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (ty3 g
+(CHead c3 (Bind b) x) t3 x2)).(ty3_conv g c3 (THead (Bind b) u t3) (THead
+(Bind b) u x0) (ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H8)
+e H9) b t3 x0 H13 x1 H14) (THead (Bind b) x t2) (THead (Bind b) x t3)
+(ty3_bind g c3 x t0 (H1 i u0 c3 x (fsubst0_both i u0 c u x H12 c3 H8) e H9) b
+t2 t3 (H3 (S i) u0 (CHead c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c
+(Bind b) u) t2 (CHead c3 (Bind b) x) (csubst0_both_bind b i u0 u x H12 c c3
+H8)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x2 H15)
+(pc3_fsubst0 c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead
+(Bind b) u t3)) i u0 c3 (THead (Bind b) x t3) (fsubst0_both i u0 c (THead
+(Bind b) u t3) (THead (Bind b) x t3) (subst0_fst u0 x u i H12 t3 (Bind b)) c3
+H8) e H9)))) (ty3_correct g (CHead c3 (Bind b) x) t2 t3 (H3 (S i) u0 (CHead
+c3 (Bind b) x) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3
+(Bind b) x) (csubst0_both_bind b i u0 u x H12 c c3 H8)) e (getl_head (Bind b)
+i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3 (Bind b) u)
+t3 x0 H13)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i) u0 (CHead
+c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2 (CHead c3
+(Bind b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind b) i c
+(CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex2 T
+(\lambda (t2: T).(eq T t6 (THead (Bind b) u t2))) (\lambda (t3: T).(subst0 (s
+(Bind b) i) u0 t2 t3)))).(ex2_ind T (\lambda (t7: T).(eq T t6 (THead (Bind b)
+u t7))) (\lambda (t7: T).(subst0 (s (Bind b) i) u0 t2 t7)) (ty3 g c3 t6
+(THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H11: (eq T t6 (THead (Bind
+b) u x))).(\lambda (H12: (subst0 (s (Bind b) i) u0 t2 x)).(eq_ind_r T (THead
+(Bind b) u x) (\lambda (t7: T).(ty3 g c3 t7 (THead (Bind b) u t3))) (ex_ind T
+(\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) t3 t7)) (ty3 g c3 (THead (Bind
+b) u x) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H13: (ty3 g (CHead
+c3 (Bind b) u) t3 x0)).(ty3_bind g c3 u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c
+u c3 H8) e H9) b x t3 (H3 (S i) u0 (CHead c3 (Bind b) u) x (fsubst0_both (S
+i) u0 (CHead c (Bind b) u) t2 x H12 (CHead c3 (Bind b) u) (csubst0_fst_bind b
+i c c3 u0 H8 u)) e (getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x0
+H13))) (ty3_correct g (CHead c3 (Bind b) u) x t3 (H3 (S i) u0 (CHead c3 (Bind
+b) u) x (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x H12 (CHead c3 (Bind
+b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind b) i c (CHead e
+(Bind Abbr) u0) H9 u)))) t6 H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T t6 (THead (Bind b) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t2 t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t7: T).(eq T t6 (THead (Bind b) u2 t7)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t7:
+T).(subst0 (s (Bind b) i) u0 t2 t7))) (ty3 g c3 t6 (THead (Bind b) u t3))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t6 (THead (Bind b) x0
+x1))).(\lambda (H12: (subst0 i u0 u x0)).(\lambda (H13: (subst0 (s (Bind b)
+i) u0 t2 x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t7: T).(ty3 g c3
+t7 (THead (Bind b) u t3))) (ex_ind T (\lambda (t7: T).(ty3 g (CHead c3 (Bind
+b) u) t3 t7)) (ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3))
+(\lambda (x: T).(\lambda (H14: (ty3 g (CHead c3 (Bind b) u) t3 x)).(ex_ind T
+(\lambda (t7: T).(ty3 g (CHead c3 (Bind b) u) x t7)) (ty3 g c3 (THead (Bind
+b) x0 x1) (THead (Bind b) u t3)) (\lambda (x2: T).(\lambda (H15: (ty3 g
+(CHead c3 (Bind b) u) x x2)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c3
+(Bind b) x0) t3 t7)) (ty3 g c3 (THead (Bind b) x0 x1) (THead (Bind b) u t3))
+(\lambda (x3: T).(\lambda (H16: (ty3 g (CHead c3 (Bind b) x0) t3
+x3)).(ty3_conv g c3 (THead (Bind b) u t3) (THead (Bind b) u x) (ty3_bind g c3
+u t0 (H1 i u0 c3 u (fsubst0_fst i u0 c u c3 H8) e H9) b t3 x H14 x2 H15)
+(THead (Bind b) x0 x1) (THead (Bind b) x0 t3) (ty3_bind g c3 x0 t0 (H1 i u0
+c3 x0 (fsubst0_both i u0 c u x0 H12 c3 H8) e H9) b x1 t3 (H3 (S i) u0 (CHead
+c3 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H13
+(CHead c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e
+(getl_head (Bind b) i c (CHead e (Bind Abbr) u0) H9 u)) x3 H16) (pc3_fsubst0
+c (THead (Bind b) u t3) (THead (Bind b) u t3) (pc3_refl c (THead (Bind b) u
+t3)) i u0 c3 (THead (Bind b) x0 t3) (fsubst0_both i u0 c (THead (Bind b) u
+t3) (THead (Bind b) x0 t3) (subst0_fst u0 x0 u i H12 t3 (Bind b)) c3 H8) e
+H9)))) (ty3_correct g (CHead c3 (Bind b) x0) x1 t3 (H3 (S i) u0 (CHead c3
+(Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t2 x1 H13 (CHead
+c3 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H12 c c3 H8)) e (getl_head
+(Bind b) i c (CHead e (Bind Abbr) u0) H9 u)))))) (ty3_correct g (CHead c3
+(Bind b) u) t3 x H14)))) (ty3_correct g (CHead c3 (Bind b) u) t2 t3 (H3 (S i)
+u0 (CHead c3 (Bind b) u) t2 (fsubst0_fst (S i) u0 (CHead c (Bind b) u) t2
+(CHead c3 (Bind b) u) (csubst0_fst_bind b i c c3 u0 H8 u)) e (getl_head (Bind
+b) i c (CHead e (Bind Abbr) u0) H9 u)))) t6 H11)))))) H10)) (subst0_gen_head
+(Bind b) u0 u t2 t6 i H7)))))))) c2 t5 H6))))))))))))))))))) (\lambda (c:
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (H0: (ty3 g c w u)).(\lambda (H1:
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c w c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t2 u)))))))))).(\lambda (v: T).(\lambda (t0:
+T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3:
+((\forall (i: nat).(\forall (u0: T).(\forall (c2: C).(\forall (t2:
+T).((fsubst0 i u0 c v c2 t2) \to (\forall (e: C).((getl i c (CHead e (Bind
+Abbr) u0)) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))))))).(\lambda (i:
+nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H4:
+(fsubst0 i u0 c (THead (Flat Appl) w v) c2 t2)).(fsubst0_ind i u0 c (THead
+(Flat Appl) w v) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c
+(CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3 (THead (Flat Appl) w (THead (Bind
+Abst) u t0))))))) (\lambda (t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat
+Appl) w v) t3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
+u0))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i u0 w u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead
+(Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4)))) (ty3 g c t3 (THead
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H7: (ex2 T (\lambda (u2:
+T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: T).(subst0 i u0 w
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v)))
+(\lambda (u2: T).(subst0 i u0 w u2)) (ty3 g c t3 (THead (Flat Appl) w (THead
+(Bind Abst) u t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl)
+x v))).(\lambda (H9: (subst0 i u0 w x)).(eq_ind_r T (THead (Flat Appl) x v)
+(\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead (Bind Abst) u t0))))
+(ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind Abst) u t0) t4)) (ty3 g c
+(THead (Flat Appl) x v) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x0: T).(\lambda (H10: (ty3 g c (THead (Bind Abst) u t0)
+x0)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
+(THead (Bind Abst) u t4) x0)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_:
+T).(ty3 g c u t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) u) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
+(t6: T).(ty3 g (CHead c (Bind Abst) u) t4 t6)))) (ty3 g c (THead (Flat Appl)
+x v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x1:
+T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (_: (pc3 c (THead (Bind Abst) u
+x1) x0)).(\lambda (_: (ty3 g c u x2)).(\lambda (H13: (ty3 g (CHead c (Bind
+Abst) u) t0 x1)).(\lambda (H14: (ty3 g (CHead c (Bind Abst) u) x1
+x3)).(ex_ind T (\lambda (t4: T).(ty3 g c u t4)) (ty3 g c (THead (Flat Appl) x
+v) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x4: T).(\lambda
+(H15: (ty3 g c u x4)).(ty3_conv g c (THead (Flat Appl) w (THead (Bind Abst) u
+t0)) (THead (Flat Appl) w (THead (Bind Abst) u x1)) (ty3_appl g c w u H0
+(THead (Bind Abst) u t0) x1 (ty3_bind g c u x4 H15 Abst t0 x1 H13 x3 H14))
+(THead (Flat Appl) x v) (THead (Flat Appl) x (THead (Bind Abst) u t0))
+(ty3_appl g c x u (H1 i u0 c x (fsubst0_snd i u0 c w x H9) e H6) v t0 H2)
+(pc3_fsubst0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat
+Appl) w (THead (Bind Abst) u t0)) (pc3_refl c (THead (Flat Appl) w (THead
+(Bind Abst) u t0))) i u0 c (THead (Flat Appl) x (THead (Bind Abst) u t0))
+(fsubst0_snd i u0 c (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
+(Flat Appl) x (THead (Bind Abst) u t0)) (subst0_fst u0 x w i H9 (THead (Bind
+Abst) u t0) (Flat Appl))) e H6)))) (ty3_correct g c x u (H1 i u0 c x
+(fsubst0_snd i u0 c w x H9) e H6)))))))))) (ty3_gen_bind g Abst c u t0 x0
+H10)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))) H7))
+(\lambda (H7: (ex2 T (\lambda (t2: T).(eq T t3 (THead (Flat Appl) w t2)))
+(\lambda (t2: T).(subst0 (s (Flat Appl) i) u0 v t2)))).(ex2_ind T (\lambda
+(t4: T).(eq T t3 (THead (Flat Appl) w t4))) (\lambda (t4: T).(subst0 (s (Flat
+Appl) i) u0 v t4)) (ty3 g c t3 (THead (Flat Appl) w (THead (Bind Abst) u
+t0))) (\lambda (x: T).(\lambda (H8: (eq T t3 (THead (Flat Appl) w
+x))).(\lambda (H9: (subst0 (s (Flat Appl) i) u0 v x)).(eq_ind_r T (THead
+(Flat Appl) w x) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w (THead
+(Bind Abst) u t0)))) (ty3_appl g c w u H0 x t0 (H3 (s (Flat Appl) i) u0 c x
+(fsubst0_snd (s (Flat Appl) i) u0 c v x H9) e H6)) t3 H8)))) H7)) (\lambda
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t3 (THead (Flat Appl)
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u0 v t2))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 w u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Flat Appl) i) u0 v t4))) (ty3 g c t3 (THead
+(Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H8: (eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H9: (subst0 i
+u0 w x0)).(\lambda (H10: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T
+(THead (Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c t4 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c (THead (Bind
+Abst) u t0) t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H11: (ty3 g c (THead
+(Bind Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 c (THead (Bind Abst) u t4) x)))) (\lambda (_:
+T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c u t5)))) (\lambda (t4:
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+(\lambda (t6: T).(ty3 g c3 t6 t3)) (ty3_cast g c3 x t3 (H1 i u c3 x
+(fsubst0_both i u c t2 x H10 c3 H6) e H7) t0 (H3 i u c3 t3 (fsubst0_fst i u c
+t3 c3 H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t5 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Cast) i) u t2 t3))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t6: T).(eq T t5 (THead (Flat Cast) u2 t6)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u t3 u2))) (\lambda (_: T).(\lambda (t6:
+T).(subst0 (s (Flat Cast) i) u t2 t6))) (ty3 g c3 t5 t3) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x0
+x1))).(\lambda (H10: (subst0 i u t3 x0)).(\lambda (H11: (subst0 (s (Flat
+Cast) i) u t2 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t6:
+T).(ty3 g c3 t6 t3)) (ty3_conv g c3 t3 t0 (H3 i u c3 t3 (fsubst0_fst i u c t3
+c3 H6) e H7) (THead (Flat Cast) x0 x1) x0 (ty3_cast g c3 x1 x0 (ty3_conv g c3
+x0 t0 (H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7) x1 t3 (H1 i u
+c3 x1 (fsubst0_both i u c t2 x1 H11 c3 H6) e H7) (pc3_s c3 t3 x0 (pc3_fsubst0
+c t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e
+H7))) t0 (H3 i u c3 x0 (fsubst0_both i u c t3 x0 H10 c3 H6) e H7))
+(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x0 (fsubst0_both i u c t3 x0 H10
+c3 H6) e H7)) t5 H9)))))) H8)) (subst0_gen_head (Flat Cast) u t3 t2 t5 i
+H5)))))))) c2 t4 H4)))))))))))))) c1 t1 t H))))).
+
+theorem ty3_csubst0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c1
+(CHead e (Bind Abbr) u)) \to (\forall (c2: C).((csubst0 i u c1 c2) \to (ty3 g
+c2 t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(\lambda (e: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H0: (getl i c1 (CHead e (Bind Abbr) u))).(\lambda (c2:
+C).(\lambda (H1: (csubst0 i u c1 c2)).(ty3_fsubst0 g c1 t1 t2 H i u c2 t1
+(fsubst0_fst i u c1 t1 c2 H1) e H0))))))))))).
+
+theorem ty3_subst0:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((ty3 g c t1
+t) \to (\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e
+(Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (ty3 g c t2
+t)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
+(ty3 g c t1 t)).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H0: (getl i c (CHead e (Bind Abbr) u))).(\lambda (t2: T).(\lambda (H1:
+(subst0 i u t1 t2)).(ty3_fsubst0 g c t1 t H i u c t2 (fsubst0_snd i u c t1 t2
+H1) e H0))))))))))).
+
+theorem ty3_gen_cabbr:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
+(CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u t1 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
+e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a:
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u t (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: ((\forall (e:
+C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0))
+\to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d
+a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))))).(\lambda (H4: (pc3 c0 t4 t3)).(\lambda (e: C).(\lambda (u0:
+T).(\lambda (d: nat).(\lambda (H5: (getl d c0 (CHead e (Bind Abbr)
+u0))).(\lambda (a0: C).(\lambda (H6: (csubst1 d u0 c0 a0)).(\lambda (a:
+C).(\lambda (H7: (drop (S O) d a0 a)).(let H8 \def (H3 e u0 d H5 a0 H6 a H7)
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda
+(_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H9: (subst1 d u0 u (lift (S O) d x0))).(\lambda (H10: (subst1 d
+u0 t4 (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (H1 e
+u0 d H5 a0 H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H13: (subst1 d u0 t3 (lift (S O) d
+x2))).(\lambda (_: (subst1 d u0 t (lift (S O) d x3))).(\lambda (H15: (ty3 g a
+x2 x3)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t3 (lift
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 H9
+H13 (ty3_conv g a x2 x3 H15 x0 x1 H11 (pc3_gen_cabbr c0 t4 t3 H4 e u0 d H5 a0
+H6 a H7 x1 H10 x2 H13)))))))) H12))))))) H8)))))))))))))))))))) (\lambda (c0:
+C).(\lambda (m: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
+nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
+C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
+d a0 a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (TSort
+m) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (TSort
+(next g m)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2))) (TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t:
+T).(subst1 d u (TSort m) t)) (subst1_refl d u (TSort m)) (lift (S O) d (TSort
+m)) (lift_sort m (S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t:
+T).(subst1 d u (TSort (next g m)) t)) (subst1_refl d u (TSort (next g m)))
+(lift (S O) d (TSort (next g m))) (lift_sort (next g m) (S O) d)) (ty3_sort g
+a m)))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t:
+T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0:
+T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0:
+C).((csubst1 d0 u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
+(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
+a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (H6: (lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n:
+nat).(getl n (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0))) (getl_conf_le
+d0 (CHead e (Bind Abbr) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0
+(le_S (S n) d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (ex2_ind C
+(\lambda (e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abbr) u) e2))
+(\lambda (e2: C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1
+(minus d0 n) u0 (CHead d (Bind Abbr) u) x)).(\lambda (H9: (getl n a0 x)).(let
+H10 \def (eq_ind nat (minus d0 n) (\lambda (n: nat).(csubst1 n u0 (CHead d
+(Bind Abbr) u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11
+\def (csubst1_gen_head (Bind Abbr) d x u u0 (minus d0 (S n)) H10) in
+(ex3_2_ind T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind
+Abbr) u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u
+u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t)
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind
+Abbr) x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14:
+(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c:
+C).(getl n a0 c)) H9 (CHead x1 (Bind Abbr) x0) H12) in (let H16 \def (eq_ind
+nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus n (minus d0 (S
+n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18:
+(getl n a (CHead x3 (Bind Abbr) x2))).(\lambda (H19: (drop (S O) (minus d0 (S
+n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t: T).(subst1 (minus d0 (S
+n)) u0 u t)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 e
+u0 (minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Abbr) u0) u
+(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift
+(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S
+n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S
+n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (H23: (subst1 (minus d0
+(S n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4
+x5)).(let H25 \def (eq_ind T x4 (\lambda (t: T).(ty3 g x3 t x5)) H24 x2
+(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r
+nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O t) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S
+n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) (lift (S O)
+n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
+u0 (lift (S n) O t) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x5)
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0))
+(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O)
+d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x5)) (\lambda
+(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O t) t0))
+(subst1_lift_ge t (lift (S O) (minus d0 (S n)) x5) u0 (minus d0 (S n)) (S n)
+H23 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n)))
+(lift (S n) O x5)) (lift_d x5 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus
+d0 (S n))))) (ty3_abbr g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0
+H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt
+Abbr a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11))))))
+(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0)))) (\lambda
+(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n: nat).(drop (S
+O) n a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n:
+nat).(csubst1 n u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 (\lambda
+(n: nat).(getl n c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind nat n
+(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n0 u0
+(TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0
+u0 (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c: C).(getl n c0 c)) H0 (CHead e (Bind Abbr) u0) (getl_mono c0
+(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (let H11 \def
+(f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind
+Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0
+(CHead e (Bind Abbr) u0) H9)) in ((let H12 \def (f_equal C T (\lambda (e0:
+C).(match e0 return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead
+_ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)
+(getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind Abbr) u0) H9)) in
+(\lambda (H13: (eq C d e)).(let H14 \def (eq_ind_r T u0 (\lambda (t: T).(getl
+n c0 (CHead e (Bind Abbr) t))) H10 u H12) in (let H15 \def (eq_ind_r T u0
+(\lambda (t: T).(csubst1 n t c0 a0)) H8 u H12) in (eq_ind T u (\lambda (t0:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n t0 (TLRef n) (lift
+(S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n t0 (lift (S n) O t)
+(lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
+(let H16 \def (eq_ind_r C e (\lambda (c: C).(getl n c0 (CHead c (Bind Abbr)
+u))) H14 d H13) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1
+n u (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n
+u (lift (S n) O t) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))) (lift n O u) (lift n O t) (subst1_single n u (TLRef n)
+(lift (S O) n (lift n O u)) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda
+(t0: T).(subst0 n u (TLRef n) t0)) (subst0_lref u n) (lift (S O) n (lift n O
+u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) (eq_ind_r T
+(lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 n u (lift (S n) O t) t0))
+(subst1_refl n u (lift (S n) O t)) (lift (S O) n (lift n O t)) (lift_free t n
+(S O) O n (le_n (plus O n)) (le_O_n n))) (ty3_lift g d u t H1 a O n
+(getl_conf_ge_drop Abbr a0 d u n (csubst1_getl_ge n n (le_n n) c0 a0 u H15
+(CHead d (Bind Abbr) u) H16) a H7)))) u0 H12))))) H11))) d0 H6))))) (\lambda
+(H6: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0:
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S
+n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
+(eq_ind_r nat (plus (minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O t) (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus
+(minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d0 u0 (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O t)
+(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(subst1 d0
+u0 (TLRef (plus (minus n (S O)) (S O))) t0)) (subst1_refl d0 u0 (TLRef (plus
+(minus n (S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O))))
+(lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n H6))) (eq_ind_r T
+(lift (plus (S O) n) O t) (\lambda (t0: T).(subst1 d0 u0 (lift (S n) O t)
+t0)) (subst1_refl d0 u0 (lift (S n) O t)) (lift (S O) d0 (lift n O t))
+(lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H6))
+(le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a
+(TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S O)) a d u
+(getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 n (le_S_n
+d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a (S O) d0
+H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 (plus d0 (S
+O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 n (le_O_n
+d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus n (S O)))) (S
+(plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S O)))))) n
+(lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) H6)))))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H0: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3
+g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0:
+nat).((getl d0 d (CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d0
+u0 d a0) \to (\forall (a: C).((drop (S O) d0 a0 a) \to (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(subst1 d0 u0 u (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d0 u0 t (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda
+(u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Abbr)
+u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 a0)).(\lambda (a:
+C).(\lambda (H5: (drop (S O) d0 a0 a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O)
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H6:
+(lt n d0)).(let H7 \def (eq_ind nat (minus d0 n) (\lambda (n: nat).(getl n
+(CHead d (Bind Abst) u) (CHead e (Bind Abbr) u0))) (getl_conf_le d0 (CHead e
+(Bind Abbr) u0) c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0 (le_S (S n)
+d0 H6))) (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (ex2_ind C (\lambda
+(e2: C).(csubst1 (minus d0 n) u0 (CHead d (Bind Abst) u) e2)) (\lambda (e2:
+C).(getl n a0 e2)) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0
+(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0
+u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2)))) (\lambda (x: C).(\lambda (H8: (csubst1 (minus d0 n) u0
+(CHead d (Bind Abst) u) x)).(\lambda (H9: (getl n a0 x)).(let H10 \def
+(eq_ind nat (minus d0 n) (\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abst)
+u) x)) H8 (S (minus d0 (S n))) (minus_x_Sy d0 n H6)) in (let H11 \def
+(csubst1_gen_head (Bind Abst) d x u u0 (minus d0 (S n)) H10) in (ex3_2_ind T
+C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 (Bind Abst) u2))))
+(\lambda (u2: T).(\lambda (_: C).(subst1 (minus d0 (S n)) u0 u u2))) (\lambda
+(_: T).(\lambda (c2: C).(csubst1 (minus d0 (S n)) u0 d c2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
+(x0: T).(\lambda (x1: C).(\lambda (H12: (eq C x (CHead x1 (Bind Abst)
+x0))).(\lambda (H13: (subst1 (minus d0 (S n)) u0 u x0)).(\lambda (H14:
+(csubst1 (minus d0 (S n)) u0 d x1)).(let H15 \def (eq_ind C x (\lambda (c:
+C).(getl n a0 c)) H9 (CHead x1 (Bind Abst) x0) H12) in (let H16 \def (eq_ind
+nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus n (minus d0 (S
+n)))) (lt_plus_minus n d0 H6)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T x0 (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S n)) x1 e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H17: (eq T x0 (lift (S O) (minus d0 (S n)) x2))).(\lambda (H18:
+(getl n a (CHead x3 (Bind Abst) x2))).(\lambda (H19: (drop (S O) (minus d0 (S
+n)) x1 x3)).(let H20 \def (eq_ind T x0 (\lambda (t: T).(subst1 (minus d0 (S
+n)) u0 u t)) H13 (lift (S O) (minus d0 (S n)) x2) H17) in (let H21 \def (H2 e
+u0 (minus d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Abbr) u0) u
+(minus d0 (S n)) H7) x1 H14 x3 H19) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 (minus d0 (S n)) u0 u (lift (S O) (minus d0 (S n))
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (minus d0 (S n)) u0 t (lift
+(S O) (minus d0 (S n)) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x3 y1
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n)
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S
+n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H22: (subst1 (minus d0 (S
+n)) u0 u (lift (S O) (minus d0 (S n)) x4))).(\lambda (_: (subst1 (minus d0 (S
+n)) u0 t (lift (S O) (minus d0 (S n)) x5))).(\lambda (H24: (ty3 g x3 x4
+x5)).(let H25 \def (eq_ind T x4 (\lambda (t: T).(ty3 g x3 t x5)) H24 x2
+(subst1_confluence_lift u x4 u0 (minus d0 (S n)) H22 x2 H20)) in (eq_ind_r
+nat (plus (minus d0 (S n)) (S n)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 n0 u0 (lift (S n) O u) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (S
+n) (minus d0 (S n))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) (lift (S O)
+n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (plus (minus d0 (S n)) (S n))
+u0 (lift (S n) O u) (lift (S O) (plus (S n) (minus d0 (S n))) y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S n) O x2)
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(subst1 d0 u0 (TLRef n) t0))
+(subst1_refl d0 u0 (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O)
+d0 H6)) (eq_ind_r T (lift (S n) O (lift (S O) (minus d0 (S n)) x2)) (\lambda
+(t0: T).(subst1 (plus (minus d0 (S n)) (S n)) u0 (lift (S n) O u) t0))
+(subst1_lift_ge u (lift (S O) (minus d0 (S n)) x2) u0 (minus d0 (S n)) (S n)
+H20 O (le_O_n (minus d0 (S n)))) (lift (S O) (plus (S n) (minus d0 (S n)))
+(lift (S n) O x2)) (lift_d x2 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus
+d0 (S n))))) (ty3_abst g n a x3 x2 H18 x5 H25)) d0 (le_plus_minus (S n) d0
+H6)) d0 (le_plus_minus_sym (S n) d0 H6)))))))) H21)))))))) (getl_drop_conf_lt
+Abst a0 x1 x0 n H15 a (S O) (minus d0 (S n)) H16))))))))) H11))))))
+(csubst1_getl_lt d0 n H6 c0 a0 u0 H4 (CHead d (Bind Abst) u) H0)))) (\lambda
+(H6: (eq nat n d0)).(let H7 \def (eq_ind_r nat d0 (\lambda (n: nat).(drop (S
+O) n a0 a)) H5 n H6) in (let H8 \def (eq_ind_r nat d0 (\lambda (n:
+nat).(csubst1 n u0 c0 a0)) H4 n H6) in (let H9 \def (eq_ind_r nat d0 (\lambda
+(n: nat).(getl n c0 (CHead e (Bind Abbr) u0))) H3 n H6) in (eq_ind nat n
+(\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 n0 u0
+(TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 n0
+u0 (lift (S n) O u) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))))) (let H10 \def (eq_ind C (CHead d (Bind Abst) u)
+(\lambda (c: C).(getl n c0 c)) H0 (CHead e (Bind Abbr) u0) (getl_mono c0
+(CHead d (Bind Abst) u) n H0 (CHead e (Bind Abbr) u0) H9)) in (let H11 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (ee: C).(match ee return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e
+(Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind
+Abbr) u0) H9)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 n u0 (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 n u0 (lift (S n) O u) (lift (S O) n y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) H11))) d0 H6))))) (\lambda (H6: (lt d0
+n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0: nat).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind
+nat (plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda
+(_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus
+(minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d0 u0 (TLRef n0) (lift (S O) d0 y1)))) (\lambda
+(_: T).(\lambda (y2: T).(subst1 d0 u0 (lift (S n) O u) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d0 u0 (TLRef (plus (minus n (S O))
+(S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d0 u0
+(lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))) (TLRef (minus n (S O))) (lift n O u) (eq_ind_r T (TLRef (plus
+(minus n (S O)) (S O))) (\lambda (t0: T).(subst1 d0 u0 (TLRef (plus (minus n
+(S O)) (S O))) t0)) (subst1_refl d0 u0 (TLRef (plus (minus n (S O)) (S O))))
+(lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O)
+d0 (lt_le_minus d0 n H6))) (eq_ind_r T (lift (plus (S O) n) O u) (\lambda
+(t0: T).(subst1 d0 u0 (lift (S n) O u) t0)) (subst1_refl d0 u0 (lift (S n) O
+u)) (lift (S O) d0 (lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O
+n) (le_S (S d0) (plus O n) H6)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
+O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u)))
+(ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst)
+u) a0 (csubst1_getl_ge d0 n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4
+(CHead d (Bind Abst) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0)
+(\lambda (n0: nat).(le n0 n)) H6 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1)
+n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H6)))) (plus (S O) (minus n
+(S O))) (plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
+(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n
+(le_lt_trans O d0 n (le_O_n d0) H6))))))))))))))))))))) (\lambda (c0:
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1:
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a:
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: ((\forall
+(e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u)
+(CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 (CHead c0 (Bind
+b) u) a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(subst1 d u0 t3 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 t4 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda
+(_: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\forall (e:
+C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) (CHead e
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 (CHead c0 (Bind b) u)
+a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d u0 t4 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 t0 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda
+(u0: T).(\lambda (d: nat).(\lambda (H6: (getl d c0 (CHead e (Bind Abbr)
+u0))).(\lambda (a0: C).(\lambda (H7: (csubst1 d u0 c0 a0)).(\lambda (a:
+C).(\lambda (H8: (drop (S O) d a0 a)).(let H9 \def (H1 e u0 d H6 a0 H7 a H8)
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 u (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S
+O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda
+(x0: T).(\lambda (x1: T).(\lambda (H10: (subst1 d u0 u (lift (S O) d
+x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d x1))).(\lambda (H12: (ty3 g a
+x0 x1)).(let H13 \def (H5 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind
+Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u
+(lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b) x0) (drop_skip_bind (S O) d
+a0 a H8 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (S
+d) u0 t4 (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (S
+d) u0 t0 (lift (S O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g
+(CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(subst1 d u0 (THead (Bind b) u t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u t4) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H14: (subst1 (S d) u0 t4 (lift (S O) (S d)
+x2))).(\lambda (_: (subst1 (S d) u0 t0 (lift (S O) (S d) x3))).(\lambda (H16:
+(ty3 g (CHead a (Bind b) x0) x2 x3)).(let H17 \def (H3 e u0 (S d) (getl_head
+(Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d
+x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b)
+x0) (drop_skip_bind (S O) d a0 a H8 b x0)) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: (subst1 (S d) u0 t3 (lift (S
+O) (S d) x4))).(\lambda (H19: (subst1 (S d) u0 t4 (lift (S O) (S d)
+x5))).(\lambda (H20: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H21 \def
+(eq_ind T x5 (\lambda (t: T).(ty3 g (CHead a (Bind b) x0) x4 t)) H20 x2
+(subst1_confluence_lift t4 x5 u0 (S d) H19 x2 H14)) in (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(THead (Bind b) x0 x4) (THead (Bind b) x0 x2) (eq_ind_r T (THead (Bind b)
+(lift (S O) d x0) (lift (S O) (S d) x4)) (\lambda (t5: T).(subst1 d u0 (THead
+(Bind b) u t3) t5)) (subst1_head u0 u (lift (S O) d x0) d H10 (Bind b) t3
+(lift (S O) (S d) x4) H18) (lift (S O) d (THead (Bind b) x0 x4)) (lift_bind b
+x0 x4 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S
+d) x2)) (\lambda (t5: T).(subst1 d u0 (THead (Bind b) u t4) t5)) (subst1_head
+u0 u (lift (S O) d x0) d H10 (Bind b) t4 (lift (S O) (S d) x2) H14) (lift (S
+O) d (THead (Bind b) x0 x2)) (lift_bind b x0 x2 (S O) d)) (ty3_bind g a x0 x1
+H12 b x4 x2 H21 x3 H16)))))))) H17))))))) H13)))))))
+H9))))))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall (u0:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
+C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u
+t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
+d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda
+(u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr)
+u0))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u0 c0 a0)).(\lambda (a:
+C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H3 e u0 d H4 a0 H5 a H6)
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u
+t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w
+v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H8: (subst1 d u0 v (lift (S O) d x0))).(\lambda (H9: (subst1 d
+u0 (THead (Bind Abst) u t) (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0
+x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w (lift (S O) d
+x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d x3))).(\lambda (H14: (ty3 g
+a x2 x3)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O)
+d x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s (Bind Abst) d) u0 t
+t3))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat
+Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4
+x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind
+Abst) d) u0 t x5)).(let H18 \def (sym_equal T (lift (S O) d x1) (THead (Bind
+Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1
+(THead (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T x4 (lift (S
+O) d y)))) (\lambda (_: T).(\lambda (z: T).(eq T x5 (lift (S O) (S d) z))))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w
+v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead
+(Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x6: T).(\lambda (x7:
+T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x6 x7))).(\lambda (H20: (eq T
+x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5 (lift (S O) (S d) x7))).(let
+H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1 (s (Bind Abst) d) u0 t t0))
+H17 (lift (S O) (S d) x7) H21) in (let H23 \def (eq_ind T x4 (\lambda (t:
+T).(subst1 d u0 u t)) H16 (lift (S O) d x6) H20) in (let H24 \def (eq_ind T
+x1 (\lambda (t: T).(ty3 g a x0 t)) H10 (THead (Bind Abst) x6 x7) H19) in (let
+H25 \def (eq_ind T x6 (\lambda (t: T).(ty3 g a x0 (THead (Bind Abst) t x7)))
+H24 x3 (subst1_confluence_lift u x6 u0 d H23 x3 H13)) in (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead (Flat
+Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) (lift (S
+O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead (Flat Appl)
+w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v (lift (S O)
+d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat Appl x2 x0 (S
+O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift (S O) d (THead
+(Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat
+Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind Abst) x3 x7))
+(eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S d) x7))
+(\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) t0))
+(subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t (lift
+(S O) (S d) x7) H22) (lift (S O) d (THead (Bind Abst) x3 x7)) (lift_bind Abst
+x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead (Bind Abst) x3
+x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) (ty3_appl g a x2
+x3 H14 x0 x7 H25))))))))))) (lift_gen_bind Abst x4 x5 x1 (S O) d H18))))))))
+(subst1_gen_head (Bind Abst) u0 u t (lift (S O) d x1) d H9))))))) H11)))))))
+H7))))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (t0:
+T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: ((\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0:
+C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e:
+C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
+Abbr) u))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a:
+C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6)
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda (_: (subst1 d u
+t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u
+d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d
+u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4
+t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u t3 (lift (S O) d
+x2))).(\lambda (H13: (subst1 d u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g
+a x2 x3)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0
+(subst1_confluence_lift t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda
+(y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) x0 x2)
+x0 (eq_ind_r T (THead (Flat Cast) (lift (S O) d x0) (lift (S O) d x2))
+(\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) t)) (subst1_head u t4
+(lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) H12) (lift (S O) d
+(THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) H8 (ty3_cast g a
+x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
+
+theorem ty3_gen_cvoid:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c
+(CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(eq T t1 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t2 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead
+e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))))))))))) (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 t3 t)).(\lambda (H1: ((\forall (e:
+C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to
+(\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c0 u
+t4)).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
+d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (H4: (pc3 c0 t4
+t3)).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H5: (getl d
+c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H6: (drop (S O) d c0
+a)).(let H7 \def (H3 e u0 d H5 a H6) in (ex3_2_ind T T (\lambda (y1:
+T).(\lambda (_: T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a
+y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H8: (eq T u (lift (S O) d x0))).(\lambda (H9:
+(eq T t4 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def
+(eq_ind T t4 (\lambda (t: T).(pc3 c0 t t3)) H4 (lift (S O) d x1) H9) in (let
+H12 \def (eq_ind T t4 (\lambda (t: T).(ty3 g c0 u t)) H2 (lift (S O) d x1)
+H9) in (let H13 \def (eq_ind T u (\lambda (t: T).(ty3 g c0 t (lift (S O) d
+x1))) H12 (lift (S O) d x0) H8) in (eq_ind_r T (lift (S O) d x0) (\lambda
+(t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H1 e u0
+d H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t3 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15:
+(eq T t3 (lift (S O) d x2))).(\lambda (H16: (eq T t (lift (S O) d
+x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t (\lambda (t:
+T).(ty3 g c0 t3 t)) H0 (lift (S O) d x3) H16) in (let H19 \def (eq_ind T t3
+(\lambda (t: T).(ty3 g c0 t (lift (S O) d x3))) H18 (lift (S O) d x2) H15) in
+(let H20 \def (eq_ind T t3 (\lambda (t: T).(pc3 c0 (lift (S O) d x1) t)) H11
+(lift (S O) d x2) H15) in (eq_ind_r T (lift (S O) d x2) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) d x0) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) x0 x2 (refl_equal T (lift
+(S O) d x0)) (refl_equal T (lift (S O) d x2)) (ty3_conv g a x2 x3 H17 x0 x1
+H10 (pc3_gen_lift c0 x1 x2 (S O) d H20 a H6))) t3 H15))))))))) H14)) u
+H8))))))))) H7)))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda
+(e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c0 (CHead e
+(Bind Void) u))).(\lambda (a: C).(\lambda (_: (drop (S O) d c0
+a)).(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TSort m) (lift
+(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (TSort (next g m))
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(TSort m) (TSort (next g m)) (eq_ind_r T (TSort m) (\lambda (t: T).(eq T
+(TSort m) t)) (refl_equal T (TSort m)) (lift (S O) d (TSort m)) (lift_sort m
+(S O) d)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(eq T (TSort (next g
+m)) t)) (refl_equal T (TSort (next g m))) (lift (S O) d (TSort (next g m)))
+(lift_sort (next g m) (S O) d)) (ty3_sort g a m)))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0: nat).((getl
+d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d0 d a) \to
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0:
+T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e (Bind Void)
+u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0 a)).(lt_eq_gt_e n d0
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (H5: (lt
+n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n: nat).(getl n (CHead
+d (Bind Abbr) u) (CHead e (Bind Void) u0))) (getl_conf_le d0 (CHead e (Bind
+Void) u0) c0 H3 (CHead d (Bind Abbr) u) n H0 (le_S_n n d0 (le_S (S n) d0
+H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7 \def (eq_ind nat
+d0 (\lambda (n: nat).(drop (S O) n c0 a)) H4 (S (plus n (minus d0 (S n))))
+(lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
+C).(getl n a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H8:
+(eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9: (getl n a (CHead x1
+(Bind Abbr) x0))).(\lambda (H10: (drop (S O) (minus d0 (S n)) d x1)).(let H11
+\def (eq_ind T u (\lambda (t0: T).(\forall (e: C).(\forall (u: T).(\forall
+(d0: nat).((getl d0 d (CHead e (Bind Void) u)) \to (\forall (a: C).((drop (S
+O) d0 d a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t0 (lift (S
+O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))))))))) H2 (lift (S O)
+(minus d0 (S n)) x0) H8) in (let H12 \def (eq_ind T u (\lambda (t0: T).(ty3 g
+d t0 t)) H1 (lift (S O) (minus d0 (S n)) x0) H8) in (let H13 \def (H11 e u0
+(minus d0 (S n)) (getl_gen_S (Bind Abbr) d (CHead e (Bind Void) u0) u (minus
+d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq
+T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n)) y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H14: (eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus
+d0 (S n)) x2))).(\lambda (H15: (eq T t (lift (S O) (minus d0 (S n))
+x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let H17 \def (eq_ind T t (\lambda (t:
+T).(ty3 g d (lift (S O) (minus d0 (S n)) x0) t)) H12 (lift (S O) (minus d0 (S
+n)) x3) H15) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x3) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O)
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H18 \def
+(eq_ind_r T x2 (\lambda (t: T).(ty3 g x1 t x3)) H16 x0 (lift_inj x0 x2 (S O)
+(minus d0 (S n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n)))
+(lift (S n) O x3)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
+T t0 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S O) n0 (lift (S n) O x3)) (lift (S O) d0
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x3))
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(TLRef n) (lift (S n) O x3) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T
+(TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n))
+(lift_lref_lt n (S O) d0 H5)) (refl_equal T (lift (S O) d0 (lift (S n) O
+x3))) (ty3_abbr g n a x1 x0 H9 x3 H18)) (plus (S n) (minus d0 (S n)))
+(le_plus_minus (S n) d0 H5)) (lift (S n) O (lift (S O) (minus d0 (S n)) x3))
+(lift_d x3 (S O) (S n) (minus d0 (S n)) O (le_O_n (minus d0 (S n)))))) t
+H15))))))) H13))))))))) (getl_drop_conf_lt Abbr c0 d u n H0 a (S O) (minus d0
+(S n)) H7))))) (\lambda (H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0
+(\lambda (n: nat).(drop (S O) n c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat
+d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind Void) u0))) H3 n H5) in
+(eq_ind nat n (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (TLRef n) (lift (S O) n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
+T (lift (S n) O t) (lift (S O) n0 y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2))))) (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
+(\lambda (c: C).(getl n c0 c)) H0 (CHead e (Bind Void) u0) (getl_mono c0
+(CHead d (Bind Abbr) u) n H0 (CHead e (Bind Void) u0) H7)) in (let H9 \def
+(eq_ind C (CHead d (Bind Abbr) u) (\lambda (ee: C).(match ee return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e
+(Bind Void) u0) (getl_mono c0 (CHead d (Bind Abbr) u) n H0 (CHead e (Bind
+Void) u0) H7)) in (False_ind (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq
+T (TLRef n) (lift (S O) n y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift
+(S n) O t) (lift (S O) n y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))) H9))) d0 H5)))) (\lambda (H5: (lt d0 n)).(eq_ind_r nat (S (plus O
+(minus n (S O)))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (lift (S n) O t) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))) (eq_ind nat (plus (S O) (minus n (S O))) (\lambda
+(n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift
+(S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift
+(S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
+(eq_ind_r nat (plus (minus n (S O)) (S O)) (\lambda (n0: nat).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t) (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef (plus (minus n (S O)) (S O)))
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t)
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(TLRef (minus n (S O))) (lift n O t) (eq_ind_r T (TLRef (plus (minus n (S O))
+(S O))) (\lambda (t0: T).(eq T (TLRef (plus (minus n (S O)) (S O))) t0))
+(refl_equal T (TLRef (plus (minus n (S O)) (S O)))) (lift (S O) d0 (TLRef
+(minus n (S O)))) (lift_lref_ge (minus n (S O)) (S O) d0 (lt_le_minus d0 n
+H5))) (eq_ind_r T (lift (plus (S O) n) O t) (\lambda (t0: T).(eq T (lift (S
+n) O t) t0)) (refl_equal T (lift (S n) O t)) (lift (S O) d0 (lift n O t))
+(lift_free t n (S O) O d0 (le_S_n d0 (plus O n) (le_S (S d0) (plus O n) H5))
+(le_O_n d0))) (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a
+(TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S O)) a d u
+(getl_drop_conf_ge n (CHead d (Bind Abbr) u) c0 H0 a (S O) d0 H4 (eq_ind_r
+nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O))
+(plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0)
+H5)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus n (S O)))) (S
+(plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S O)))))) n
+(lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) H5)))))))))))))))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(H0: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3
+g d u t)).(\lambda (H2: ((\forall (e: C).(\forall (u0: T).(\forall (d0:
+nat).((getl d0 d (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O)
+d0 d a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O)
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d0 y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
+(Bind Void) u0))).(\lambda (a: C).(\lambda (H4: (drop (S O) d0 c0
+a)).(lt_eq_gt_e n d0 (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
+n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O
+u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))
+(\lambda (H5: (lt n d0)).(let H6 \def (eq_ind nat (minus d0 n) (\lambda (n:
+nat).(getl n (CHead d (Bind Abst) u) (CHead e (Bind Void) u0))) (getl_conf_le
+d0 (CHead e (Bind Void) u0) c0 H3 (CHead d (Bind Abst) u) n H0 (le_S_n n d0
+(le_S (S n) d0 H5))) (S (minus d0 (S n))) (minus_x_Sy d0 n H5)) in (let H7
+\def (eq_ind nat d0 (\lambda (n: nat).(drop (S O) n c0 a)) H4 (S (plus n
+(minus d0 (S n)))) (lt_plus_minus n d0 H5)) in (ex3_2_ind T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift (S O) (minus d0 (S n)) v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl n a (CHead e0 (Bind Abst) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop (S O) (minus d0 (S n)) d e0))) (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+C).(\lambda (H8: (eq T u (lift (S O) (minus d0 (S n)) x0))).(\lambda (H9:
+(getl n a (CHead x1 (Bind Abst) x0))).(\lambda (H10: (drop (S O) (minus d0 (S
+n)) d x1)).(let H11 \def (eq_ind T u (\lambda (t0: T).(\forall (e:
+C).(\forall (u: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u))
+\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T t0 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T t (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))))))))) H2 (lift (S O) (minus d0 (S n)) x0) H8) in (let H12 \def
+(eq_ind T u (\lambda (t0: T).(ty3 g d t0 t)) H1 (lift (S O) (minus d0 (S n))
+x0) H8) in (eq_ind_r T (lift (S O) (minus d0 (S n)) x0) (\lambda (t0:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O t0) (lift (S O)
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H13 \def
+(H11 e u0 (minus d0 (S n)) (getl_gen_S (Bind Abst) d (CHead e (Bind Void) u0)
+u (minus d0 (S n)) H6) x1 H10) in (ex3_2_ind T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (lift (S O) (minus d0 (S n)) x0) (lift (S O) (minus d0 (S n))
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) (minus d0 (S n))
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g x1 y1 y2))) (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O (lift (S O) (minus d0 (S
+n)) x0)) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T (lift (S O)
+(minus d0 (S n)) x0) (lift (S O) (minus d0 (S n)) x2))).(\lambda (H15: (eq T
+t (lift (S O) (minus d0 (S n)) x3))).(\lambda (H16: (ty3 g x1 x2 x3)).(let
+H17 \def (eq_ind T t (\lambda (t: T).(ty3 g d (lift (S O) (minus d0 (S n))
+x0) t)) H12 (lift (S O) (minus d0 (S n)) x3) H15) in (let H18 \def (eq_ind_r
+T x2 (\lambda (t: T).(ty3 g x1 t x3)) H16 x0 (lift_inj x0 x2 (S O) (minus d0
+(S n)) H14)) in (eq_ind T (lift (S O) (plus (S n) (minus d0 (S n))) (lift (S
+n) O x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
+(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))
+(eq_ind nat d0 (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
+T (lift (S O) n0 (lift (S n) O x0)) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S O) d0 (lift (S n) O x0)) (lift (S O) d0
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef n) (lift (S
+n) O x0) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift (S O) d0 (TLRef n)) (lift_lref_lt n (S O) d0
+H5)) (refl_equal T (lift (S O) d0 (lift (S n) O x0))) (ty3_abst g n a x1 x0
+H9 x3 H18)) (plus (S n) (minus d0 (S n))) (le_plus_minus (S n) d0 H5)) (lift
+(S n) O (lift (S O) (minus d0 (S n)) x0)) (lift_d x0 (S O) (S n) (minus d0 (S
+n)) O (le_O_n (minus d0 (S n)))))))))))) H13)) u H8))))))))
+(getl_drop_conf_lt Abst c0 d u n H0 a (S O) (minus d0 (S n)) H7))))) (\lambda
+(H5: (eq nat n d0)).(let H6 \def (eq_ind_r nat d0 (\lambda (n: nat).(drop (S
+O) n c0 a)) H4 n H5) in (let H7 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl
+n c0 (CHead e (Bind Void) u0))) H3 n H5) in (eq_ind nat n (\lambda (n0:
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O)
+n0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O)
+n0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H8 \def
+(eq_ind C (CHead d (Bind Abst) u) (\lambda (c: C).(getl n c0 c)) H0 (CHead e
+(Bind Void) u0) (getl_mono c0 (CHead d (Bind Abst) u) n H0 (CHead e (Bind
+Void) u0) H7)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
+(ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
+False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: K).Prop) with
+[(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])])) I (CHead e (Bind Void) u0) (getl_mono c0 (CHead d
+(Bind Abst) u) n H0 (CHead e (Bind Void) u0) H7)) in (False_ind (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n) (lift (S O) n y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) n y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) H9))) d0 H5)))) (\lambda
+(H5: (lt d0 n)).(eq_ind_r nat (S (plus O (minus n (S O)))) (\lambda (n0:
+nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O)
+d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O)
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind nat
+(plus (S O) (minus n (S O))) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r nat (plus (minus n (S
+O)) (S O)) (\lambda (n0: nat).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq
+T (TLRef n0) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
+(lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (TLRef
+(plus (minus n (S O)) (S O))) (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T (lift (S n) O u) (lift (S O) d0 y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (TLRef (minus n (S O))) (lift n O u)
+(eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (t0: T).(eq T
+(TLRef (plus (minus n (S O)) (S O))) t0)) (refl_equal T (TLRef (plus (minus n
+(S O)) (S O)))) (lift (S O) d0 (TLRef (minus n (S O)))) (lift_lref_ge (minus
+n (S O)) (S O) d0 (lt_le_minus d0 n H5))) (eq_ind_r T (lift (plus (S O) n) O
+u) (\lambda (t0: T).(eq T (lift (S n) O u) t0)) (refl_equal T (lift (S n) O
+u)) (lift (S O) d0 (lift n O u)) (lift_free u n (S O) O d0 (le_S_n d0 (plus O
+n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
+O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u)))
+(ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst)
+u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le
+n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n
+(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O)))
+(plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal
+nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n
+(le_O_n d0) H5))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda
+(t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall
+(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (b: B).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H2: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: ((\forall (e:
+C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) (CHead e
+(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 (Bind b) u) a)
+\to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (t0: T).(\lambda (H4:
+(ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\forall (e: C).(\forall
+(u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Void)
+u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 (Bind b) u) a) \to (ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda
+(d: nat).(\lambda (H6: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a:
+C).(\lambda (H7: (drop (S O) d c0 a)).(let H8 \def (H1 e u0 d H6 a H7) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T u (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (THead (Bind b) u t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H9: (eq T u (lift (S O) d x0))).(\lambda (H10: (eq T t (lift (S
+O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 \def (eq_ind T t (\lambda
+(t: T).(ty3 g c0 u t)) H0 (lift (S O) d x1) H10) in (let H13 \def (eq_ind T u
+(\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H12 (lift (S O) d x0) H9) in
+(let H14 \def (eq_ind T u (\lambda (t: T).(\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d (CHead c0 (Bind b) t) (CHead e (Bind Void) u))
+\to (\forall (a: C).((drop (S O) d (CHead c0 (Bind b) t) a) \to (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))))))))) H5 (lift (S O) d x0) H9) in (let H15 \def
+(eq_ind T u (\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) t4 t0)) H4 (lift (S
+O) d x0) H9) in (let H16 \def (eq_ind T u (\lambda (t: T).(\forall (e:
+C).(\forall (u: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) t) (CHead e
+(Bind Void) u)) \to (\forall (a: C).((drop (S O) d (CHead c0 (Bind b) t) a)
+\to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))))))))) H3 (lift (S O) d x0) H9) in
+(let H17 \def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) t3
+t4)) H2 (lift (S O) d x0) H9) in (eq_ind_r T (lift (S O) d x0) (\lambda (t5:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) t5 t3)
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b)
+t5 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (let H18 \def (H16 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind
+Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d
+c0 a H7 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3
+(lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S
+O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b)
+x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind
+b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2:
+T).(eq T (THead (Bind b) (lift (S O) d x0) t4) (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H19: (eq T t3 (lift (S O) (S d) x2))).(\lambda (H20: (eq T t4
+(lift (S O) (S d) x3))).(\lambda (H21: (ty3 g (CHead a (Bind b) x0) x2
+x3)).(let H22 \def (eq_ind T t4 (\lambda (t: T).(\forall (e: C).(\forall (u:
+T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) (lift (S O) d x0)) (CHead
+e (Bind Void) u)) \to (\forall (a: C).((drop (S O) d0 (CHead c0 (Bind b)
+(lift (S O) d x0)) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t
+(lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O)
+d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))))))))) H14
+(lift (S O) (S d) x3) H20) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t5:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S
+O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
+(THead (Bind b) (lift (S O) d x0) t5) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) (S d) x2)
+(\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead
+(Bind b) (lift (S O) d x0) t5) (lift (S O) d y1)))) (\lambda (_: T).(\lambda
+(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H23
+\def (H22 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H6
+(lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H7 b
+x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) (S
+d) x3) (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0
+(lift (S O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead a
+(Bind b) x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0)
+(lift (S O) (S d) x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2:
+T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H24: (eq T
+(lift (S O) (S d) x3) (lift (S O) (S d) x4))).(\lambda (_: (eq T t0 (lift (S
+O) (S d) x5))).(\lambda (H26: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H27
+\def (eq_ind_r T x4 (\lambda (t: T).(ty3 g (CHead a (Bind b) x0) t x5)) H26
+x3 (lift_inj x3 x4 (S O) (S d) H24)) in (eq_ind T (lift (S O) d (THead (Bind
+b) x0 x2)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+t5 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
+b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead
+(Bind b) x0 x3)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (lift (S O) d (THead (Bind b) x0 x2)) (lift (S O) d y1)))) (\lambda
+(_: T).(\lambda (y2: T).(eq T t5 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (lift (S O) d (THead (Bind b) x0 x2)) (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Bind b)
+x0 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))) (THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (refl_equal T (lift (S O)
+d (THead (Bind b) x0 x2))) (refl_equal T (lift (S O) d (THead (Bind b) x0
+x3))) (ty3_bind g a x0 x1 H11 b x2 x3 H21 x5 H27)) (THead (Bind b) (lift (S
+O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 (S O) d)) (THead (Bind b)
+(lift (S O) d x0) (lift (S O) (S d) x2)) (lift_bind b x0 x2 (S O) d))))))))
+H23)) t3 H19) t4 H20))))))) H18)) u H9)))))))))))) H8)))))))))))))))))))))
+(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w
+u)).(\lambda (H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl
+d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T u (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (v: T).(\lambda (t:
+T).(\lambda (H2: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3:
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e:
+C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind
+Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def
+(H3 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
+v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind
+Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w
+v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T v (lift (S O) d x0))).(\lambda (H8: (eq T (THead (Bind
+Abst) u t) (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def
+(eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift
+(S O) d x0) H7) in (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d
+y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead
+(Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2
+T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x2 x3))).(\lambda (H12: (eq T u
+(lift (S O) d x2))).(\lambda (H13: (eq T t (lift (S O) (S d) x3))).(let H14
+\def (eq_ind T x1 (\lambda (t: T).(ty3 g a x0 t)) H9 (THead (Bind Abst) x2
+x3) H11) in (eq_ind_r T (lift (S O) (S d) x3) (\lambda (t0: T).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d
+x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) w (THead (Bind Abst) u t0)) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H15 \def (eq_ind T u (\lambda
+(t: T).(\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e
+(Bind Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))))))))) H1 (lift (S O) d x2) H12) in (eq_ind_r T
+(lift (S O) d x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind
+Abst) t0 (lift (S O) (S d) x3))) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H16 \def (H15 e u0 d H4 a H5) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x2) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d x0)) (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H17: (eq T w (lift (S O) d x4))).(\lambda (H18:
+(eq T (lift (S O) d x2) (lift (S O) d x5))).(\lambda (H19: (ty3 g a x4
+x5)).(eq_ind_r T (lift (S O) d x4) (\lambda (t0: T).(ex3_2 T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (lift (S O) d x0)) (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) t0 (THead
+(Bind Abst) (lift (S O) d x2) (lift (S O) (S d) x3))) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H20 \def (eq_ind_r
+T x5 (\lambda (t: T).(ty3 g a x4 t)) H19 x2 (lift_inj x2 x5 (S O) d H18)) in
+(eq_ind T (lift (S O) d (THead (Bind Abst) x2 x3)) (\lambda (t0: T).(ex3_2 T
+T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) (lift (S O) d x4)
+(lift (S O) d x0)) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq
+T (THead (Flat Appl) (lift (S O) d x4) t0) (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead (Flat
+Appl) x4 x0)) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T t0 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
+(THead (Flat Appl) (lift (S O) d x4) (lift (S O) d (THead (Bind Abst) x2
+x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2))))) (eq_ind T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2
+x3))) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+(lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (lift (S O) d (THead (Flat Appl) x4 x0)) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Flat Appl) x4
+(THead (Bind Abst) x2 x3))) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x4 x0) (THead (Flat Appl) x4
+(THead (Bind Abst) x2 x3)) (refl_equal T (lift (S O) d (THead (Flat Appl) x4
+x0))) (refl_equal T (lift (S O) d (THead (Flat Appl) x4 (THead (Bind Abst) x2
+x3)))) (ty3_appl g a x4 x2 H20 x0 x3 H14)) (THead (Flat Appl) (lift (S O) d
+x4) (lift (S O) d (THead (Bind Abst) x2 x3))) (lift_flat Appl x4 (THead (Bind
+Abst) x2 x3) (S O) d)) (THead (Flat Appl) (lift (S O) d x4) (lift (S O) d
+x0)) (lift_flat Appl x4 x0 (S O) d)) (THead (Bind Abst) (lift (S O) d x2)
+(lift (S O) (S d) x3)) (lift_bind Abst x2 x3 (S O) d))) w H17)))))) H16)) u
+H12)) t H13))))))) (lift_gen_bind Abst u t x1 (S O) d H8)) v H7)))))))
+H6))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H0: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall
+(u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1
+y2)))))))))))).(\lambda (t0: T).(\lambda (H2: (ty3 g c0 t4 t0)).(\lambda (H3:
+((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind
+Void) u)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda
+(y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda
+(d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Void) u))).(\lambda (a:
+C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def (H3 e u d H4 a H5) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) (\lambda (y1:
+T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda
+(_: T).(eq T (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7:
+(eq T t4 (lift (S O) d x0))).(\lambda (H8: (eq T t0 (lift (S O) d
+x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T t0 (\lambda (t:
+T).(ty3 g c0 t4 t)) H2 (lift (S O) d x1) H8) in (let H11 \def (eq_ind T t4
+(\lambda (t: T).(ty3 g c0 t (lift (S O) d x1))) H10 (lift (S O) d x0) H7) in
+(let H12 \def (eq_ind T t4 (\lambda (t: T).(\forall (e: C).(\forall (u:
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u)) \to (\forall (a:
+C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T
+t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O)
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))))))))) H1 (lift
+(S O) d x0) H7) in (let H13 \def (eq_ind T t4 (\lambda (t: T).(ty3 g c0 t3
+t)) H0 (lift (S O) d x0) H7) in (eq_ind_r T (lift (S O) d x0) (\lambda (t:
+T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) t t3)
+(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def
+(H12 e u d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
+t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d
+x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))
+(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Cast) (lift (S
+O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
+(lift (S O) d x0) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3
+g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift
+(S O) d x2))).(\lambda (H16: (eq T (lift (S O) d x0) (lift (S O) d
+x3))).(\lambda (H17: (ty3 g a x2 x3)).(let H18 \def (eq_ind T t3 (\lambda (t:
+T).(ty3 g c0 t (lift (S O) d x0))) H13 (lift (S O) d x2) H15) in (eq_ind_r T
+(lift (S O) d x2) (\lambda (t: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_:
+T).(eq T (THead (Flat Cast) (lift (S O) d x0) t) (lift (S O) d y1))))
+(\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x0) (lift (S O) d y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H19 \def (eq_ind_r
+T x3 (\lambda (t: T).(ty3 g a x2 t)) H17 x0 (lift_inj x0 x3 (S O) d H16)) in
+(eq_ind T (lift (S O) d (THead (Flat Cast) x0 x2)) (\lambda (t: T).(ex3_2 T T
+(\lambda (y1: T).(\lambda (_: T).(eq T t (lift (S O) d y1)))) (\lambda (_:
+T).(\lambda (y2: T).(eq T (lift (S O) d x0) (lift (S O) d y2)))) (\lambda
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1:
+T).(\lambda (_: T).(eq T (lift (S O) d (THead (Flat Cast) x0 x2)) (lift (S O)
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d x0) (lift (S O)
+d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat
+Cast) x0 x2) x0 (refl_equal T (lift (S O) d (THead (Flat Cast) x0 x2)))
+(refl_equal T (lift (S O) d x0)) (ty3_cast g a x2 x0 H19 x1 H9)) (THead (Flat
+Cast) (lift (S O) d x0) (lift (S O) d x2)) (lift_flat Cast x0 x2 (S O) d)))
+t3 H15))))))) H14)) t4 H7)))))))))) H6)))))))))))))))) c t1 t2 H))))).
inductive csub3 (g:G): C \to (C \to Prop) \def
| csub3_sort: \forall (n: nat).(csub3 g (CSort n) (CSort n))
-| csub3_head: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (k: K).(\forall (u: T).(csub3 g (CHead c1 k u) (CHead c2 k u))))))
-| csub3_void: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csub3 g (CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
-| csub3_abst: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (u: T).(\forall (t: T).((ty3 g c2 u t) \to (csub3 g (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))).
-
-axiom csub3_gen_abbr: \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g (CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) .
-
-axiom csub3_gen_abst: \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g (CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) .
-
-axiom csub3_gen_bind: \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) .
-
-axiom csub3_refl: \forall (g: G).(\forall (c: C).(csub3 g c c)) .
-
-axiom csub3_clear_conf: \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c2 e2)))))))) .
-
-axiom csub3_drop_flat: \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Flat f) u)))))))))))) .
-
-axiom csub3_drop_abbr: \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))))))))))) .
-
-axiom csub3_drop_abst: \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) .
-
-axiom csub3_getl_abbr: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall (n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))))) .
-
-axiom csub3_getl_abst: \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall (n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) .
-
-axiom csub3_pr2: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pr2 c2 t1 t2))))))) .
-
-axiom csub3_pc3: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1 t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t1 t2))))))) .
-
-axiom csub3_ty3: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (ty3 g c2 t1 t2))))))) .
-
-axiom csub3_ty3_ld: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1 t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2)))))))) .
-
-axiom ty3_sred_wcpr0_pr0: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) \to (ty3 g c2 t2 t))))))))) .
-
-axiom ty3_sred_pr1: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) .
-
-axiom ty3_sred_pr2: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) .
-
-axiom ty3_sred_pr3: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t))))))) .
-
-axiom ty3_cred_pr2: \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1 v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) .
-
-axiom ty3_cred_pr3: \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1 v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2))))))))) .
-
-axiom ty3_gen__le_S_minus: \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to (le d (S (minus n h)))))) .
-
-axiom ty3_gen_lift: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: T).(ty3 g e t1 t2))))))))))) .
-
-axiom ty3_tred: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2))))))) .
-
-axiom ty3_sconv_pc3: \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 u2) \to (pc3 c t1 t2))))))))) .
-
-axiom ty3_sred_back: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2 t) \to (ty3 g c t1 t))))))))) .
-
-axiom ty3_sconv: \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1 u2) \to (ty3 g c u1 t2))))))))) .
-
-axiom ty3_tau0: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to (\forall (t2: T).((tau0 g c u t2) \to (ty3 g c u t2))))))) .
-
-axiom ty3_arity: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity g c t2 (asucc g a1)))))))) .
-
-axiom ty3_predicative: \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: Prop).P))))))) .
-
-axiom ty3_acyclic: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to ((pc3 c u t) \to (\forall (P: Prop).P)))))) .
-
-axiom ty3_sn3: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to (sn3 c t))))) .
-
-axiom pc3_dec: \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c u1 u2) ((pc3 c u1 u2) \to (\forall (P: Prop).P)))))))))) .
-
-axiom pc3_abst_dec: \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P))))))))))) .
-
-axiom ty3_inference: \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to (\forall (P: Prop).P)))))) .
+| csub3_head: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall
+(k: K).(\forall (u: T).(csub3 g (CHead c1 k u) (CHead c2 k u))))))
+| csub3_void: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall
+(b: B).((not (eq B b Void)) \to (\forall (u1: T).(\forall (u2: T).(csub3 g
+(CHead c1 (Bind Void) u1) (CHead c2 (Bind b) u2))))))))
+| csub3_abst: \forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall
+(u: T).(\forall (t: T).((ty3 g c2 u t) \to (csub3 g (CHead c1 (Bind Abst) t)
+(CHead c2 (Bind Abbr) u))))))).
+
+theorem csub3_gen_abbr:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g
+(CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2
+(Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
+(H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H return
+(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead
+e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C c2
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) with
+[(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind
+Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort
+n) (\lambda (e: C).(match e return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr)
+v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C
+c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1)))
+| (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u)
+(CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3
+\def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
+(CHead e1 (Bind Abbr) v) H1) in ((let H4 \def (f_equal C K (\lambda (e:
+C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead
+_ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in ((let
+H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind
+Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to
+(ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind
+Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3
+g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v)))
+(\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq T u v)).(eq_ind T v
+(\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to ((csub3 g e1 c0) \to
+(ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 (Bind Abbr) v)
+c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: C).((csub3 g e1 c0) \to
+(ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r
+C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) H (CHead c0 (Bind
+Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abbr) v)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C
+(CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v H7))) k (sym_eq K k
+(Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void
+c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void)
+u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2)
+c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match
+e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _)
+\Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind ((eq C
+(CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to
+(ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1)
+\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind
+Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def
+(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e1
+(Bind Abbr) v) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u) c2) \to
+((csub3 g c1 c0) \to ((ty3 g c0 u t) \to (ex2 C (\lambda (e2: C).(eq C c2
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0
+H1)))]) in (H0 (refl_equal C (CHead e1 (Bind Abbr) v)) (refl_equal C
+c2))))))).
+
+theorem csub3_gen_abst:
+ \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g
+(CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda
+(e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
+e2 v2 v1)))))))))
+\def
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda
+(H: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H return
+(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c (CHead
+e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: C).(eq C
+c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T
+(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1)))))))))) with [(csub3_sort n) \Rightarrow (\lambda
+(H0: (eq C (CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H1: (eq C (CSort
+n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H0) in (False_ind ((eq C
+(CSort n) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst)
+v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda
+(v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
+H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead
+c1 k u) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CHead c0 k u)
+c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead
+c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H4 \def (f_equal C K (\lambda
+(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1)
+in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_:
+C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead
+c1 k u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K
+k (Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c
+c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1)))
+(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2:
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
+v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst)
+(\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1
+c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1)))
+(\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2:
+T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2
+v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C
+(CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1
+e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0
+(Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c:
+C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
+e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2
+(\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind
+Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind
+Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2)))
+(ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1)
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1
+e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C
+(\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1)))
+(\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1))
+H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1
+(sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2)
+\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind
+Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def
+(eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1
+(Bind Abst) v1) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to
+((csub3 g c1 c0) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda (e2: C).(eq
+C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C
+T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t
+H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind
+Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t)
+(CHead e1 (Bind Abst) v1) H2) in ((let H5 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead
+c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1)
+H2) in (eq_ind C e1 (\lambda (c: C).((eq T t v1) \to ((eq C (CHead c0 (Bind
+Abbr) u) c2) \to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (or (ex2 C (\lambda
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1
+e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) (\lambda (H6: (eq T t
+v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to
+((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to (or (ex2 C (\lambda (e2: C).(eq C
+c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T
+(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u)
+c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to
+((ty3 g c0 u v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
+e2 v2 v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u
+v1)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind
+Abst) v1) c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda
+(e2: C).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda
+(e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
+e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead
+c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0
+u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1
+H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead
+e1 (Bind Abst) v1)) (refl_equal C c2))))))).
+
+theorem csub3_gen_bind:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
+(v1: T).((csub3 g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
+(v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def
+(match H return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c
+c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1
+e2)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
+(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def
+(eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I
+(CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2_3 B C
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1
+e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq
+C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u)
+c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead
+c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda
+(e: C).(match e return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in
+((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k
+u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind
+b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind
+b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3
+g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind
+T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0)
+\to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1)
+c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def
+(eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead
+c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1
+(refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k
+(sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) |
+(csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1
+(Bind Void) u1) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind
+b) u2) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow
+t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def
+(f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B) with [(CSort
+_) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead c1
+(Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1)
+(CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1)
+\to ((eq T u1 v1) \to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0)
+\to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7:
+(eq B Void b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C
+(CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda
+(_: T).((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B
+b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0
+(Bind b) u2) c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g
+e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda
+(e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10:
+(csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C
+c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b)
+u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1
+(Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b)
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10)))))
+c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4))
+H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead
+c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t)
+\Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in
+((let H5 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H6
+\def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B
+Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3
+g c c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7:
+(eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to ((eq C (CHead
+c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) \to (ex2_3 B C
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1
+e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C
+(CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to
+(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind Abbr) u)
+c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to
+((ty3 g c0 u v1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: (csub3 g e1
+c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def (eq_ind_r C c2 (\lambda (c:
+C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind Abbr) u) H9) in
+(let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 g (CHead e1 (Bind b) v1)
+(CHead c0 (Bind Abbr) u))) H12 Abst H7) in (ex2_3_intro B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g
+e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H10))))) c2 H9))
+t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) H3 H0
+H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C
+c2)))))))).
+
+theorem csub3_refl:
+ \forall (g: G).(\forall (c: C).(csub3 g c c))
+\def
+ \lambda (g: G).(\lambda (c: C).(C_ind (\lambda (c0: C).(csub3 g c0 c0))
+(\lambda (n: nat).(csub3_sort g n)) (\lambda (c0: C).(\lambda (H: (csub3 g c0
+c0)).(\lambda (k: K).(\lambda (t: T).(csub3_head g c0 c0 H k t))))) c)).
+
+theorem csub3_clear_conf:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2))
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1
+c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c
+e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c0
+e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n)
+e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csub3 g e1 e2))
+(\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3
+e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
+(clear (CHead c3 k u) e1)).((match k return (\lambda (k0: K).((clear (CHead
+c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2:
+C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\lambda (H3:
+(clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda
+(c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead
+c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind
+b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind
+b) u) (csub3_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1
+(clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: (clear
+(CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 u H3))
+in (ex2_ind C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4
+e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead
+c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csub3 g e1 x)).(\lambda
+(H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda
+(e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u)))))
+H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g
+c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda
+(e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (b:
+B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Void) u1)
+e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2))))
+(ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Void) u1) e2))
+(\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)) (CHead c4 (Bind b) u2)
+(csub3_void g c3 c4 H0 b H2 u1 u2) (clear_bind b c4 u2)) e1 (clear_gen_bind
+Void c3 e1 u1 H3)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0:
+(csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C
+(\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4
+e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c4 u
+t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Abst) t)
+e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u)
+e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) t) e2))
+(\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr)
+u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 (clear_gen_bind
+Abst c3 e1 t H3))))))))))) c1 c2 H)))).
+
+theorem csub3_drop_flat:
+ \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall
+(c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1
+(CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
+\def
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1:
+C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f)
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1
+(Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H
+(CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2
+\def (match H1 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ?
+c c0)).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f)
+u))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
+(CHead d1 (Flat f) u))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def
+(eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I
+(CHead d1 (Flat f) u) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2
+(Flat f) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) \Rightarrow (\lambda
+(H1: (eq C (CHead c1 k u0) (CHead d1 (Flat f) u))).(\lambda (H2: (eq C (CHead
+c0 k u0) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow
+t])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in ((let H4 \def (f_equal C K
+(\lambda (e: C).(match e return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1
+(Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow
+c])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in (eq_ind C d1 (\lambda (c:
+C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to
+((csub3 g c c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq K k (Flat
+f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0
+u0) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u)))))))) (\lambda (H7: (eq
+T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Flat f) t) c2) \to
+((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Flat f) u))))))) (\lambda (H8: (eq C (CHead c0
+(Flat f) u) c2)).(eq_ind C (CHead c0 (Flat f) u) (\lambda (c: C).((csub3 g d1
+c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c
+(CHead d2 (Flat f) u)))))) (\lambda (H9: (csub3 g d1 c0)).(ex_intro2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c0 (Flat
+f) u) (CHead d2 (Flat f) u))) c0 H9 (drop_refl (CHead c0 (Flat f) u)))) c2
+H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1
+H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow
+(\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead d1 (Flat f)
+u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C
+(CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C
+(CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead
+d2 (Flat f) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1)
+\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f)
+u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def (eq_ind
+C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C
+(CHead c0 (Bind Abbr) u0) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead
+d2 (Flat f) u))))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal C (CHead d1 (Flat
+f) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall
+(c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u:
+T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f)
+u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1
+c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall
+(u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Flat f)
+u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1:
+(drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(let H2 \def (match H1
+return (\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0:
+C).(\lambda (_: (drop n n2 c c0)).((eq nat n (S n0)) \to ((eq nat n2 O) \to
+((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2
+(Flat f) u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat
+O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort
+n1))).(\lambda (H4: (eq C c (CHead d1 (Flat f) u))).((let H5 \def (eq_ind nat
+O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq
+nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CSort n1) (CHead d2 (Flat f) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h c
+e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq
+nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C
+e (CHead d1 (Flat f) u))).((let H6 \def (f_equal nat nat (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n)
+\Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq
+nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat
+f) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))))))
+(\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) (CSort
+n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _)
+\Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Flat
+f) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))
+H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1
+u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d)
+O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda
+(H5: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda
+(n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u0)) (CSort
+n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n (r k d) c
+e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0)
+O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq nat (S d)
+O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1))
+\to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c
+e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0)
+O (CSort n1) (CHead d2 (Flat f) u))))))) H7))) h (sym_eq nat h (S n0) H2) H3
+H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal
+C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0:
+C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall
+(d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead
+d2 (Flat f) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u:
+T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead
+d1 (Flat f) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b:
+B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S
+n0) O (CHead c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0)))
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CHead c3 (Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4:
+(csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f)
+u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind
+b) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop
+(Bind b) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda
+(u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead
+c0 (Flat f0) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f0) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g
+d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead
+x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1
+(Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u:
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
+u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S
+n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f)
+u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5:
+(csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x
+(Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead
+d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3:
+C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u:
+T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f)
+u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u
+t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0
+(Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3
+g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3
+(CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst)
+c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
+
+theorem csub3_drop_abbr:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g
+c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind
+Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+n O c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1:
+C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u:
+T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr)
+u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1
+c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1
+(Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H
+(CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in
+(let H2 \def (match H1 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_:
+(csub3 ? c c0)).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C c0 c2) \to (ex2
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2
+(Bind Abbr) u))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C
+(CSort n) (CHead d1 (Bind Abbr) u))).(\lambda (H1: (eq C (CSort n) c2)).((let
+H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abbr) u) H0) in (False_ind ((eq C (CSort n) c2)
+\to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2
+(CHead d2 (Bind Abbr) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0)
+\Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr)
+u))).(\lambda (H2: (eq C (CHead c0 k u0) c2)).((let H3 \def (f_equal C T
+(\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1
+(Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e
+return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _)
+\Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H5
+\def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c: C).((eq K k (Bind
+Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to ((csub3 g c c0)
+\to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2
+(CHead d2 (Bind Abbr) u))))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind
+K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 u0) c2)
+\to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda (H7: (eq T u0
+u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to
+((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H8: (eq C (CHead c0
+(Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c:
+C).((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H9: (csub3 g d1
+c0)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O
+O (CHead c0 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c0 H9 (drop_refl (CHead
+c0 (Bind Abbr) u)))) c2 H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind
+Abbr) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0
+H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1)
+(CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2)
+c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match
+e return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k
+_) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _)
+\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C
+(CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead
+d2 (Bind Abbr) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1)
+\Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind
+Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def
+(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1
+(Bind Abbr) u) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u0) c2) \to
+((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) H4)) H3 H0
+H1)))]) in (H2 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C
+c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2:
+C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda
+(n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O
+(CSort n1) (CHead d1 (Bind Abbr) u))).(let H2 \def (match H1 return (\lambda
+(n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_:
+(drop n n2 c c0)).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort
+n1)) \to ((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2
+(Bind Abbr) u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq
+nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort
+n1))).(\lambda (H4: (eq C c (CHead d1 (Bind Abbr) u))).((let H5 \def (eq_ind
+nat O (\lambda (e: nat).(match e return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq
+nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Bind Abbr) u)) \to
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CSort n1) (CHead d2 (Bind Abbr) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h
+c e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq
+nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C
+e (CHead d1 (Bind Abbr) u))).((let H6 \def (f_equal nat nat (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).nat) with [O \Rightarrow h | (S n)
+\Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq
+nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind
+Abbr) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr)
+u))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0)
+(CSort n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _
+_) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1
+(Bind Abbr) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr)
+u)))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c
+e H1 u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat
+(S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort
+n1))).(\lambda (H5: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind
+nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n
+(r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))
+\to ((drop n (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))))))
+(\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0:
+nat).(match e0 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0)
+(r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))
+\to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))))
+H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S
+n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1
+(Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3
+g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0
+(CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda
+(k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0:
+T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abbr) u0)) \to (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda (b: B).(\lambda (u:
+T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0
+(Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csub3
+g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead
+x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0
+(CHead d1 (Bind Abbr) u0) u n0 H3)))))))) (\lambda (f: F).(\lambda (u:
+T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0
+(Flat f) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csub3
+g d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Bind Abbr)
+u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+(S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop
+(Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop
+(Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) k)))))) (\lambda (c0:
+C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall
+(d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead
+d2 (Bind Abbr) u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b
+Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u:
+T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind
+Abbr) u))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2
+(Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6:
+(drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2
+(Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u) H6
+u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abbr)
+u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1:
+(csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0)
+O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda
+(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda
+(u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind
+Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2
+(Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda
+(H6: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
+(CHead d2 (Bind Abbr) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind
+Abbr) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1
+(Bind Abbr) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
+
+theorem csub3_drop_abst:
+ \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g
+c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind
+Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1:
+C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t:
+T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst)
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda
+(c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (t:
+T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind
+C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abst) t)
+(drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1
+return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csub3 ? c c0)).((eq C c
+(CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with [(csub3_sort n) \Rightarrow
+(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) t))).(\lambda (H2: (eq C
+(CSort n) c2)).((let H3 \def (eq_ind C (CSort n) (\lambda (e: C).(match e
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abst) t) H1) in (False_ind ((eq C
+(CSort n) c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t)))))) H3)) H2))) | (csub3_head c0 c3 H1 k u) \Rightarrow (\lambda (H2: (eq
+C (CHead c0 k u) (CHead d1 (Bind Abst) t))).(\lambda (H3: (eq C (CHead c3 k
+u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in ((let H5 \def (f_equal C K
+(\lambda (e: C).(match e return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k u) (CHead d1 (Bind
+Abst) t) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in (eq_ind C d1 (\lambda (c:
+C).((eq K k (Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to
+((csub3 g c c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g
+d2 u0 t)))))))))) (\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst)
+(\lambda (k0: K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1
+c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O
+O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))
+(\lambda (H8: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind
+Abst) t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (eq C (CHead c3 (Bind Abst) t)
+c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))
+(\lambda (H10: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))
+(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O
+(CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H10 (drop_refl (CHead
+c3 (Bind Abst) t))))) c2 H9)) u (sym_eq T u t H8))) k (sym_eq K k (Bind Abst)
+H7))) c0 (sym_eq C c0 d1 H6))) H5)) H4)) H3 H1))) | (csub3_void c0 c3 H1 b H2
+u1 u2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Void) u1) (CHead d1
+(Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) c2)).((let H5
+\def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow
+(match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst
+\Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])]))
+I (CHead d1 (Bind Abst) t) H3) in (False_ind ((eq C (CHead c3 (Bind b) u2)
+c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst)
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H4 H1 H2))) | (csub3_abst
+c0 c3 H1 u t0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Abst) t0)
+(CHead d1 (Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind Abbr) u)
+c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow t0 | (CHead _ _ t) \Rightarrow t])) (CHead
+c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H3) in ((let H6 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) t0)
+(CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to
+((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0)
+\to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O
+c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))))
+(\lambda (H7: (eq T t0 t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3
+(Bind Abbr) u) c2) \to ((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr)
+u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda
+(H8: (eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u)
+(\lambda (c: C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst)
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda
+(d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (csub3 g d1
+c3)).(\lambda (H10: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))
+(ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H9
+(drop_refl (CHead c3 (Bind Abbr) u)) H10)))) c2 H8)) t0 (sym_eq T t0 t H7)))
+c0 (sym_eq C c0 d1 H6))) H5)) H4 H1 H2)))]) in (H2 (refl_equal C (CHead d1
+(Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H:
+((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1:
+C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda
+(c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c
+(CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1:
+C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind
+Abst) t))).(let H2 \def (match H1 return (\lambda (n: nat).(\lambda (n2:
+nat).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop n n2 c c0)).((eq nat
+n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1
+(Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl c)
+\Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O
+O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Bind
+Abst) t))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C
+c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2
+C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H2 H3 H4))))) |
+(drop_drop k h c e H1 u) \Rightarrow (\lambda (H2: (eq nat (S h) (S
+n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u) (CSort
+n1))).(\lambda (H5: (eq C e (CHead d1 (Bind Abst) t))).((let H6 \def (f_equal
+nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: nat).nat) with [O
+\Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0
+(\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) \to ((eq
+C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n) O c e) \to (or (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1)
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort
+n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2
+u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u)
+(CSort n1))).(let H9 \def (eq_ind C (CHead c k u) (\lambda (e0: C).(match e0
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _
+_) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1
+(Bind Abst) t)) \to ((drop (r k n0) O c e) \to (or (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2
+(Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2
+(Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))
+H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u)
+\Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d)
+O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u)) (CSort n1))).(\lambda
+(H5: (eq C (CHead e k u) (CHead d1 (Bind Abst) t))).(eq_ind nat (S n0)
+(\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u))
+(CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop n
+(r k d) c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0:
+T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H6: (eq nat (S d)
+O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return
+(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
+I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u)) (CSort n1))
+\to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop (S n0) (r k d)
+c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+(S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t)))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5
+H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C
+(CSort n1)) (refl_equal C (CHead d1 (Bind Abst) t)))))))) (\lambda (c0:
+C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall
+(d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3
+(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall
+(d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abst)
+t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+(S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (b: B).(\lambda (u:
+T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0
+(Bind b) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C
+T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind
+Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))
+(\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O
+(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 g d1
+x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O
+(CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst)
+t))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4))
+(\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u))))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0:
+T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 x0)).(\lambda (H6: (drop
+n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3
+(CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t
+(drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) (\lambda
+(f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop
+(S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O
+(CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead
+d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda
+(x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x
+(Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead
+x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop
+(S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3
+g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind
+Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O
+(CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda
+(u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5:
+(csub3 g d1 x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr)
+x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u)
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead
+c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead
+c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr)
+x1) H6 u) H7))))))) H4)) (H2 d1 t (drop_gen_drop (Flat f) c0 (CHead d1 (Bind
+Abst) t) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda
+(H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop
+(S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_:
+(not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1:
+C).(\lambda (t: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1)
+(CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst)
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind
+Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5:
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3
+(CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b)
+u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda
+(H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2)
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t))) x H6
+(drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 u2)))))) H5)) (\lambda
+(H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop
+n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+(S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u))))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda
+(x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0
+(Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b)
+u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead
+c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3
+g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b)
+u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u2)
+H8))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind
+Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda
+(H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop
+(S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda
+(d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda
+(H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst)
+t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop
+n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead
+d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O
+(CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0:
+T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))))
+(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2
+(Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
+(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1
+x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3
+(Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop
+(S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2
+(Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0)
+H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))))
+(\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
+(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
+(csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr)
+x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u)
+(CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead
+c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0:
+T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind
+Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst)
+c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
+
+theorem csub3_getl_abbr:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
+(n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g
+c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n
+c2 (CHead d2 (Bind Abbr) u)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e:
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u)))
+(\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x:
+C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
+Abbr) u))).((match x return (\lambda (c: C).((drop n O c1 c) \to ((clear c
+(CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C
+(\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind
+Abbr) u))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n O c1
+(CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abbr)
+u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n0 H4 (\forall (c2: C).((csub3
+g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl
+n c2 (CHead d2 (Bind Abbr) u))))))))) | (CHead c k t) \Rightarrow (\lambda
+(H3: (drop n O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead
+d1 (Bind Abbr) u))).((match k return (\lambda (k0: K).((drop n O c1 (CHead c
+k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2:
+C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) with [(Bind b)
+\Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t))).(\lambda (H6:
+(clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal
+C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u)
+(CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in
+((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c
+(CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e:
+C).(match e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead
+_ _ t) \Rightarrow t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t)
+(clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B
+Abbr b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csub3 g
+c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop n O c1 (CHead c
+(Bind b) t))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop n
+O c1 (CHead c (Bind b) u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c
+(\lambda (c: C).(drop n O c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in
+(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2
+(CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x0: C).(\lambda
+(H16: (csub3 g d1 x0)).(\lambda (H17: (drop n O c2 (CHead x0 (Bind Abbr)
+u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n
+c2 (CHead d2 (Bind Abbr) u))) x0 H16 (getl_intro n c2 (CHead x0 (Bind Abbr)
+u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 u)))))) (csub3_drop_abbr
+g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda
+(H5: (drop n O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat
+f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda
+(c0: C).((drop n O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g c0
+c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2
+(CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda (n0: nat).(\forall (x0:
+C).((drop n0 O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g x0 c2)
+\to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2
+(CHead d2 (Bind Abbr) u))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0
+(CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let
+H10 \def (eq_ind C x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t)
+(drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c
+(CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f
+t) in (let H11 \def (csub3_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1
+(Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind
+Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3
+g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(x1: C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13:
+(clear c2 x1)).(let H14 \def (csub3_gen_abbr g d1 x1 u H12) in (ex2_ind C
+(\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3
+g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: C).(\lambda (H15: (eq C x1
+(CHead x2 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x2)).(let H17 \def
+(eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15)
+in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abbr) u))) x2 H16 (getl_intro O c2 (CHead x2 (Bind Abbr) u)
+c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda
+(H8: ((\forall (x: C).((drop n0 O x (CHead c (Flat f) t)) \to (\forall (c2:
+C).((csub3 g x c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x0: C).(\lambda
+(H9: (drop (S n0) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10:
+(csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n0 H9) in
+(ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0
+(CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
+T).(drop n0 O e (CHead c (Flat f) t))))) (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda
+(x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2
+(Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) t))).(let H14
+\def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C
+(\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2:
+C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x4: C).(\lambda
+(H15: (csub3 g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2
+x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2
+e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0)
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x5: B).(\lambda (x6: C).(\lambda
+(x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19:
+(csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16
+(CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (ex2_ind
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2
+(Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x8: C).(\lambda
+(H22: (csub3 g d1 x8)).(\lambda (H23: (getl n0 x6 (CHead x8 (Bind Abbr)
+u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S
+n0) c2 (CHead d2 (Bind Abbr) u))) x8 H22 (getl_clear_bind x5 c2 x6 x7 H20
+(CHead x8 (Bind Abbr) u) n0 H23))))) H21)))))))) H17))))) H14)))))))
+H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+
+theorem csub3_getl_abst:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall
+(n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g
+c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda
+(n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e:
+C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t)))
+(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1
+x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).((match x return
+(\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to
+(\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t)))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n
+O c1 (CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst)
+t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csub3
+g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t)))))))))) | (CHead c k t0) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c
+k t0))).(\lambda (H4: (clear (CHead c k t0) (CHead d1 (Bind Abst)
+t))).((match k return (\lambda (k0: K).((drop n O c1 (CHead c k0 t0)) \to
+((clear (CHead c k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2:
+C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t)))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c
+(Bind b) t0))).(\lambda (H6: (clear (CHead c (Bind b) t0) (CHead d1 (Bind
+Abst) t))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
+(CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1
+(Bind Abst) t) t0 H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0)
+(clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) t)
+(CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in
+(\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2:
+C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t0 (\lambda (t:
+T).(drop n O c1 (CHead c (Bind b) t))) H5 t H9) in (let H14 \def (eq_ind_r B
+b (\lambda (b: B).(drop n O c1 (CHead c (Bind b) t))) H13 Abst H10) in (let
+H15 \def (eq_ind_r C c (\lambda (c: C).(drop n O c1 (CHead c (Bind Abst) t)))
+H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n
+O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n
+c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x0: C).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2
+(CHead x0 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x0 H17 (getl_intro n c2 (CHead
+x0 (Bind Abst) t) (CHead x0 (Bind Abst) t) H18 (clear_bind Abst x0 t)))))))
+H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H17: (csub3 g d1
+x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19:
+(ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u))))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H17 (getl_intro n c2
+(CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0
+x1)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8))
+H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f)
+t0))).(\lambda (H6: (clear (CHead c (Flat f) t0) (CHead d1 (Bind Abst)
+t))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c
+(Flat f) t0)) \to (\forall (c2: C).((csub3 g c0 c2) \to (or (ex2 C (\lambda
+(d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst)
+t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda
+(d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0:
+nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2:
+C).((csub3 g x0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
+n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2
+u t)))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f)
+t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C
+x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t0) (drop_gen_refl x0
+(CHead c (Flat f) t0) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind
+Abst) t) (clear_gen_flat f c (CHead d1 (Bind Abst) t) t0 H6) f t0) in (let
+H11 \def (csub3_clear_conf g (CHead c (Flat f) t0) c2 H10 (CHead d1 (Bind
+Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abst)
+t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1
+(Bind Abst) t) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def
+(csub3_gen_abst g d1 x1 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x1
+(CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T
+(\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u:
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x1 (CHead
+e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda
+(e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2
+(Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) t))).(\lambda
+(H17: (csub3 g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: C).(clear c2
+c)) H13 (CHead x2 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))))
+(ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2:
+C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
+C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g
+d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x2 H17
+(getl_intro O c2 (CHead x2 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15))
+(\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead
+e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda
+(e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
+e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O
+c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H16: (eq C x1 (CHead x2
+(Bind Abbr) x3))).(\lambda (H17: (csub3 g d1 x2)).(\lambda (H18: (ty3 g x2 x3
+t)).(let H19 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2
+(Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u:
+T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3
+g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x2 x3 H17 (getl_intro
+O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H19) H18)))))))) H15))
+H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x: C).((drop
+n0 O x (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x c2) \to (or
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x0:
+C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t0))).(\lambda (c2:
+C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c
+(Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e:
+C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_:
+B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t0))))) (or
+(ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead
+d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0
+(CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f)
+t0))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3)
+H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t))))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2
+(Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def
+(csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (or (ex2
+C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead
+d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x5:
+B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind
+x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda
+(c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2
+H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl n0 x6 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
+n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2
+u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl
+(S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda
+(_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2
+(CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u
+t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2:
+C).(getl n0 x6 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2:
+C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t)))
+(or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2
+(CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_:
+T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead
+d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))
+(\lambda (x8: C).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6
+(CHead x8 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1
+d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T
+(\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda
+(u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda
+(u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2))
+(\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x8 H23
+(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) t) n0 H24)))))) H22))
+(\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2)))
+(\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u))))
+(\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda
+(d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u:
+T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u:
+T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (H23: (csub3 g d1
+x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H25:
+(ty3 g x8 x9 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda
+(d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl
+(S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g
+d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1
+d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr)
+u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x8 x9 H23
+(getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n0 H24) H25)))))))
+H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1
+H2)))) H0))))))).
+
+theorem csub3_pr2:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
+t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pr2 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((csub3 g c c2) \to (pr2 c2 t t0)))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2:
+C).(\lambda (_: (csub3 g c c2)).(pr2_free c2 t3 t4 H0))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
+C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abbr g c d u i H0
+c2 H3) in (ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl
+i c2 (CHead d2 (Bind Abbr) u))) (pr2 c2 t3 t) (\lambda (x: C).(\lambda (_:
+(csub3 g d x)).(\lambda (H6: (getl i c2 (CHead x (Bind Abbr) u))).(pr2_delta
+c2 x u i H6 t3 t4 H1 t H2)))) H4)))))))))))))) c1 t1 t2 H))))).
+
+theorem csub3_pc3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
+t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (pc3 c1 t1 t2)).(pc3_ind_left c1 (\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((csub3 g c1 c2) \to (pc3 c2 t t0))))) (\lambda (t:
+T).(\lambda (c2: C).(\lambda (_: (csub3 g c1 c2)).(pc3_refl c2 t)))) (\lambda
+(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0 t3)).(\lambda (t4:
+T).(\lambda (_: (pc3 c1 t3 t4)).(\lambda (H2: ((\forall (c2: C).((csub3 g c1
+c2) \to (pc3 c2 t3 t4))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c1
+c2)).(pc3_pr2_u c2 t3 t0 (csub3_pr2 g c1 t0 t3 H0 c2 H3) t4 (H2 c2
+H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c1 t0
+t3)).(\lambda (t4: T).(\lambda (_: (pc3 c1 t0 t4)).(\lambda (H2: ((\forall
+(c2: C).((csub3 g c1 c2) \to (pc3 c2 t0 t4))))).(\lambda (c2: C).(\lambda
+(H3: (csub3 g c1 c2)).(pc3_t t0 c2 t3 (pc3_pr2_x c2 t3 t0 (csub3_pr2 g c1 t0
+t3 H0 c2 H3)) t4 (H2 c2 H3)))))))))) t1 t2 H))))).
+
+theorem csub3_ty3:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda
+(c: C).(\lambda (t0: T).(\lambda (t: T).(\lambda (_: (ty3 g c t0 t)).(\lambda
+(H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t0 t))))).(\lambda (u:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
+C).((csub3 g c c2) \to (ty3 g c2 u t3))))).(\lambda (H4: (pc3 c t3
+t0)).(\lambda (c2: C).(\lambda (H5: (csub3 g c c2)).(ty3_conv g c2 t0 t (H1
+c2 H5) u t3 (H3 c2 H5) (csub3_pc3 g c t3 t0 H4 c2 H5)))))))))))))) (\lambda
+(c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (csub3 g c
+c2)).(ty3_sort g c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda
+(t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((csub3 g
+d c2) \to (ty3 g c2 u t))))).(\lambda (c2: C).(\lambda (H3: (csub3 g c
+c2)).(let H4 \def (csub3_getl_abbr g c d u n H0 c2 H3) in (ex2_ind C (\lambda
+(d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr)
+u))) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x: C).(\lambda (H5:
+(csub3 g d x)).(\lambda (H6: (getl n c2 (CHead x (Bind Abbr) u))).(ty3_abbr g
+n c2 x u H6 t (H2 x H5))))) H4)))))))))))) (\lambda (n: nat).(\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
+Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2:
+((\forall (c2: C).((csub3 g d c2) \to (ty3 g c2 u t))))).(\lambda (c2:
+C).(\lambda (H3: (csub3 g c c2)).(let H4 \def (csub3_getl_abst g c d u n H0
+c2 H3) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2:
+C).(getl n c2 (CHead d2 (Bind Abst) u)))) (ex3_2 C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2
+u0 u)))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (H5: (ex2 C (\lambda
+(d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst)
+u))))).(ex2_ind C (\lambda (d2: C).(csub3 g d d2)) (\lambda (d2: C).(getl n
+c2 (CHead d2 (Bind Abst) u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda
+(x: C).(\lambda (H6: (csub3 g d x)).(\lambda (H7: (getl n c2 (CHead x (Bind
+Abst) u))).(ty3_abst g n c2 x u H7 t (H2 x H6))))) H5)) (\lambda (H5: (ex3_2
+C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2:
+C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2:
+C).(\lambda (u0: T).(ty3 g d2 u0 u))))).(ex3_2_ind C T (\lambda (d2:
+C).(\lambda (_: T).(csub3 g d d2))) (\lambda (d2: C).(\lambda (u0: T).(getl n
+c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2
+u0 u))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (_: (csub3 g d x0)).(\lambda (H7: (getl n c2 (CHead x0 (Bind
+Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 u)).(ty3_abbr g n c2 x0 x1 H7 u
+H8)))))) H5)) H4)))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c2: C).((csub3 g c
+c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t0 t3)).(\lambda (H3: ((\forall
+(c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t0 t3))))).(\lambda
+(t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5:
+((\forall (c2: C).((csub3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3
+t4))))).(\lambda (c2: C).(\lambda (H6: (csub3 g c c2)).(ty3_bind g c2 u t (H1
+c2 H6) b t0 t3 (H3 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b) u))
+t4 (H5 (CHead c2 (Bind b) u) (csub3_head g c c2 H6 (Bind b)
+u)))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
+C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t0 t3)).(\lambda
+(H1: ((\forall (c2: C).((csub3 g c c2) \to (ty3 g c2 t0 t3))))).(\lambda (t4:
+T).(\lambda (_: (ty3 g c t3 t4)).(\lambda (H3: ((\forall (c2: C).((csub3 g c
+c2) \to (ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (csub3 g c
+c2)).(ty3_cast g c2 t0 t3 (H1 c2 H4) t4 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
+
+theorem csub3_ty3_ld:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (v: T).((ty3 g c u
+v) \to (\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind Abst) v) t1
+t2) \to (ty3 g (CHead c (Bind Abbr) u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (H:
+(ty3 g c u v)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead
+c (Bind Abst) v) t1 t2)).(csub3_ty3 g (CHead c (Bind Abst) v) t1 t2 H0 (CHead
+c (Bind Abbr) u) (csub3_abst g c c (csub3_refl g c) u v H))))))))).
+
+theorem ty3_sred_wcpr0_pr0:
+ \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
+t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2)
+\to (ty3 g c2 t2 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
+(H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
+(t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to
+(ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0:
+T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
+c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u:
+T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2
+t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0
+c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2
+H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2
+H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2:
+C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m)
+t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m))))
+(ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda
+(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
+Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2:
+((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g
+c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2:
+T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3:
+T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda
+(u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2
+(TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5:
+(getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda
+(H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)))))))
+(wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n
+H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0:
+T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2)
+\to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2:
+C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n)
+t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u)))
+(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind
+Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind
+Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g
+c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u
+(pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S
+n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n)
+O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1
+H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n
+H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda
+(_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to
+(\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b:
+B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b)
+u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2)
+\to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t4:
+T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall
+(c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2)
+\to (ty3 g c2 t2 t4))))))).(\lambda (c2: C).(\lambda (H6: (wcpr0 c
+c2)).(\lambda (t5: T).(\lambda (H7: (pr0 (THead (Bind b) u t2) t5)).(let H8
+\def (match H7 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t
+t0)).((eq T t (THead (Bind b) u t2)) \to ((eq T t0 t5) \to (ty3 g c2 t5
+(THead (Bind b) u t3))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq
+T t4 (THead (Bind b) u t2))).(\lambda (H8: (eq T t4 t5)).(eq_ind T (THead
+(Bind b) u t2) (\lambda (t: T).((eq T t t5) \to (ty3 g c2 t5 (THead (Bind b)
+u t3)))) (\lambda (H9: (eq T (THead (Bind b) u t2) t5)).(eq_ind T (THead
+(Bind b) u t2) (\lambda (t: T).(ty3 g c2 t (THead (Bind b) u t3))) (ty3_bind
+g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u)
+(wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)) t4 (H5
+(CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3
+(pr0_refl t3))) t5 H9)) t4 (sym_eq T t4 (THead (Bind b) u t2) H7) H8))) |
+(pr0_comp u1 u2 H7 t4 t5 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1
+t4) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead k u2 t5) t5)).((let
+H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t)
+\Rightarrow t])) (THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H12 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t]))
+(THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T K
+(\lambda (e: T).(match e return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k]))
+(THead k u1 t4) (THead (Bind b) u t2) H9) in (eq_ind K (Bind b) (\lambda (k0:
+K).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T (THead k0 u2 t5) t5) \to ((pr0
+u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))))
+(\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to
+((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to (ty3 g
+c2 t5 (THead (Bind b) u t3))))))) (\lambda (H15: (eq T t4 t2)).(eq_ind T t2
+(\lambda (t: T).((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 u u2) \to ((pr0 t
+t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) (\lambda (H16: (eq T (THead
+(Bind b) u2 t5) t5)).(eq_ind T (THead (Bind b) u2 t5) (\lambda (t: T).((pr0 u
+u2) \to ((pr0 t2 t5) \to (ty3 g c2 t (THead (Bind b) u t3))))) (\lambda (H17:
+(pr0 u u2)).(\lambda (H18: (pr0 t2 t5)).(ex_ind T (\lambda (t: T).(ty3 g
+(CHead c2 (Bind b) u) t4 t)) (ty3 g c2 (THead (Bind b) u2 t5) (THead (Bind b)
+u t3)) (\lambda (x: T).(\lambda (H19: (ty3 g (CHead c2 (Bind b) u) t4
+x)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u2) t3 t)) (ty3 g c2
+(THead (Bind b) u2 t5) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H20:
+(ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3)
+(THead (Bind b) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t3 t4
+(H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3
+(pr0_refl t3)) x H19) (THead (Bind b) u2 t5) (THead (Bind b) u2 t3) (ty3_bind
+g c2 u2 t0 (H1 c2 H6 u2 H17) b t5 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c
+c2 H6 u u2 H17 (Bind b)) t5 H18) x0 H20) (pc3_pr2_x c2 (THead (Bind b) u2 t3)
+(THead (Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind b)
+t3))))) (ty3_correct g (CHead c2 (Bind b) u2) t5 t3 (H3 (CHead c2 (Bind b)
+u2) (wcpr0_comp c c2 H6 u u2 H17 (Bind b)) t5 H18))))) (ty3_correct g (CHead
+c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u
+(pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))) t5 H16)) t4 (sym_eq T t4 t2
+H15))) u1 (sym_eq T u1 u H14))) k (sym_eq K k (Bind b) H13))) H12)) H11)) H10
+H7 H8))) | (pr0_beta u0 v1 v2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (THead (Bind b) u
+t2))).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t5)).((let H11 \def
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u t2) H9) in (False_ind ((eq T (THead (Bind Abbr)
+v2 t5) t5) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b)
+u t3))))) H11)) H10 H7 H8))) | (pr0_upsilon b0 H7 v1 v2 H8 u1 u2 H9 t4 t5
+H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b0)
+u1 t4)) (THead (Bind b) u t2))).(\lambda (H12: (eq T (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t5)) t5)).((let H13 \def (eq_ind T
+(THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
+(THead (Bind b) u t2) H11) in (False_ind ((eq T (THead (Bind b0) u2 (THead
+(Flat Appl) (lift (S O) O v2) t5)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v1
+v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u
+t3))))))) H13)) H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t4 t5 H8 w H9)
+\Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t4) (THead (Bind b) u
+t2))).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t5)).((let H12 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H13 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H14 \def
+(f_equal T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort
+_) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abbr])])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in
+(eq_ind B Abbr (\lambda (b: B).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T
+(THead (Bind Abbr) u2 w) t5) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O
+u2 t5 w) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))))) (\lambda (H15: (eq T
+u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind
+Abbr) u2 w) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to
+(ty3 g c2 t5 (THead (Bind Abbr) u t3)))))))) (\lambda (H16: (eq T t4
+t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) t5) \to
+((pr0 u u2) \to ((pr0 t t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead
+(Bind Abbr) u t3))))))) (\lambda (H17: (eq T (THead (Bind Abbr) u2 w)
+t5)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u u2) \to ((pr0
+t2 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t (THead (Bind Abbr) u t3))))))
+(\lambda (H18: (pr0 u u2)).(\lambda (H19: (pr0 t2 t5)).(\lambda (H20: (subst0
+O u2 t5 w)).(let H21 \def (eq_ind_r B b (\lambda (b: B).(\forall (c2:
+C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) \to
+(ty3 g c2 t2 t4)))))) H5 Abbr H14) in (let H22 \def (eq_ind_r B b (\lambda
+(b: B).(ty3 g (CHead c (Bind b) u) t3 t4)) H4 Abbr H14) in (let H23 \def
+(eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u)
+c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3)))))) H3 Abbr H14)
+in (let H24 \def (eq_ind_r B b (\lambda (b: B).(ty3 g (CHead c (Bind b) u) t2
+t3)) H2 Abbr H14) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr)
+u) t4 t)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3))
+(\lambda (x: T).(\lambda (H25: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(ex_ind
+T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t)) (ty3 g c2 (THead
+(Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (H26:
+(ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u
+t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u))
+Abbr t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u)
+(Bind Abbr)) t3 (pr0_refl t3)) x H25) (THead (Bind Abbr) u2 w) (THead (Bind
+Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 u2 H18) Abbr w t3 (ty3_subst0 g
+(CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c
+c2 H6 u u2 H18 (Bind Abbr)) t5 H19) c2 u2 O (getl_refl Abbr c2 u2) w H20) x0
+H26) (pc3_pr2_x c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3)
+(pr2_head_1 c2 u u2 (pr2_free c2 u u2 H18) (Bind Abbr) t3))))) (ty3_correct g
+(CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c
+c2 H6 u u2 H18 (Bind Abbr)) t5 H19))))) (ty3_correct g (CHead c2 (Bind Abbr)
+u) t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u)
+(Bind Abbr)) t3 (pr0_refl t3))))))))))) t5 H17)) t4 (sym_eq T t4 t2 H16))) u1
+(sym_eq T u1 u H15))) b H14)) H13)) H12)) H11 H7 H8 H9))) | (pr0_zeta b0 H7
+t4 t5 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O)
+O t4)) (THead (Bind b) u t2))).(\lambda (H10: (eq T t5 t5)).((let H11 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t:
+T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (TLRef _)
+\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
+\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
+(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
+| (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d)
+t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2)
+H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead
+_ t _) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind
+b) u t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O)
+O t4)) (THead (Bind b) u t2) H9) in (eq_ind B b (\lambda (b1: B).((eq T u0 u)
+\to ((eq T (lift (S O) O t4) t2) \to ((eq T t5 t5) \to ((not (eq B b1 Abst))
+\to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) (\lambda (H14:
+(eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t4) t2) \to
+((eq T t5 t5) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5
+(THead (Bind b) u t3))))))) (\lambda (H15: (eq T (lift (S O) O t4)
+t2)).(eq_ind T (lift (S O) O t4) (\lambda (_: T).((eq T t5 t5) \to ((not (eq
+B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))
+(\lambda (H16: (eq T t5 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B b
+Abst)) \to ((pr0 t4 t) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) (\lambda
+(H17: (not (eq B b Abst))).(\lambda (H18: (pr0 t4 t5)).(let H19 \def
+(eq_ind_r T t2 (\lambda (t: T).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u)
+c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 t3)))))) H3 (lift (S O)
+O t4) H15) in (let H20 \def (eq_ind_r T t2 (\lambda (t: T).(ty3 g (CHead c
+(Bind b) u) t t3)) H2 (lift (S O) O t4) H15) in (ex_ind T (\lambda (t:
+T).(ty3 g (CHead c2 (Bind b) u) t4 t)) (ty3 g c2 t5 (THead (Bind b) u t3))
+(\lambda (x: T).(\lambda (H4: (ty3 g (CHead c2 (Bind b) u) t4 x)).(B_ind
+(\lambda (b: B).((not (eq B b Abst)) \to ((ty3 g (CHead c2 (Bind b) u) t3 t4)
+\to ((ty3 g (CHead c2 (Bind b) u) t4 x) \to ((ty3 g (CHead c2 (Bind b) u)
+(lift (S O) O t5) t3) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda
+(H21: (not (eq B Abbr Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Abbr) u)
+t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(\lambda (H22:
+(ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3)).(let H \def
+(ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3 H22 c2 u O
+(getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2
+(Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t5)
+(lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S
+O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t5
+(THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0:
+(subst1 O u (lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (subst1 O u
+t3 (lift (S O) O x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind
+T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 (lift_inj x0 t5 (S O) O
+(subst1_gen_lift_eq t5 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat
+(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S
+O)) (plus_comm O (S O))) H0))) in (ty3_conv g c2 (THead (Bind Abbr) u t3)
+(THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr
+t3 t4 H2 x H5) t5 x1 H24 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t
+(THead (Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2
+c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free
+c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1))
+(pr0_delta1 u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H3))) x1
+(pr3_pr2 c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead
+(Bind Abbr) u (lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1)
+u)))))))))))) H)))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_:
+(ty3 g (CHead c2 (Bind Abst) u) t3 t4)).(\lambda (_: (ty3 g (CHead c2 (Bind
+Abst) u) t4 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O
+t5) t3)).(let H \def (match (H21 (refl_equal B Abst)) return (\lambda (_:
+False).(ty3 g c2 t5 (THead (Bind Abst) u t3))) with []) in H))))) (\lambda
+(H21: (not (eq B Void Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Void) u)
+t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Void) u) t4 x)).(\lambda (H22:
+(ty3 g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3)).(let H \def
+(ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3 H22 c2 u O
+(getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2 (drop_refl c2) u)) in
+(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) O t5) (lift
+(S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) O y2))))
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t5 (THead
+(Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: (eq T
+(lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (eq T t3 (lift (S O) O
+x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind T t3 (\lambda
+(t: T).(ty3 g (CHead c2 (Bind Void) u) t t4)) H2 (lift (S O) O x1) H3) in
+(eq_ind_r T (lift (S O) O x1) (\lambda (t: T).(ty3 g c2 t5 (THead (Bind Void)
+u t))) (let H25 \def (eq_ind_r T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5
+(lift_inj t5 x0 (S O) O H0)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S
+O) O x1)) (THead (Bind Void) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl
+u)) Void (lift (S O) O x1) t4 H24 x H5) t5 x1 H25 (pc3_pr2_x c2 x1 (THead
+(Bind Void) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Void) u (lift (S
+O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u))))) t3 H3)))))))
+H)))))) b H17 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u)
+(Bind b)) t3 (pr0_refl t3)) H4 (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6
+u u (pr0_refl u) (Bind b)) (lift (S O) O t5) (pr0_lift t4 t5 H18 (S O) O)))))
+(ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u)
+(wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))))) t5
+(sym_eq T t5 t5 H16))) t2 H15)) u0 (sym_eq T u0 u H14))) b0 (sym_eq B b0 b
+H13))) H12)) H11)) H10 H7 H8))) | (pr0_epsilon t4 t5 H7 u0) \Rightarrow
+(\lambda (H8: (eq T (THead (Flat Cast) u0 t4) (THead (Bind b) u
+t2))).(\lambda (H9: (eq T t5 t5)).((let H10 \def (eq_ind T (THead (Flat Cast)
+u0 t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H8) in (False_ind ((eq
+T t5 t5) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) H10)) H9
+H7)))]) in (H8 (refl_equal T (THead (Bind b) u t2)) (refl_equal T
+t5)))))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2)
+\to (\forall (t2: T).((pr0 w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v:
+T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u
+t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2:
+T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0)))))))).(\lambda (c2:
+C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead
+(Flat Appl) w v) t2)).(let H6 \def (match H5 return (\lambda (t: T).(\lambda
+(t1: T).(\lambda (_: (pr0 t t1)).((eq T t (THead (Flat Appl) w v)) \to ((eq T
+t1 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))))
+with [(pr0_refl t0) \Rightarrow (\lambda (H5: (eq T t0 (THead (Flat Appl) w
+v))).(\lambda (H6: (eq T t0 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda
+(t: T).((eq T t t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst)
+u t0))))) (\lambda (H7: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead
+(Flat Appl) w v) (\lambda (t: T).(ty3 g c2 t (THead (Flat Appl) w (THead
+(Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2
+H4 v (pr0_refl v))) t2 H7)) t0 (sym_eq T t0 (THead (Flat Appl) w v) H5) H6)))
+| (pr0_comp u1 u2 H5 t1 t0 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1
+t1) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead k u2 t0) t2)).((let
+H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t)
+\Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H10
+\def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _)
+\Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H11
+\def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in (eq_ind K
+(Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t1 v) \to ((eq T (THead
+k0 u2 t0) t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H12: (eq T u1 w)).(eq_ind
+T w (\lambda (t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) u2 t0) t2) \to
+((pr0 t u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
+(Bind Abst) u t0)))))))) (\lambda (H13: (eq T t1 v)).(eq_ind T v (\lambda (t:
+T).((eq T (THead (Flat Appl) u2 t0) t2) \to ((pr0 w u2) \to ((pr0 t t0) \to
+(ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda
+(H14: (eq T (THead (Flat Appl) u2 t0) t2)).(eq_ind T (THead (Flat Appl) u2
+t0) (\lambda (t: T).((pr0 w u2) \to ((pr0 v t0) \to (ty3 g c2 t (THead (Flat
+Appl) w (THead (Bind Abst) u t0)))))) (\lambda (H15: (pr0 w u2)).(\lambda
+(H16: (pr0 v t0)).(ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u
+t0) t)) (ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x: T).(\lambda (H17: (ty3 g c2 (THead (Bind Abst) u
+t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_:
+T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t:
+T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3))))
+(ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind Abst) u
+t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2
+(THead (Bind Abst) u x0) x)).(\lambda (H19: (ty3 g c2 u x1)).(\lambda (H20:
+(ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H21: (ty3 g (CHead c2 (Bind
+Abst) u) x0 x2)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u
+t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2
+H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H19 Abst
+t0 x0 H20 x2 H21)) (THead (Flat Appl) u2 t0) (THead (Flat Appl) u2 (THead
+(Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 H15) t0 t0 (H3 c2 H4 t0
+H16)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) (THead
+(Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 c2 w u2 (pr2_free c2 w u2
+H15) (Flat Appl) (THead (Bind Abst) u t0))))))))))) (ty3_gen_bind g Abst c2 u
+t0 x H17)))) (ty3_correct g c2 v (THead (Bind Abst) u t0) (H3 c2 H4 v
+(pr0_refl v)))))) t2 H14)) t1 (sym_eq T t1 v H13))) u1 (sym_eq T u1 w H12)))
+k (sym_eq K k (Flat Appl) H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u0 v1 v2
+H5 t1 t0 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead
+(Bind Abbr) v2 t0) t2)).((let H9 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u0
+t1) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t1) | (THead _ _ t)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead
+(Flat Appl) w v) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
+(Bind Abst) u0 t1)) (THead (Flat Appl) w v) H7) in (eq_ind T w (\lambda (t:
+T).((eq T (THead (Bind Abst) u0 t1) v) \to ((eq T (THead (Bind Abbr) v2 t0)
+t2) \to ((pr0 t v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))))) (\lambda (H11: (eq T (THead (Bind Abst) u0
+t1) v)).(eq_ind T (THead (Bind Abst) u0 t1) (\lambda (_: T).((eq T (THead
+(Bind Abbr) v2 t0) t2) \to ((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2
+(THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H12: (eq T
+(THead (Bind Abbr) v2 t0) t2)).(eq_ind T (THead (Bind Abbr) v2 t0) (\lambda
+(t: T).((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))) (\lambda (H13: (pr0 w v2)).(\lambda (H14: (pr0
+t1 t0)).(let H15 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: C).((wcpr0
+c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead (Bind Abst) u
+t0))))))) H3 (THead (Bind Abst) u0 t1) H11) in (let H16 \def (eq_ind_r T v
+(\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead (Bind Abst)
+u0 t1) H11) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0)
+t)) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind
+Abst) u t0))) (\lambda (x: T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u
+t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_:
+T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t:
+T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3))))
+(ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u
+t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2
+(THead (Bind Abst) u x0) x)).(\lambda (H17: (ty3 g c2 u x1)).(\lambda (H18:
+(ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H19: (ty3 g (CHead c2 (Bind
+Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda
+(_: T).(pc3 c2 (THead (Bind Abst) u0 t2) (THead (Bind Abst) u t0)))))
+(\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u0 t)))) (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u0) t0
+t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2
+(Bind Abst) u0) t2 t3)))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat
+Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (H0: (pc3 c2 (THead (Bind Abst) u0 x3) (THead (Bind Abst) u
+t0))).(\lambda (H20: (ty3 g c2 u0 x4)).(\lambda (H21: (ty3 g (CHead c2 (Bind
+Abst) u0) t0 x3)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abst) u0) x3
+x5)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u: T).(pc3 (CHead c2
+(Bind b) u) x3 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w
+(THead (Bind Abst) u t0))) (\lambda (H23: (pc3 c2 u0 u)).(\lambda (H24:
+((\forall (b: B).(\forall (u: T).(pc3 (CHead c2 (Bind b) u) x3
+t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
+(Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
+(pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H17 Abst t0 x0
+H18 x2 H19)) (THead (Bind Abbr) v2 t0) (THead (Bind Abbr) v2 x3) (ty3_bind g
+c2 v2 u (H1 c2 H4 v2 H13) Abbr t0 x3 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2
+u0 x4 H20 v2 u (H1 c2 H4 v2 H13) (pc3_s c2 u u0 H23)) t0 x3 H21) x5
+(csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H20 v2 u (H1 c2 H4 v2 H13)
+(pc3_s c2 u u0 H23)) x3 x5 H22)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead
+(Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H24 Abbr v2)) (THead
+(Flat Appl) w (THead (Bind Abst) u t0)) (pc3_pr2_x c2 (THead (Bind Abbr) v2
+t0) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_free c2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0)) (THead (Bind Abbr) v2 t0) (pr0_beta u w v2
+H13 t0 t0 (pr0_refl t0)))))))) (pc3_gen_abst c2 u0 u x3 t0 H0)))))))))
+(ty3_gen_bind g Abst c2 u0 t0 (THead (Bind Abst) u t0) (H15 c2 H4 (THead
+(Bind Abst) u0 t0) (pr0_comp u0 u0 (pr0_refl u0) t1 t0 H14 (Bind
+Abst)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H2)))) (ty3_correct g c2
+(THead (Bind Abst) u0 t1) (THead (Bind Abst) u t0) (H15 c2 H4 (THead (Bind
+Abst) u0 t1) (pr0_refl (THead (Bind Abst) u0 t1))))))))) t2 H12)) v H11)) v1
+(sym_eq T v1 w H10))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7
+t1 t0 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind
+b) u1 t1)) (THead (Flat Appl) w v))).(\lambda (H10: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t0)) t2)).((let H11 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow (THead (Bind b) u1 t1) | (TLRef _) \Rightarrow (THead (Bind b) u1
+t1) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1
+t1)) (THead (Flat Appl) w v) H9) in ((let H12 \def (f_equal T T (\lambda (e:
+T).(match e return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef
+_) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t1)) (THead (Flat Appl) w v) H9) in (eq_ind T w (\lambda
+(t: T).((eq T (THead (Bind b) u1 t1) v) \to ((eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to ((pr0 t
+v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w
+(THead (Bind Abst) u t0)))))))))) (\lambda (H13: (eq T (THead (Bind b) u1 t1)
+v)).(eq_ind T (THead (Bind b) u1 t1) (\lambda (_: T).((eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to
+((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H14: (eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2)).(eq_ind T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (\lambda (t: T).((not (eq B b
+Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t
+(THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H15: (not (eq
+B b Abst))).(\lambda (H16: (pr0 w v2)).(\lambda (H17: (pr0 u1 u2)).(\lambda
+(H18: (pr0 t1 t0)).(let H19 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2:
+C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead
+(Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t1) H13) in (let H20 \def
+(eq_ind_r T v (\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead
+(Bind b) u1 t1) H13) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind
+Abst) u t0) t)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x:
+T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H3 \def H2 in
+(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2
+(THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
+T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
+(CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda
+(t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead
+(Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H22: (ty3 g c2
+u x1)).(\lambda (H23: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H24:
+(ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t2) (THead
+(Bind Abst) u t0))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g
+c2 u2 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2
+(Bind b) u2) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3
+g (CHead c2 (Bind b) u2) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
+(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H0: (pc3 c2
+(THead (Bind b) u2 x3) (THead (Bind Abst) u t0))).(\lambda (H25: (ty3 g c2 u2
+x4)).(\lambda (H26: (ty3 g (CHead c2 (Bind b) u2) t0 x3)).(\lambda (_: (ty3 g
+(CHead c2 (Bind b) u2) x3 x5)).(let H28 \def (eq_ind T (lift (S O) O (THead
+(Bind Abst) u t0)) (\lambda (t: T).(pc3 (CHead c2 (Bind b) u2) x3 t))
+(pc3_gen_not_abst b H15 c2 x3 t0 u2 u H0) (THead (Bind Abst) (lift (S O) O u)
+(lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (let H29 \def
+(eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(ty3 g
+(CHead c2 (Bind b) u2) t (lift (S O) O x))) (ty3_lift g c2 (THead (Bind Abst)
+u t0) x H2 (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2
+(drop_refl c2) u2)) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O)
+t0)) (lift_bind Abst u t0 (S O) O)) in (ex4_3_ind T T T (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind
+Abst) (lift (S O) O u) t2) (lift (S O) O x))))) (\lambda (_: T).(\lambda (t:
+T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c2
+(Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) t0) t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead (CHead c2
+(Bind b) u2) (Bind Abst) (lift (S O) O u)) t2 t3)))) (ty3 g c2 (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead
+(Bind Abst) u t0))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (x8:
+T).(\lambda (_: (pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O
+u) x6) (lift (S O) O x))).(\lambda (H31: (ty3 g (CHead c2 (Bind b) u2) (lift
+(S O) O u) x7)).(\lambda (H32: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind
+Abst) (lift (S O) O u)) (lift (S O) (S O) t0) x6)).(\lambda (H33: (ty3 g
+(CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) x6 x8)).(ty3_conv
+g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w
+(THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead
+(Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H22 Abst t0 x0 H23 x2 H24)) (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u)
+(lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x4 H25 b (THead (Flat Appl) (lift
+(S O) O v2) t0) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift
+(S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) u2) (lift
+(S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H16) (CHead c2
+(Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t0 (lift
+(S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift
+(S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) O u) x6)
+(ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O)
+(S O) t0) x6 H32 x8 H33) t0 x3 H26 H28)) (THead (Flat Appl) (lift (S O) O v2)
+(THead (Bind Abst) (lift (S O) O u) x6)) (ty3_appl g (CHead c2 (Bind b) u2)
+(lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H16)
+(CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2)
+u2)) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) x6 (ty3_bind
+g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) (S O) t0)
+x6 H32 x8 H33))) (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda
+(t: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t))
+(THead (Flat Appl) w (THead (Bind Abst) u t0)))) (pc3_pc1 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u
+t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_pr0_u2 (THead
+(Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))))
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O (THead
+(Bind Abst) u t0)))) (pr0_upsilon b H15 v2 v2 (pr0_refl v2) u2 u2 (pr0_refl
+u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O (THead (Bind Abst)
+u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl)
+w (THead (Bind Abst) u t0)) (pc1_s (THead (Flat Appl) v2 (THead (Bind b) u2
+(lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind
+Abst) u t0)) (pc1_head w v2 (pc1_pr0_r w v2 H16) (THead (Bind Abst) u t0)
+(THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead
+(Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0)))
+(pr0_zeta b H15 (THead (Bind Abst) u t0) (THead (Bind Abst) u t0) (pr0_refl
+(THead (Bind Abst) u t0)) u2)) (Flat Appl)))) c2) (THead (Bind Abst) (lift (S
+O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O))))))))))
+(ty3_gen_bind g Abst (CHead c2 (Bind b) u2) (lift (S O) O u) (lift (S O) (S
+O) t0) (lift (S O) O x) H29))))))))))) (ty3_gen_bind g b c2 u2 t0 (THead
+(Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) u2 t0) (pr0_comp u1 u2 H17 t1 t0
+H18 (Bind b)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H3))))) (ty3_correct g
+c2 (THead (Bind b) u2 t0) (THead (Bind Abst) u t0) (H19 c2 H4 (THead (Bind b)
+u2 t0) (pr0_comp u1 u2 H17 t1 t0 H18 (Bind b))))))))))) t2 H14)) v H13)) v1
+(sym_eq T v1 w H12))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t1 t0 H6
+w0 H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t1) (THead (Flat
+Appl) w v))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w0) t2)).((let H10
+\def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
+(THead (Flat Appl) w v) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w0)
+t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to ((subst0 O u2 t0 w0) \to (ty3 g c2
+t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) H10)) H9 H5 H6 H7))) |
+(pr0_zeta b H5 t1 t0 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Bind b)
+u0 (lift (S O) O t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T t0
+t2)).((let H9 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t1)) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T t0
+t2) \to ((not (eq B b Abst)) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat
+Appl) w (THead (Bind Abst) u t0)))))) H9)) H8 H5 H6))) | (pr0_epsilon t1 t0
+H5 u0) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u0 t1) (THead (Flat
+Appl) w v))).(\lambda (H7: (eq T t0 t2)).((let H8 \def (eq_ind T (THead (Flat
+Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v)
+H6) in (False_ind ((eq T t0 t2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead
+(Flat Appl) w (THead (Bind Abst) u t0))))) H8)) H7 H5)))]) in (H6 (refl_equal
+T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c:
+C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda
+(H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to
+(ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3
+t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0
+t3 t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c
+c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let
+H6 \def (match H5 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr0 t
+t0)).((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c2 t4
+t3)))))) with [(pr0_refl t) \Rightarrow (\lambda (H5: (eq T t (THead (Flat
+Cast) t3 t2))).(\lambda (H6: (eq T t t4)).(eq_ind T (THead (Flat Cast) t3 t2)
+(\lambda (t0: T).((eq T t0 t4) \to (ty3 g c2 t4 t3))) (\lambda (H7: (eq T
+(THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda
+(t0: T).(ty3 g c2 t0 t3)) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0
+(H3 c2 H4 t3 (pr0_refl t3))) t4 H7)) t (sym_eq T t (THead (Flat Cast) t3 t2)
+H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T
+(THead k u1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead k u2
+t5) t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead
+_ _ t) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in
+((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t
+_) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in ((let
+H11 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _)
+\Rightarrow k])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in (eq_ind K
+(Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to ((eq T t4 t2) \to ((eq T
+(THead k0 u2 t5) t4) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4
+t3))))))) (\lambda (H12: (eq T u1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4
+t2) \to ((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t u2) \to ((pr0 t4 t5)
+\to (ty3 g c2 t4 t3)))))) (\lambda (H13: (eq T t4 t2)).(eq_ind T t2 (\lambda
+(t: T).((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t3 u2) \to ((pr0 t t5)
+\to (ty3 g c2 t4 t3))))) (\lambda (H14: (eq T (THead (Flat Cast) u2 t5)
+t4)).(eq_ind T (THead (Flat Cast) u2 t5) (\lambda (t: T).((pr0 t3 u2) \to
+((pr0 t2 t5) \to (ty3 g c2 t t3)))) (\lambda (H15: (pr0 t3 u2)).(\lambda
+(H16: (pr0 t2 t5)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead
+(Flat Cast) u2 t5) u2 (ty3_cast g c2 t5 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2
+H15) t5 t3 (H1 c2 H4 t5 H16) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H15))) t0
+(H3 c2 H4 u2 H15)) (pc3_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H15))))) t4 H14))
+t4 (sym_eq T t4 t2 H13))) u1 (sym_eq T u1 t3 H12))) k (sym_eq K k (Flat Cast)
+H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow
+(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (THead
+(Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t4)).((let
+H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
+\Rightarrow (match f return (\lambda (_: F).Prop) with [Appl \Rightarrow True
+| Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind
+((eq T (THead (Bind Abbr) v2 t5) t4) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to
+(ty3 g c2 t4 t3)))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4
+t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b)
+u1 t4)) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t5)) t4)).((let H11 \def (eq_ind T
+(THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow
+False])])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t4) \to ((not (eq B b
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4
+t3)))))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t4 t5 H6 w H7)
+\Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t4) (THead (Flat Cast)
+t3 t2))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t4)).((let H10 \def
+(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3
+t2) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2)
+\to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t4 t3))))) H10)) H9 H5
+H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: (eq T (THead
+(Bind b) u (lift (S O) O t4)) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T
+t5 t4)).((let H9 \def (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
+\Rightarrow False])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T
+t5 t4) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))) H9))
+H8 H5 H6))) | (pr0_epsilon t4 t5 H5 u) \Rightarrow (\lambda (H6: (eq T (THead
+(Flat Cast) u t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5
+t4)).((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _
+t) \Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in
+((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in
+(eq_ind T t3 (\lambda (_: T).((eq T t4 t2) \to ((eq T t5 t4) \to ((pr0 t4 t5)
+\to (ty3 g c2 t4 t3))))) (\lambda (H10: (eq T t4 t2)).(eq_ind T t2 (\lambda
+(t: T).((eq T t5 t4) \to ((pr0 t t5) \to (ty3 g c2 t4 t3)))) (\lambda (H11:
+(eq T t5 t4)).(eq_ind T t4 (\lambda (t: T).((pr0 t2 t) \to (ty3 g c2 t4 t3)))
+(\lambda (H12: (pr0 t2 t4)).(H1 c2 H4 t4 H12)) t5 (sym_eq T t5 t4 H11))) t4
+(sym_eq T t4 t2 H10))) u (sym_eq T u t3 H9))) H8)) H7 H5)))]) in (H6
+(refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1
+t1 t H))))).
+
+theorem ty3_sred_pr1:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1
+t2)).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (t3:
+T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5: T).(\lambda (_:
+(pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c t3 t)
+\to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3: (ty3 g
+c t4 t)).(H2 g t (ty3_sred_wcpr0_pr0 g c t4 t H3 c (wcpr0_refl c) t3
+H0))))))))))) t1 t2 H)))).
+
+theorem ty3_sred_pr2:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
+G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g:
+G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3
+t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
+G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0
+(ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
+H2)))))))))))))) c t1 t2 H)))).
+
+theorem ty3_sred_pr3:
+ \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
+(g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
+\def
+ \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall
+(t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g:
+G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
+(_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c
+t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3:
+(ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2
+H)))).
+
+theorem ty3_cred_pr2:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind
+b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0:
+C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b:
+B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b)
+t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0
+(Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0
+(pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda
+(t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0:
+T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0
+t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead
+c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl
+c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead
+c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0)
+(CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1
+v2 H))))).
+
+theorem ty3_cred_pr3:
+ \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1
+v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
+(Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b:
+B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to
+(ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b)
+t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1
+t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b:
+B).(\forall (t1: T).(\forall (t4: T).((ty3 g (CHead c (Bind b) t2) t1 t4) \to
+(ty3 g (CHead c (Bind b) t3) t1 t4))))))).(\lambda (b: B).(\lambda (t0:
+T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b
+t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
+
+theorem ty3_gen__le_S_minus:
+ \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
+(le d (S (minus n h))))))
+\def
+ \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
+d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
+\def (eq_ind nat n (\lambda (n: nat).(le d n)) H0 (plus (minus n h) h)
+(le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h)
+(le_minus d n h H))))))).
+
+theorem ty3_gen_lift:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop
+h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2:
+T).(ty3 g e t1 t2)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T
+(lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\forall (e: C).((drop h d c e)
+\to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: T).(ty3 g
+e t1 t2))))) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(unintro nat d
+(\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall (e: C).((drop h n c e)
+\to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) (\lambda (t2: T).(ty3 g
+e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall (x0: nat).((eq T y
+(lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to (ex2 T (\lambda (t2:
+T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t t2)))))))) (ty3_ind
+g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall
+(x1: nat).((eq T t (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) t0)) (\lambda (t2: T).(ty3 g e
+x0 t2))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda
+(_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall (x: T).(\forall (x0: nat).((eq T
+t2 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x
+t2)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
+t3)).(\lambda (H4: ((\forall (x: T).(\forall (x0: nat).((eq T u (lift h x0
+x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\lambda (H5:
+(pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T u
+(lift h x1 x0))).(\lambda (e: C).(\lambda (H7: (drop h x1 c0 e)).(let H8 \def
+(eq_ind T u (\lambda (t: T).(\forall (x: T).(\forall (x0: nat).((eq T t (lift
+h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2))))))))) H4
+(lift h x1 x0) H6) in (let H9 \def (eq_ind T u (\lambda (t: T).(ty3 g c0 t
+t3)) H3 (lift h x1 x0) H6) in (let H10 \def (H8 x0 x1 (refl_equal T (lift h
+x1 x0)) e H7) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3))
+(\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1
+t4) t2)) (\lambda (t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (H11:
+(pc3 c0 (lift h x1 x2) t3)).(\lambda (H12: (ty3 g e x0 x2)).(ex_intro2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0
+t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 H5) H12)))) H10)))))))))))))))))))
+(\lambda (c0: C).(\lambda (m: nat).(\lambda (x0: T).(\lambda (x1:
+nat).(\lambda (H1: (eq T (TSort m) (lift h x1 x0))).(\lambda (e: C).(\lambda
+(_: (drop h x1 c0 e)).(eq_ind_r T (TSort m) (\lambda (t: T).(ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e
+t t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort (next g
+m)))) (\lambda (t2: T).(ty3 g e (TSort m) t2)) (TSort (next g m)) (eq_ind_r T
+(TSort (next g m)) (\lambda (t: T).(pc3 c0 t (TSort (next g m)))) (pc3_refl
+c0 (TSort (next g m))) (lift h x1 (TSort (next g m))) (lift_sort (next g m) h
+x1)) (ty3_sort g e m)) x0 (lift_gen_sort h x1 m x0 H1))))))))) (\lambda (n:
+nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n
+c0 (CHead d0 (Bind Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u
+t)).(\lambda (H3: ((\forall (x: T).(\forall (x0: nat).((eq T u (lift h x0 x))
+\to (\forall (e: C).((drop h x0 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0
+(lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\lambda (x0:
+T).(\lambda (x1: nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda
+(e: C).(\lambda (H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n
+H4) in (let H6 \def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0
+t2))) (\lambda (H7: (land (lt n x1) (eq T x0 (TLRef n)))).(and_ind (lt n x1)
+(eq T x0 (TLRef n)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S
+n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n
+x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda
+(t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n:
+nat).(drop h n c0 e)) H5 (S (plus n (minus x1 (S n)))) (lt_plus_minus n x1
+H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus
+x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind
+Abbr) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0
+e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t)))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3:
+C).(\lambda (H11: (eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl
+n e (CHead x3 (Bind Abbr) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0
+x3)).(let H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x: T).(\forall (x0:
+nat).((eq T t0 (lift h x0 x)) \to (\forall (e: C).((drop h x0 d0 e) \to (ex2
+T (\lambda (t2: T).(pc3 d0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x
+t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u
+(\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2) H11) in
+(let H16 \def (H14 x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n))
+x2)) x3 H13) in (ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n))
+t2) t)) (\lambda (t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))
+(\lambda (x4: T).(\lambda (H17: (pc3 d0 (lift h (minus x1 (S n)) x4)
+t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S
+n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift
+(S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda
+(t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t)))
+(\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift
+(S n) O (lift h (minus x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n)
+O t))) (pc3_lift c0 d0 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus
+x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4))
+(lift_d x4 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g
+n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)))))))))
+(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9)))
+H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
+h))))).(and_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0
+(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
+T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+(lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
+(S (minus n h)) O t) (eq_ind_r T (lift (plus h (S (minus n h))) O t) (\lambda
+(t0: T).(pc3 c0 t0 (lift (S n) O t))) (eq_ind nat (S (plus h (minus n h)))
+(\lambda (n0: nat).(pc3 c0 (lift n0 O t) (lift (S n) O t))) (eq_ind nat n
+(\lambda (n0: nat).(pc3 c0 (lift (S n0) O t) (lift (S n) O t))) (pc3_refl c0
+(lift (S n) O t)) (plus h (minus n h)) (le_plus_minus h n (le_trans_plus_r x1
+h n H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1
+(lift (S (minus n h)) O t)) (lift_free t (S (minus n h)) h O x1 (le_trans x1
+(S (minus n h)) (plus O (S (minus n h))) (ty3_gen__le_S_minus x1 h n H8)
+(le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abbr g (minus n h) e d0 u
+(getl_drop_conf_ge n (CHead d0 (Bind Abbr) u) c0 H1 e h x1 H5 H8) t H2)) x0
+H9))) H7)) H6)))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d0: C).(\lambda (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abst)
+u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall
+(x: T).(\forall (x0: nat).((eq T u (lift h x0 x)) \to (\forall (e: C).((drop
+h x0 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x0 t2) t)) (\lambda
+(t2: T).(ty3 g e x t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda
+(H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda (H5: (drop h
+x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 \def H_x in
+(or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 h) n) (eq T
+x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift
+(S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: (land (lt n x1)
+(eq T x0 (TLRef n)))).(and_ind (lt n x1) (eq T x0 (TLRef n)) (ex2 T (\lambda
+(t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e
+x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 (TLRef
+n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (let
+H10 \def (eq_ind nat x1 (\lambda (n: nat).(drop h n c0 e)) H5 (S (plus n
+(minus x1 (S n)))) (lt_plus_minus n x1 H8)) in (ex3_2_ind T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (minus x1 (S n)) v)))) (\lambda (v:
+T).(\lambda (e0: C).(getl n e (CHead e0 (Bind Abst) v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e (TLRef
+n) t2))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H11: (eq T u (lift h
+(minus x1 (S n)) x2))).(\lambda (H12: (getl n e (CHead x3 (Bind Abst)
+x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0 x3)).(let H14 \def (eq_ind T
+u (\lambda (t0: T).(\forall (x: T).(\forall (x0: nat).((eq T t0 (lift h x0
+x)) \to (\forall (e: C).((drop h x0 d0 e) \to (ex2 T (\lambda (t2: T).(pc3 d0
+(lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x t2))))))))) H3 (lift h (minus
+x1 (S n)) x2) H11) in (let H15 \def (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0
+t)) H2 (lift h (minus x1 (S n)) x2) H11) in (eq_ind_r T (lift h (minus x1 (S
+n)) x2) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift
+(S n) O t0))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (let H16 \def (H14
+x2 (minus x1 (S n)) (refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in
+(ex2_ind T (\lambda (t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda
+(t2: T).(ty3 g x3 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+(lift (S n) O (lift h (minus x1 (S n)) x2)))) (\lambda (t2: T).(ty3 g e
+(TLRef n) t2))) (\lambda (x4: T).(\lambda (_: (pc3 d0 (lift h (minus x1 (S
+n)) x4) t)).(\lambda (H18: (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus
+x1 (S n))) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2)
+(lift (S n) O (lift h (minus n0 (S n)) x2)))) (\lambda (t2: T).(ty3 g e
+(TLRef n) t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h (plus (S n)
+(minus x1 (S n))) t2) (lift (S n) O (lift h (minus (plus (S n) (minus x1 (S
+n))) (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)) (lift (S n) O x2)
+(eq_ind_r T (lift (S n) O (lift h (minus x1 (S n)) x2)) (\lambda (t0: T).(pc3
+c0 t0 (lift (S n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n))
+x2)))) (eq_ind nat x1 (\lambda (n0: nat).(pc3 c0 (lift (S n) O (lift h (minus
+x1 (S n)) x2)) (lift (S n) O (lift h (minus n0 (S n)) x2)))) (pc3_refl c0
+(lift (S n) O (lift h (minus x1 (S n)) x2))) (plus (S n) (minus x1 (S n)))
+(le_plus_minus (S n) x1 H8)) (lift h (plus (S n) (minus x1 (S n))) (lift (S
+n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O (le_O_n (minus x1 (S n)))))
+(ty3_abst g n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)) u
+H11)))))))) (getl_drop_conf_lt Abst c0 d0 u n H1 e h (minus x1 (S n)) H10)))
+x0 H9))) H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
+h))))).(and_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
+T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0
+(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
+T).(ty3 g e t0 t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
+(lift (S n) O u))) (\lambda (t2: T).(ty3 g e (TLRef (minus n h)) t2)) (lift
+(S (minus n h)) O u) (eq_ind_r T (lift (plus h (S (minus n h))) O u) (\lambda
+(t0: T).(pc3 c0 t0 (lift (S n) O u))) (eq_ind nat (S (plus h (minus n h)))
+(\lambda (n0: nat).(pc3 c0 (lift n0 O u) (lift (S n) O u))) (eq_ind nat n
+(\lambda (n0: nat).(pc3 c0 (lift (S n0) O u) (lift (S n) O u))) (pc3_refl c0
+(lift (S n) O u)) (plus h (minus n h)) (le_plus_minus h n (le_trans_plus_r x1
+h n H8))) (plus h (S (minus n h))) (plus_n_Sm h (minus n h))) (lift h x1
+(lift (S (minus n h)) O u)) (lift_free u (S (minus n h)) h O x1 (le_trans x1
+(S (minus n h)) (plus O (S (minus n h))) (ty3_gen__le_S_minus x1 h n H8)
+(le_n (plus O (S (minus n h))))) (le_O_n x1))) (ty3_abst g (minus n h) e d0 u
+(getl_drop_conf_ge n (CHead d0 (Bind Abst) u) c0 H1 e h x1 H5 H8) t H2)) x0
+H9))) H7)) H6)))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
+T).(\lambda (H1: (ty3 g c0 u t)).(\lambda (H2: ((\forall (x: T).(\forall (x0:
+nat).((eq T u (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x
+t2)))))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H3:
+(ty3 g (CHead c0 (Bind b) u) t2 t3)).(\lambda (H4: ((\forall (x: T).(\forall
+(x0: nat).((eq T t2 (lift h x0 x)) \to (\forall (e: C).((drop h x0 (CHead c0
+(Bind b) u) e) \to (ex2 T (\lambda (t2: T).(pc3 (CHead c0 (Bind b) u) (lift h
+x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\lambda (t0:
+T).(\lambda (H5: (ty3 g (CHead c0 (Bind b) u) t3 t0)).(\lambda (H6: ((\forall
+(x: T).(\forall (x0: nat).((eq T t3 (lift h x0 x)) \to (\forall (e: C).((drop
+h x0 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t2: T).(pc3 (CHead c0
+(Bind b) u) (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 g e x
+t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H7: (eq T (THead
+(Bind b) u t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H8: (drop h x1 c0
+e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b)
+y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 T (\lambda (t4:
+T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3))) (\lambda (t4: T).(ty3 g e
+x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T x0 (THead
+(Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1 x2))).(\lambda (H11: (eq
+T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda (t4:
+T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind b) u t3)))
+(\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def (eq_ind T t2 (\lambda (t:
+T).(\forall (x: T).(\forall (x0: nat).((eq T t (lift h x0 x)) \to (\forall
+(e: C).((drop h x0 (CHead c0 (Bind b) u) e) \to (ex2 T (\lambda (t2: T).(pc3
+(CHead c0 (Bind b) u) (lift h x0 t2) t3)) (\lambda (t2: T).(ty3 g e x
+t2))))))))) H4 (lift h (S x1) x3) H11) in (let H13 \def (eq_ind T t2 (\lambda
+(t: T).(ty3 g (CHead c0 (Bind b) u) t t3)) H3 (lift h (S x1) x3) H11) in (let
+H14 \def (eq_ind T u (\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) (lift h (S
+x1) x3) t3)) H13 (lift h x1 x2) H10) in (let H15 \def (eq_ind T u (\lambda
+(t: T).(\forall (x: T).(\forall (x0: nat).((eq T (lift h (S x1) x3) (lift h
+x0 x)) \to (\forall (e: C).((drop h x0 (CHead c0 (Bind b) t) e) \to (ex2 T
+(\lambda (t2: T).(pc3 (CHead c0 (Bind b) t) (lift h x0 t2) t3)) (\lambda (t2:
+T).(ty3 g e x t2))))))))) H12 (lift h x1 x2) H10) in (let H16 \def (eq_ind T
+u (\lambda (t: T).(\forall (x: T).(\forall (x0: nat).((eq T t3 (lift h x0 x))
+\to (\forall (e: C).((drop h x0 (CHead c0 (Bind b) t) e) \to (ex2 T (\lambda
+(t2: T).(pc3 (CHead c0 (Bind b) t) (lift h x0 t2) t0)) (\lambda (t2: T).(ty3
+g e x t2))))))))) H6 (lift h x1 x2) H10) in (let H17 \def (eq_ind T u
+(\lambda (t: T).(ty3 g (CHead c0 (Bind b) t) t3 t0)) H5 (lift h x1 x2) H10)
+in (let H18 \def (eq_ind T u (\lambda (t0: T).(\forall (x: T).(\forall (x0:
+nat).((eq T t0 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2
+T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x
+t2))))))))) H2 (lift h x1 x2) H10) in (let H19 \def (eq_ind T u (\lambda (t0:
+T).(ty3 g c0 t0 t)) H1 (lift h x1 x2) H10) in (eq_ind_r T (lift h x1 x2)
+(\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind
+b) t4 t3))) (\lambda (t5: T).(ty3 g e (THead (Bind b) x2 x3) t5)))) (let H20
+\def (H18 x2 x1 (refl_equal T (lift h x1 x2)) e H8) in (ex2_ind T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3)))
+(\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4:
+T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H22: (ty3 g e x2
+x4)).(let H23 \def (H15 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
+(Bind b) x2) (drop_skip_bind h x1 c0 e H8 b x2)) in (ex2_ind T (\lambda (t4:
+T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda
+(t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0
+(lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e
+(THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H24: (pc3 (CHead c0
+(Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H25: (ty3 g (CHead
+e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t4: T).(ty3 g (CHead e (Bind b)
+x2) x5 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
+(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))
+(\lambda (x6: T).(\lambda (H26: (ty3 g (CHead e (Bind b) x2) x5
+x6)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
+(lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))
+(THead (Bind b) x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S
+x1) x5)) (\lambda (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h x1 x2) t3)))
+(pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H24) (lift h x1
+(THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H22 b
+x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 H25)))))
+H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h x1
+H7)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
+T).(\lambda (H1: (ty3 g c0 w u)).(\lambda (H2: ((\forall (x: T).(\forall (x0:
+nat).((eq T w (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T
+(\lambda (t2: T).(pc3 c0 (lift h x0 t2) u)) (\lambda (t2: T).(ty3 g e x
+t2)))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v (THead
+(Bind Abst) u t))).(\lambda (H4: ((\forall (x: T).(\forall (x0: nat).((eq T v
+(lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x0 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e
+x t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead
+(Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1 c0
+e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T (\lambda (t2:
+T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t))))
+(\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
+(H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq T w (lift h x1
+x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T (THead (Flat Appl)
+x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead
+(Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e t0 t2))))
+(let H10 \def (eq_ind T v (\lambda (t0: T).(\forall (x: T).(\forall (x0:
+nat).((eq T t0 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to (ex2
+T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) (THead (Bind Abst) u t))) (\lambda
+(t2: T).(ty3 g e x t2))))))))) H4 (lift h x1 x3) H9) in (let H11 \def (eq_ind
+T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 (lift h x1 x3)
+H9) in (let H12 \def (eq_ind T w (\lambda (t: T).(\forall (x: T).(\forall
+(x0: nat).((eq T t (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) u)) (\lambda (t2: T).(ty3 g e
+x t2))))))))) H2 (lift h x1 x2) H8) in (let H13 \def (eq_ind T w (\lambda (t:
+T).(ty3 g c0 t u)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2)
+(\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat
+Appl) t0 (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat
+Appl) x2 x3) t2)))) (let H14 \def (H12 x2 x1 (refl_equal T (lift h x1 x2)) e
+H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2:
+T).(ty3 g e x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead
+(Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g
+e (THead (Flat Appl) x2 x3) t2))) (\lambda (x4: T).(\lambda (H15: (pc3 c0
+(lift h x1 x4) u)).(\lambda (H16: (ty3 g e x2 x4)).(let H17 \def (H10 x3 x1
+(refl_equal T (lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0
+(lift h x1 t2) (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x3 t2))
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
+x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
+x2 x3) t2))) (\lambda (x5: T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead
+(Bind Abst) u t))).(\lambda (H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda
+(u1: T).(\lambda (t2: T).(pr3 e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1:
+T).(\lambda (_: T).(pr3 c0 u (lift h x1 u1)))) (\lambda (_: T).(\lambda (t2:
+T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S
+x1) t2)))))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat
+Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e
+(THead (Flat Appl) x2 x3) t2))) (\lambda (x6: T).(\lambda (x7: T).(\lambda
+(H20: (pr3 e x5 (THead (Bind Abst) x6 x7))).(\lambda (H21: (pr3 c0 u (lift h
+x1 x6))).(\lambda (H22: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c0 (Bind
+b) u) t (lift h (S x1) x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0))
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
+x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
+x2 x3) t2))) (\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(ex4_3_ind T T
+T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 e (THead (Bind Abst)
+x6 t2) x8)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g e x6
+t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead e (Bind
+Abst) x6) x7 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g
+(CHead e (Bind Abst) x6) t2 t3)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
+t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda
+(t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda
+(x10: T).(\lambda (x11: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9)
+x8)).(\lambda (H25: (ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind
+Abst) x6) x7 x9)).(\lambda (H27: (ty3 g (CHead e (Bind Abst) x6) x9
+x11)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
+(lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead
+(Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))
+(eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst)
+x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead
+(Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7))
+(THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h
+(S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t)))
+(pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind
+Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6))
+(lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind
+Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1
+(THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead
+(Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4
+H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6)
+(pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind
+Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9
+H26 x11 H27) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7)
+H20))))))))))) (ty3_gen_bind g Abst e x6 x7 x8 (ty3_sred_pr3 e x5 (THead
+(Bind Abst) x6 x7) H20 g x8 H23))))) (ty3_correct g e x3 x5 H19)))))))
+(pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0
+H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda
+(H2: ((\forall (x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x)) \to
+(\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
+x0 t2) t3)) (\lambda (t2: T).(ty3 g e x t2)))))))))).(\lambda (t0:
+T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x: T).(\forall
+(x0: nat).((eq T t3 (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 g e
+x t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead
+(Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6: (drop h x1
+c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
+Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h x1 y0))))
+(\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda
+(t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
+x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 x2))).(\lambda (H9: (eq T t2 (lift
+h x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
+(\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e t
+t4)))) (let H10 \def (eq_ind T t3 (\lambda (t: T).(\forall (x: T).(\forall
+(x0: nat).((eq T t (lift h x0 x)) \to (\forall (e: C).((drop h x0 c0 e) \to
+(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2) t0)) (\lambda (t2: T).(ty3 g e
+x t2))))))))) H4 (lift h x1 x2) H8) in (let H11 \def (eq_ind T t3 (\lambda
+(t: T).(ty3 g c0 t t0)) H3 (lift h x1 x2) H8) in (let H12 \def (eq_ind T t3
+(\lambda (t: T).(\forall (x: T).(\forall (x0: nat).((eq T t2 (lift h x0 x))
+\to (\forall (e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0
+(lift h x0 t2) t)) (\lambda (t2: T).(ty3 g e x t2))))))))) H2 (lift h x1 x2)
+H8) in (let H13 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t2 t)) H1 (lift h
+x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t: T).(ex2 T (\lambda (t4:
+T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2
+x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t: T).(ty3 g c0 t (lift h x1
+x2))) H13 (lift h x1 x3) H9) in (let H15 \def (eq_ind T t2 (\lambda (t:
+T).(\forall (x: T).(\forall (x0: nat).((eq T t (lift h x0 x)) \to (\forall
+(e: C).((drop h x0 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x0 t2)
+(lift h x1 x2))) (\lambda (t2: T).(ty3 g e x t2))))))))) H12 (lift h x1 x3)
+H9) in (let H16 \def (H15 x3 x1 (refl_equal T (lift h x1 x3)) e H6) in
+(ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda
+(t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift
+h x1 x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda
+(x4: T).(\lambda (H17: (pc3 c0 (lift h x1 x4) (lift h x1 x2))).(\lambda (H18:
+(ty3 g e x3 x4)).(let H19 \def (H10 x2 x1 (refl_equal T (lift h x1 x2)) e H6)
+in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4:
+T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1
+x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4))) (\lambda (x5:
+T).(\lambda (_: (pc3 c0 (lift h x1 x5) t0)).(\lambda (H21: (ty3 g e x2
+x5)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (lift h x1 x2)))
+(\lambda (t4: T).(ty3 g e (THead (Flat Cast) x2 x3) t4)) x2 (pc3_refl c0
+(lift h x1 x2)) (ty3_cast g e x3 x2 (ty3_conv g e x2 x5 H21 x3 x4 H18
+(pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21))))) H19))))) H16)))) t3
+H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1 H5))))))))))))))) c y x
+H0))))) H))))))).
+
+theorem ty3_tred:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T
+(\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1:
+(ty3 g c t1 x)).(ty3_conv g c t2 x (ty3_sred_pr3 c t1 t2 H0 g x H1) u t1 H
+(pc3_pr3_r c t1 t2 H0)))) (ty3_correct g c u t1 H)))))))).
+
+theorem ty3_sconv_pc3:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
+u2) \to (pc3 c t1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x:
+T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_unique g c x
+t1 (ty3_sred_pr3 c u1 x H3 g t1 H) t2 (ty3_sred_pr3 c u2 x H4 g t2 H0)))))
+H2)))))))))).
+
+theorem ty3_sred_back:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c
+t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2
+t) \to (ty3 g c t1 t)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda
+(H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
+(t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t
+t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g
+c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t
+H1)))) (ty3_correct g c t2 t H1)))))))))).
+
+theorem ty3_sconv:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
+u2) \to (ty3 g c u1 t2)))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
+(t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda
+(x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back
+g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
+
+theorem ty3_tau0:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
+t1) \to (\forall (t2: T).((tau0 g c u t2) \to (ty3 g c u t2)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
+(ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_:
+T).(\forall (t2: T).((tau0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda
+(_: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda
+(u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3:
+((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (_:
+(pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0
+H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda
+(H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 return (\lambda (c:
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c
+c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m)
+t2)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H0: (eq C c0
+c0)).(\lambda (H1: (eq T (TSort n) (TSort m))).(\lambda (H2: (eq T (TSort
+(next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to
+((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H3:
+(eq T (TSort n) (TSort m))).(let H4 \def (f_equal T nat (\lambda (e:
+T).(match e return (\lambda (_: T).nat) with [(TSort n) \Rightarrow n |
+(TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort m)
+H3) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to
+(ty3 g c0 (TSort m) t2))) (\lambda (H5: (eq T (TSort (next g m)) t2)).(eq_ind
+T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0
+m) t2 H5)) n (sym_eq nat n m H4)))) c0 (sym_eq C c0 c0 H0) H1 H2)))) |
+(tau0_abbr c0 d v i H0 w H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda
+(H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: (eq T (lift (S i) O w)
+t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TSort m)) \to ((eq T
+(lift (S i) O w) t2) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v
+w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (TLRef i) (TSort
+m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H5) in
+(False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr)
+v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0
+c0 H2) H3 H4 H0 H1)))) | (tau0_abst c0 d v i H0 w H1) \Rightarrow (\lambda
+(H2: (eq C c0 c0)).(\lambda (H3: (eq T (TLRef i) (TSort m))).(\lambda (H4:
+(eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i)
+(TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d (Bind
+Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5:
+(eq T (TLRef i) (TSort m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m)
+H5) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0 (CHead d (Bind
+Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq
+C c0 c0 H2) H3 H4 H0 H1)))) | (tau0_bind b c0 v t1 t0 H0) \Rightarrow
+(\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Bind b) v t1) (TSort
+m))).(\lambda (H3: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c:
+C).((eq T (THead (Bind b) v t1) (TSort m)) \to ((eq T (THead (Bind b) v t0)
+t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) t2)))))
+(\lambda (H4: (eq T (THead (Bind b) v t1) (TSort m))).(let H5 \def (eq_ind T
+(THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TSort m) H4) in (False_ind ((eq T (THead (Bind b) v
+t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m)
+t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | (tau0_appl c0 v t1 t0 H0)
+\Rightarrow (\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Flat
+Appl) v t1) (TSort m))).(\lambda (H3: (eq T (THead (Flat Appl) v t0)
+t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TSort m))
+\to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0
+(TSort m) t2))))) (\lambda (H4: (eq T (THead (Flat Appl) v t1) (TSort
+m))).(let H5 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H4) in
+(False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to
+(ty3 g c0 (TSort m) t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) |
+(tau0_cast c0 v1 v2 H0 t1 t0 H1) \Rightarrow (\lambda (H2: (eq C c0
+c0)).(\lambda (H3: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(\lambda (H4:
+(eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T
+(THead (Flat Cast) v1 t1) (TSort m)) \to ((eq T (THead (Flat Cast) v2 t0) t2)
+\to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TSort m) t2))))))
+(\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(let H6 \def
+(eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T
+(THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0)
+\to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1))))])
+in (H1 (refl_equal C c0) (refl_equal T (TSort m)) (refl_equal T t2)))))))
+(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
+(H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3
+g d u0 t)).(\lambda (H2: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0
+t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def
+(match H3 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda
+(_: (tau0 ? c t t0)).((eq C c c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2)
+\to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c0 n0) \Rightarrow
+(\lambda (H3: (eq C c0 c0)).(\lambda (H4: (eq T (TSort n0) (TLRef
+n))).(\lambda (H5: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_:
+C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g
+c0 (TLRef n) t2)))) (\lambda (H6: (eq T (TSort n0) (TLRef n))).(let H7 \def
+(eq_ind T (TSort n0) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow False])) I (TLRef n) H6) in (False_ind ((eq T (TSort (next g
+n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5))))
+| (tau0_abbr c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0
+c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S
+i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to
+((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d0 (Bind Abbr) v)) \to
+((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef
+i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n
+| (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n
+(\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0
+(Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda
+(H10: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t:
+T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0
+(TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda
+(H12: (tau0 g d0 v w)).(let H13 \def (eq_ind C (CHead d (Bind Abbr) u0)
+(\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0
+(CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (let H14 \def
+(f_equal C C (\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort
+_) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
+d0 (Bind Abbr) v) H11)) in ((let H15 \def (f_equal C T (\lambda (e: C).(match
+e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) (getl_mono
+c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (\lambda
+(H16: (eq C d d0)).(let H17 \def (eq_ind_r T v (\lambda (t: T).(getl n c0
+(CHead d0 (Bind Abbr) t))) H13 u0 H15) in (let H18 \def (eq_ind_r T v
+(\lambda (t: T).(tau0 g d0 t w)) H12 u0 H15) in (let H19 \def (eq_ind_r C d0
+(\lambda (c: C).(getl n c0 (CHead c (Bind Abbr) u0))) H17 d H16) in (let H20
+\def (eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H18 d H16) in (ty3_abbr
+g n c0 d u0 H19 w (H2 w H20)))))))) H14))))) t2 H10)) i (sym_eq nat i n
+H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i H3 w H4)
+\Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef
+n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c:
+C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c
+(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n)
+t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T
+nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i]))
+(TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S
+n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w)
+\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O v)
+t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 (CHead d0 (Bind
+Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11:
+(getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0 v w)).(let H2
+\def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n c0 c)) H0
+(CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead
+d0 (Bind Abst) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abbr) u0)
+(\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow
+False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abst) v)
+(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in
+(False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H13))))) t2 H10)) i (sym_eq
+nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1
+t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead
+(Bind b) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Bind b) v t0)
+t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TLRef n)) \to
+((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to
+(ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef
+n))).(let H8 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in
+(False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b)
+v) t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6
+H3)))) | (tau0_appl c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0
+c0)).(\lambda (H5: (eq T (THead (Flat Appl) v t1) (TLRef n))).(\lambda (H6:
+(eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T
+(THead (Flat Appl) v t1) (TLRef n)) \to ((eq T (THead (Flat Appl) v t0) t2)
+\to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T
+(THead (Flat Appl) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Flat
+Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Flat Appl) v
+t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq
+C c0 c0 H4) H5 H6 H3)))) | (tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow
+(\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1)
+(TLRef n))).(\lambda (H7: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0
+(\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead
+(Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g
+c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef
+n))).(let H9 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match
+e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in
+(False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to
+((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0
+H5) H6 H7 H3 H4))))]) in (H4 (refl_equal C c0) (refl_equal T (TLRef n))
+(refl_equal T t2))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 t)).(\lambda (_: ((\forall
+(t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda
+(H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 return (\lambda (c:
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c
+c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) \to (ty3 g c0 (TLRef n)
+t2)))))))) with [(tau0_sort c0 n0) \Rightarrow (\lambda (H3: (eq C c0
+c0)).(\lambda (H4: (eq T (TSort n0) (TLRef n))).(\lambda (H5: (eq T (TSort
+(next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) (TLRef n))
+\to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) (\lambda
+(H6: (eq T (TSort n0) (TLRef n))).(let H7 \def (eq_ind T (TSort n0) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True
+| (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef
+n) H6) in (False_ind ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n)
+t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) | (tau0_abbr c0 d0 v i H3 w H4)
+\Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef
+n))).(\lambda (H7: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c:
+C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c
+(CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n)
+t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T
+nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i]))
+(TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S
+n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w)
+\to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O w)
+t2)).(eq_ind T (lift (S n) O w) (\lambda (t: T).((getl n c0 (CHead d0 (Bind
+Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11:
+(getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H2
+\def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: C).(getl n c0 c)) H0
+(CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+d0 (Bind Abbr) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abst) u0)
+(\lambda (ee: C).(match ee return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_:
+K).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: B).Prop)
+with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow
+False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abbr) v)
+(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in
+(False_ind (ty3 g c0 (TLRef n) (lift (S n) O w)) H13))))) t2 H10)) i (sym_eq
+nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i
+H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef
+i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0
+(\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2)
+\to ((getl i c (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0
+(TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def
+(f_equal T nat (\lambda (e: T).(match e return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _)
+\Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0:
+nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v))
+\to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T
+(lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0
+(CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t))))
+(\lambda (H11: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H12: (tau0 g
+d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c:
+C).(getl n c0 c)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (let H13 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u0)
+(CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead
+d0 (Bind Abst) v) H11)) in ((let H14 \def (f_equal C T (\lambda (e: C).(match
+e return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) (getl_mono
+c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (\lambda
+(H15: (eq C d d0)).(let H16 \def (eq_ind_r T v (\lambda (t: T).(getl n c0
+(CHead d0 (Bind Abst) t))) H2 u0 H14) in (let H17 \def (eq_ind_r T v (\lambda
+(t: T).(tau0 g d0 t w)) H12 u0 H14) in (eq_ind T u0 (\lambda (t: T).(ty3 g c0
+(TLRef n) (lift (S n) O t))) (let H18 \def (eq_ind_r C d0 (\lambda (c:
+C).(getl n c0 (CHead c (Bind Abst) u0))) H16 d H15) in (let H19 \def
+(eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H17 d H15) in (ty3_abst g n
+c0 d u0 H18 t H1))) v H14))))) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0
+(sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 t0 H3) \Rightarrow
+(\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Bind b) v t1) (TLRef
+n))).(\lambda (H6: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c:
+C).((eq T (THead (Bind b) v t1) (TLRef n)) \to ((eq T (THead (Bind b) v t0)
+t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2)))))
+(\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef n))).(let H8 \def (eq_ind T
+(THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop)
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
+_) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Bind b) v
+t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n)
+t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_appl c0 v t1 t0 H3)
+\Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Flat
+Appl) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Flat Appl) v t0)
+t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TLRef n))
+\to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0
+(TLRef n) t2))))) (\lambda (H7: (eq T (THead (Flat Appl) v t1) (TLRef
+n))).(let H8 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in
+(False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to
+(ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) |
+(tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0
+c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(\lambda (H7:
+(eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T
+(THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead (Flat Cast) v2 t0) t2)
+\to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))))
+(\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let H9 \def
+(eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T
+(THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0)
+\to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4))))])
+in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) (refl_equal T
+t2))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda
+(H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 u0 t2) \to
+(ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3:
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 t3)).(\lambda (H3: ((\forall
+(t3: T).((tau0 g (CHead c0 (Bind b) u0) t2 t3) \to (ty3 g (CHead c0 (Bind b)
+u0) t2 t3))))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t3
+t0)).(\lambda (_: ((\forall (t2: T).((tau0 g (CHead c0 (Bind b) u0) t3 t2)
+\to (ty3 g (CHead c0 (Bind b) u0) t3 t2))))).(\lambda (t4: T).(\lambda (H6:
+(tau0 g c0 (THead (Bind b) u0 t2) t4)).(let H7 \def (match H6 return (\lambda
+(c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C
+c c0) \to ((eq T t (THead (Bind b) u0 t2)) \to ((eq T t0 t4) \to (ty3 g c0
+(THead (Bind b) u0 t2) t4)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda
+(H6: (eq C c0 c0)).(\lambda (H7: (eq T (TSort n) (THead (Bind b) u0
+t2))).(\lambda (H8: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_:
+C).((eq T (TSort n) (THead (Bind b) u0 t2)) \to ((eq T (TSort (next g n)) t4)
+\to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H9: (eq T (TSort n)
+(THead (Bind b) u0 t2))).(let H10 \def (eq_ind T (TSort n) (\lambda (e:
+T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead
+(Bind b) u0 t2) H9) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0
+(THead (Bind b) u0 t2) t4)) H10))) c0 (sym_eq C c0 c0 H6) H7 H8)))) |
+(tau0_abbr c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda
+(H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (lift (S
+i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Bind b) u0
+t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c (CHead d (Bind Abbr) v))
+\to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda
+(H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (TLRef
+i) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (lift (S i) O w)
+t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g
+c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6
+H7)))) | (tau0_abst c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0
+c0)).(\lambda (H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10:
+(eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i)
+(THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c (CHead
+d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2)
+t4)))))) (\lambda (H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12
+\def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H11) in
+(False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst)
+v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0
+(sym_eq C c0 c0 H8) H9 H10 H6 H7)))) | (tau0_bind b0 c0 v t4 t5 H6)
+\Rightarrow (\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Bind b0)
+v t4) (THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Bind b0) v t5)
+t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b0) v t4) (THead (Bind
+b) u0 t2)) \to ((eq T (THead (Bind b0) v t5) t4) \to ((tau0 g (CHead c (Bind
+b0) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq
+T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(let H11 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t]))
+(THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H12 \def (f_equal
+T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t]))
+(THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H13 \def (f_equal
+T B (\lambda (e: T).(match e return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match
+k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow b0])])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in
+(eq_ind B b (\lambda (b1: B).((eq T v u0) \to ((eq T t4 t2) \to ((eq T (THead
+(Bind b1) v t5) t4) \to ((tau0 g (CHead c0 (Bind b1) v) t4 t5) \to (ty3 g c0
+(THead (Bind b) u0 t2) t4)))))) (\lambda (H14: (eq T v u0)).(eq_ind T u0
+(\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind b) t t5) t4) \to ((tau0
+g (CHead c0 (Bind b) t) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))
+(\lambda (H15: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind
+b) u0 t5) t4) \to ((tau0 g (CHead c0 (Bind b) u0) t t5) \to (ty3 g c0 (THead
+(Bind b) u0 t2) t4)))) (\lambda (H16: (eq T (THead (Bind b) u0 t5)
+t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t: T).((tau0 g (CHead c0
+(Bind b) u0) t2 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t))) (\lambda (H17:
+(tau0 g (CHead c0 (Bind b) u0) t2 t5)).(let H_y \def (H3 t5 H17) in (ex_ind T
+(\lambda (t: T).(ty3 g (CHead c0 (Bind b) u0) t5 t)) (ty3 g c0 (THead (Bind
+b) u0 t2) (THead (Bind b) u0 t5)) (\lambda (x: T).(\lambda (H1: (ty3 g (CHead
+c0 (Bind b) u0) t5 x)).(ty3_bind g c0 u0 t H0 b t2 t5 H_y x H1)))
+(ty3_correct g (CHead c0 (Bind b) u0) t2 t5 H_y)))) t4 H16)) t4 (sym_eq T t4
+t2 H15))) v (sym_eq T v u0 H14))) b0 (sym_eq B b0 b H13))) H12)) H11))) c0
+(sym_eq C c0 c0 H7) H8 H9 H6)))) | (tau0_appl c0 v t4 t5 H6) \Rightarrow
+(\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Flat Appl) v t4)
+(THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Flat Appl) v t5)
+t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t4) (THead
+(Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c t4
+t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq T (THead
+(Flat Appl) v t4) (THead (Bind b) u0 t2))).(let H11 \def (eq_ind T (THead
+(Flat Appl) v t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H10) in
+(False_ind ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to
+(ty3 g c0 (THead (Bind b) u0 t2) t4))) H11))) c0 (sym_eq C c0 c0 H7) H8 H9
+H6)))) | (tau0_cast c0 v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H8: (eq C c0
+c0)).(\lambda (H9: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0
+t2))).(\lambda (H10: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0
+(\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2)) \to
+((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c t4
+t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (THead
+(Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (THead
+(Flat Cast) v1 t4) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H11) in
+(False_ind ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 v1 v2) \to
+((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0
+(sym_eq C c0 c0 H8) H9 H10 H6 H7))))]) in (H7 (refl_equal C c0) (refl_equal T
+(THead (Bind b) u0 t2)) (refl_equal T t4)))))))))))))))))) (\lambda (c0:
+C).(\lambda (w: T).(\lambda (u0: T).(\lambda (H0: (ty3 g c0 w u0)).(\lambda
+(_: ((\forall (t2: T).((tau0 g c0 w t2) \to (ty3 g c0 w t2))))).(\lambda (v:
+T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v (THead (Bind Abst) u0
+t))).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 v t2) \to (ty3 g c0 v
+t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead (Flat Appl) w v)
+t2)).(let H5 \def (match H4 return (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c c0) \to ((eq T t (THead (Flat
+Appl) w v)) \to ((eq T t0 t2) \to (ty3 g c0 (THead (Flat Appl) w v)
+t2)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H4: (eq C c0
+c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Appl) w v))).(\lambda (H6:
+(eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n)
+(THead (Flat Appl) w v)) \to ((eq T (TSort (next g n)) t2) \to (ty3 g c0
+(THead (Flat Appl) w v) t2)))) (\lambda (H7: (eq T (TSort n) (THead (Flat
+Appl) w v))).(let H8 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w
+v) H7) in (False_ind ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (THead (Flat
+Appl) w v) t2)) H8))) c0 (sym_eq C c0 c0 H4) H5 H6)))) | (tau0_abbr c0 d v0 i
+H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef
+i) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (lift (S i) O w0)
+t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Appl) w v))
+\to ((eq T (lift (S i) O w0) t2) \to ((getl i c (CHead d (Bind Abbr) v0)) \to
+((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda
+(H9: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (TLRef
+i) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (lift (S i) O w0)
+t2) \to ((getl i c0 (CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3
+g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4
+H5)))) | (tau0_abst c0 d v0 i H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0
+c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H8:
+(eq T (lift (S i) O v0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i)
+(THead (Flat Appl) w v)) \to ((eq T (lift (S i) O v0) t2) \to ((getl i c
+(CHead d (Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat
+Appl) w v) t2)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Appl) w
+v))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w
+v) H9) in (False_ind ((eq T (lift (S i) O v0) t2) \to ((getl i c0 (CHead d
+(Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v)
+t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b c0 v0 t1
+t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead
+(Bind b) v0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Bind b)
+v0 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v0 t1) (THead
+(Flat Appl) w v)) \to ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c
+(Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda
+(H8: (eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v))).(let H9 \def
+(eq_ind T (THead (Bind b) v0 t1) (\lambda (e: T).(match e return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind
+_) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w
+v) H8) in (False_ind ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c0
+(Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H9))) c0
+(sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_appl c0 v0 t1 t0 H4) \Rightarrow
+(\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v0 t1)
+(THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0)
+t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v0 t1) (THead
+(Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t0) t2) \to ((tau0 g c t1
+t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H8: (eq T (THead
+(Flat Appl) v0 t1) (THead (Flat Appl) w v))).(let H9 \def (f_equal T T
+(\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in ((let H10 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t]))
+(THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda
+(t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) t t0) t2) \to ((tau0 g c0
+t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H11: (eq T t1
+v)).(eq_ind T v (\lambda (t: T).((eq T (THead (Flat Appl) w t0) t2) \to
+((tau0 g c0 t t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H12:
+(eq T (THead (Flat Appl) w t0) t2)).(eq_ind T (THead (Flat Appl) w t0)
+(\lambda (t: T).((tau0 g c0 v t0) \to (ty3 g c0 (THead (Flat Appl) w v) t)))
+(\lambda (H13: (tau0 g c0 v t0)).(let H_y \def (H3 t0 H13) in (let H1 \def
+(ty3_unique g c0 v t0 H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T (\lambda
+(t: T).(ty3 g c0 t0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl)
+w t0)) (\lambda (x: T).(\lambda (H3: (ty3 g c0 t0 x)).(ex_ind T (\lambda (t:
+T).(ty3 g c0 u0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w
+t0)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0 x0)).(ex_ind T (\lambda (t2:
+T).(ty3 g c0 (THead (Bind Abst) u0 t) t2)) (ty3 g c0 (THead (Flat Appl) w v)
+(THead (Flat Appl) w t0)) (\lambda (x1: T).(\lambda (H15: (ty3 g c0 (THead
+(Bind Abst) u0 t) x1)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_:
+T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t2) x1)))) (\lambda (_:
+T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u0 t)))) (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u0) t t2))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind
+Abst) u0) t2 t3)))) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w
+t0)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0
+(THead (Bind Abst) u0 x2) x1)).(\lambda (H17: (ty3 g c0 u0 x3)).(\lambda
+(H18: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(\lambda (H19: (ty3 g (CHead c0
+(Bind Abst) u0) x2 x4)).(ty3_conv g c0 (THead (Flat Appl) w t0) (THead (Flat
+Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w u0 H0 t0 x2 (ty3_sconv g
+c0 t0 x H3 (THead (Bind Abst) u0 t) (THead (Bind Abst) u0 x2) (ty3_bind g c0
+u0 x3 H17 Abst t x2 H18 x4 H19) H1)) (THead (Flat Appl) w v) (THead (Flat
+Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v t H2) (pc3_s c0
+(THead (Flat Appl) w (THead (Bind Abst) u0 t)) (THead (Flat Appl) w t0)
+(pc3_thin_dx c0 t0 (THead (Bind Abst) u0 t) H1 w Appl)))))))))) (ty3_gen_bind
+g Abst c0 u0 t x1 H15)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2))))
+(ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t0 H_y))))) t2 H12)) t1
+(sym_eq T t1 v H11))) v0 (sym_eq T v0 w H10))) H9))) c0 (sym_eq C c0 c0 H5)
+H6 H7 H4)))) | (tau0_cast c0 v1 v2 H4 t1 t0 H5) \Rightarrow (\lambda (H6: (eq
+C c0 c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w
+v))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda
+(c: C).((eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v)) \to ((eq T
+(THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to
+(ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (THead (Flat
+Cast) v1 t1) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead (Flat
+Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow
+False | (Flat f) \Rightarrow (match f return (\lambda (_: F).Prop) with [Appl
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v)
+H9) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2)
+\to ((tau0 g c0 t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0
+(sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) (refl_equal T
+(THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0:
+C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda
+(H1: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda
+(t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t2: T).((tau0
+g c0 t3 t2) \to (ty3 g c0 t3 t2))))).(\lambda (t4: T).(\lambda (H4: (tau0 g
+c0 (THead (Flat Cast) t3 t2) t4)).(let H5 \def (match H4 return (\lambda (c:
+C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (tau0 ? c t t0)).((eq C c
+c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c0
+(THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c0 n) \Rightarrow
+(\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Cast)
+t3 t2))).(\lambda (H6: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda
+(_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g
+n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H7: (eq T
+(TSort n) (THead (Flat Cast) t3 t2))).(let H8 \def (eq_ind T (TSort n)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T (TSort (next g
+n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H8))) c0 (sym_eq C c0 c0
+H4) H5 H6)))) | (tau0_abbr c0 d v i H4 w H5) \Rightarrow (\lambda (H6: (eq C
+c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda
+(H8: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef
+i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c
+(CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast)
+t3 t2) t4)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Cast) t3
+t2))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3
+t2) H9) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d
+(Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2)
+t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c0 d v i H4
+w H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i)
+(THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (lift (S i) O v) t4)).(eq_ind
+C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T
+(lift (S i) O v) t4) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v
+w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T
+(TLRef i) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (TLRef i)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
+False])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (lift (S i) O
+v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3
+g c0 (THead (Flat Cast) t3 t2) t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4
+H5)))) | (tau0_bind b c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0
+c0)).(\lambda (H6: (eq T (THead (Bind b) v t4) (THead (Flat Cast) t3
+t2))).(\lambda (H7: (eq T (THead (Bind b) v t5) t4)).(eq_ind C c0 (\lambda
+(c: C).((eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T
+(THead (Bind b) v t5) t4) \to ((tau0 g (CHead c (Bind b) v) t4 t5) \to (ty3 g
+c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Bind b) v
+t4) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Bind b) v t4)
+(\lambda (e: T).(match e return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq
+T (THead (Bind b) v t5) t4) \to ((tau0 g (CHead c0 (Bind b) v) t4 t5) \to
+(ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7
+H4)))) | (tau0_appl c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0
+c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3
+t2))).(\lambda (H7: (eq T (THead (Flat Appl) v t5) t4)).(eq_ind C c0 (\lambda
+(c: C).((eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T
+(THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Flat
+Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Flat Appl) v t4) (THead
+(Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Flat Appl) v t4) (\lambda
+(e: T).(match e return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
+\Rightarrow (match f return (\lambda (_: F).Prop) with [Appl \Rightarrow True
+| Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind
+((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to (ty3 g c0
+(THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) |
+(tau0_cast c0 v1 v2 H4 t4 t5 H5) \Rightarrow (\lambda (H6: (eq C c0
+c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3
+t2))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0
+(\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))
+\to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c
+t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T
+(THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))).(let H10 \def (f_equal
+T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t]))
+(THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in ((let H11 \def
+(f_equal T T (\lambda (e: T).(match e return (\lambda (_: T).T) with [(TSort
+_) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t]))
+(THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in (eq_ind T t3
+(\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to
+((tau0 g c0 t v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3
+t2) t4)))))) (\lambda (H12: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T
+(THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 t3 v2) \to ((tau0 g c0 t t5)
+\to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H13: (eq T (THead
+(Flat Cast) v2 t5) t4)).(eq_ind T (THead (Flat Cast) v2 t5) (\lambda (t:
+T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t5) \to (ty3 g c0 (THead (Flat Cast)
+t3 t2) t)))) (\lambda (H14: (tau0 g c0 t3 v2)).(\lambda (H15: (tau0 g c0 t2
+t5)).(let H_y \def (H1 t5 H15) in (let H_y0 \def (H3 v2 H14) in (let H3 \def
+(ty3_unique g c0 t2 t5 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 v2
+t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda
+(x: T).(\lambda (H16: (ty3 g c0 v2 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t5
+t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda
+(x0: T).(\lambda (H17: (ty3 g c0 t5 x0)).(ty3_conv g c0 (THead (Flat Cast) v2
+t5) v2 (ty3_cast g c0 t5 v2 (ty3_sconv g c0 t5 x0 H17 t3 v2 H_y0 H3) x H16)
+(THead (Flat Cast) t3 t2) t3 (ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3
+(THead (Flat Cast) v2 t5) (pc3_pr2_u c0 t5 (THead (Flat Cast) v2 t5)
+(pr2_free c0 (THead (Flat Cast) v2 t5) t5 (pr0_epsilon t5 t5 (pr0_refl t5)
+v2)) t3 H3))))) (ty3_correct g c0 t2 t5 H_y)))) (ty3_correct g c0 t3 v2
+H_y0))))))) t4 H13)) t4 (sym_eq T t4 t2 H12))) v1 (sym_eq T v1 t3 H11)))
+H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0)
+(refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1
+H))))).
+
+theorem ty3_arity:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
+t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity
+g c t2 (asucc g a1))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda
+(t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g
+c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t:
+T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity
+g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
+a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda
+(a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1)))
+(ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3
+(asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_:
+(arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A
+(\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g
+a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10:
+(arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0:
+T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1:
+A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1))))
+(\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3
+x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
+g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g
+(asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12
+g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11)))))
+H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A
+(\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0
+(TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort
+g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g
+d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1))
+(\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g
+c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr
+c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
+(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A
+(\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g
+a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1))
+(\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
+a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d
+t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind
+A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g
+c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g
+a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A
+(\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0
+(lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g
+d u x H4 (asucc g x0) H7)) (arity_lift g d u (asucc g x0) (arity_repl g d u x
+H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) H6))))))
+H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_:
+(ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 u a1))
+(\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: B).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3
+t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3
+a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g
+a1))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4
+t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4
+a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let
+H6 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
+A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead
+(Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc
+g a1)))) (\lambda (x: A).(\lambda (H7: (arity g c0 u x)).(\lambda (_: (arity
+g c0 t (asucc g x))).(let H9 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g
+(CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u)
+t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
+(\lambda (x0: A).(\lambda (H10: (arity g (CHead c0 (Bind b) u) t3
+x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x
+\def (leq_asucc g x) in (let H12 \def H_x in (ex_ind A (\lambda (a0: A).(leq
+g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1))))
+(\lambda (x1: A).(\lambda (H13: (leq g x (asucc g x1))).((match b return
+(\lambda (b0: B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda
+(a1: A).(arity g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g
+(CHead c0 (Bind b0) u) t0 (asucc g a1)))) \to ((arity g (CHead c0 (Bind b0)
+u) t3 x0) \to ((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A
+(\lambda (a1: A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1:
+A).(arity g c0 (THead (Bind b0) u t4) (asucc g a1))))))))) with [Abbr
+\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abbr) u) t4 t0)).(\lambda (_:
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t4 a1)) (\lambda
+(a1: A).(arity g (CHead c0 (Bind Abbr) u) t0 (asucc g a1))))).(\lambda (H16:
+(arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda (H17: (arity g (CHead c0
+(Bind Abbr) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0
+(THead (Bind Abbr) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr)
+u t4) (asucc g a1))) x0 (arity_bind g Abbr not_abbr_abst c0 u x H7 t3 x0 H16)
+(arity_bind g Abbr not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) | Abst
+\Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abst) u) t4 t0)).(\lambda (_:
+(ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t4 a1)) (\lambda
+(a1: A).(arity g (CHead c0 (Bind Abst) u) t0 (asucc g a1))))).(\lambda (H16:
+(arity g (CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H17: (arity g (CHead c0
+(Bind Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0
+(THead (Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst)
+u t4) (asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x
+H7 (asucc g x1) H13) t3 x0 H16) (arity_repl g c0 (THead (Bind Abst) u t4)
+(AHead x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc
+g x1) H13) t4 (asucc g x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g
+(AHead x1 x0))))))))) | Void \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind
+Void) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0
+(Bind Void) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t0
+(asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Void) u) t3
+x0)).(\lambda (H17: (arity g (CHead c0 (Bind Void) u) t4 (asucc g
+x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1))
+(\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0
+(arity_bind g Void not_void_abst c0 u x H7 t3 x0 H16) (arity_bind g Void
+not_void_abst c0 u x H7 t4 (asucc g x0) H17))))))]) H4 H5 H10 H11)))
+H12)))))) H9))))) H6))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda
+(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1:
+A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g
+a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind
+Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v a1))
+(\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))))).(let H4
+\def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1:
+A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead
+(Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w
+(THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity
+g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def H3 in
+(ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0
+(THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0
+(THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl)
+w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8:
+(arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) (asucc g
+x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in (ex3_2_ind A
+A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 a2))))
+(\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) (ex2 A (\lambda
+(a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0
+(THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1:
+A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1
+x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g
+(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_equal A (asucc g x0)
+(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in
+(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A
+x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1
+x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2
+(\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17)
+in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1
+x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v)
+a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u
+t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1
+(leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc
+g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc
+g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10)))))
+H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1:
+A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
+a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (_: (ex2 A
+(\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g
+a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1))
+(\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity
+g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g
+a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity
+g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat
+Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x
+(arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))).
+
+theorem ty3_predicative:
+ \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u:
+T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P:
+Prop).P)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (u:
+T).(\lambda (H: (ty3 g c (THead (Bind Abst) v t) u)).(\lambda (H0: (pc3 c u
+v)).(\lambda (P: Prop).(let H1 \def H in (ex4_3_ind T T T (\lambda (t2:
+T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) v t2) u))))
+(\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c v t0)))) (\lambda
+(t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) v) t
+t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c
+(Bind Abst) v) t2 t1)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (_: (pc3 c (THead (Bind Abst) v x0) u)).(\lambda (H3: (ty3 g c v
+x1)).(\lambda (_: (ty3 g (CHead c (Bind Abst) v) t x0)).(\lambda (_: (ty3 g
+(CHead c (Bind Abst) v) x0 x2)).(let H_y \def (ty3_conv g c v x1 H3 (THead
+(Bind Abst) v t) u H H0) in (let H_x \def (ty3_arity g c (THead (Bind Abst) v
+t) v H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c
+(THead (Bind Abst) v t) a1)) (\lambda (a1: A).(arity g c v (asucc g a1))) P
+(\lambda (x: A).(\lambda (H7: (arity g c (THead (Bind Abst) v t) x)).(\lambda
+(H8: (arity g c v (asucc g x))).(let H9 \def (arity_gen_abst g c v t x H7) in
+(ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A x (AHead a1 a2))))
+(\lambda (a1: A).(\lambda (_: A).(arity g c v (asucc g a1)))) (\lambda (_:
+A).(\lambda (a2: A).(arity g (CHead c (Bind Abst) v) t a2))) P (\lambda (x3:
+A).(\lambda (x4: A).(\lambda (H10: (eq A x (AHead x3 x4))).(\lambda (H11:
+(arity g c v (asucc g x3))).(\lambda (_: (arity g (CHead c (Bind Abst) v) t
+x4)).(let H13 \def (eq_ind A x (\lambda (a: A).(arity g c v (asucc g a))) H8
+(AHead x3 x4) H10) in (leq_ahead_asucc_false g x3 (asucc g x4) (arity_mono g
+c v (asucc g (AHead x3 x4)) H13 (asucc g x3) H11) P))))))) H9)))))
+H6))))))))))) (ty3_gen_bind g Abst c v t u H1)))))))))).
+
+theorem ty3_acyclic:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to ((pc3 c u t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(\lambda (H0: (pc3 c u t)).(\lambda (P: Prop).(let H_y \def
+(ty3_conv g c t u H t u H H0) in (let H_x \def (ty3_arity g c t t H_y) in
+(let H1 \def H_x in (ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda
+(a1: A).(arity g c t (asucc g a1))) P (\lambda (x: A).(\lambda (H2: (arity g
+c t x)).(\lambda (H3: (arity g c t (asucc g x))).(leq_asucc_false g x
+(arity_mono g c t (asucc g x) H3 x H2) P)))) H1)))))))))).
+
+theorem ty3_sn3:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
+u) \to (sn3 c t)))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
+(ty3 g c t u)).(let H_x \def (ty3_arity g c t u H) in (let H0 \def H_x in
+(ex2_ind A (\lambda (a1: A).(arity g c t a1)) (\lambda (a1: A).(arity g c u
+(asucc g a1))) (sn3 c t) (\lambda (x: A).(\lambda (H1: (arity g c t
+x)).(\lambda (_: (arity g c u (asucc g x))).(sc3_sn3 g x c t (sc3_arity g c t
+x H1))))) H0))))))).
+
+theorem pc3_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (pc3 c
+u1 u2) ((pc3 c u1 u2) \to (\forall (P: Prop).P))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(let H_y \def (ty3_sn3 g c u1 t1 H) in (let H_y0 \def (ty3_sn3 g c u2
+t2 H0) in (let H_x \def (nf2_sn3 c u1 H_y) in (let H1 \def H_x in (ex2_ind T
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (pc3 c u1 u2)
+((pc3 c u1 u2) \to (\forall (P: Prop).P))) (\lambda (x: T).(\lambda (H2: (pr3
+c u1 x)).(\lambda (H3: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H_y0) in (let
+H4 \def H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2
+c u)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to (\forall (P: Prop).P))) (\lambda
+(x0: T).(\lambda (H5: (pr3 c u2 x0)).(\lambda (H6: (nf2 c x0)).(let H_x1 \def
+(term_dec x x0) in (let H7 \def H_x1 in (or_ind (eq T x x0) ((eq T x x0) \to
+(\forall (P: Prop).P)) (or (pc3 c u1 u2) ((pc3 c u1 u2) \to (\forall (P:
+Prop).P))) (\lambda (H8: (eq T x x0)).(let H9 \def (eq_ind_r T x0 (\lambda
+(t: T).(nf2 c t)) H6 x H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t:
+T).(pr3 c u2 t)) H5 x H8) in (or_introl (pc3 c u1 u2) ((pc3 c u1 u2) \to
+(\forall (P: Prop).P)) (pc3_pr3_t c u1 x H2 u2 H10))))) (\lambda (H8: (((eq T
+x x0) \to (\forall (P: Prop).P)))).(or_intror (pc3 c u1 u2) ((pc3 c u1 u2)
+\to (\forall (P: Prop).P)) (\lambda (H9: (pc3 c u1 u2)).(\lambda (P:
+Prop).(let H10 \def H9 in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda
+(t: T).(pr3 c u2 t)) P (\lambda (x1: T).(\lambda (H11: (pr3 c u1
+x1)).(\lambda (H12: (pr3 c u2 x1)).(let H_x2 \def (pr3_confluence c u2 x0 H5
+x1 H12) in (let H13 \def H_x2 in (ex2_ind T (\lambda (t: T).(pr3 c x0 t))
+(\lambda (t: T).(pr3 c x1 t)) P (\lambda (x2: T).(\lambda (H14: (pr3 c x0
+x2)).(\lambda (H15: (pr3 c x1 x2)).(let H_y1 \def (nf2_pr3_unfold c x0 x2 H14
+H6) in (let H16 \def (eq_ind_r T x2 (\lambda (t: T).(pr3 c x1 t)) H15 x0
+H_y1) in (let H17 \def (nf2_pr3_confluence c x H3 x0 H6 u1 H2) in (H8 (H17
+(pr3_t x1 u1 c H11 x0 H16)) P))))))) H13)))))) H10)))))) H7)))))) H4))))))
+H1)))))))))))).
+
+theorem pc3_abst_dec:
+ \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
+u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to (or (ex4_2
+T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to (\forall (P: Prop).P)))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
+(H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
+u2 t2)).(let H1 \def (ty3_sn3 g c u1 t1 H) in (let H2 \def (ty3_sn3 g c u2 t2
+H0) in (let H_x \def (nf2_sn3 c u1 H1) in (let H3 \def H_x in (ex2_ind T
+(\lambda (u: T).(pr3 c u1 u)) (\lambda (u: T).(nf2 c u)) (or (ex4_2 T T
+(\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst) u2 u))))
+(\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u) t1)))
+(\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda
+(v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind Abst) u2 u))
+\to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (pr3 c u1
+x)).(\lambda (H5: (nf2 c x)).(let H_x0 \def (nf2_sn3 c u2 H2) in (let H6 \def
+H_x0 in (ex2_ind T (\lambda (u: T).(pr3 c u2 u)) (\lambda (u: T).(nf2 c u))
+(or (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst)
+u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u)
+t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_:
+T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind
+Abst) u2 u)) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H7: (pr3
+c u2 x0)).(\lambda (H8: (nf2 c x0)).(let H_x1 \def (abst_dec x x0) in (let H9
+\def H_x1 in (or_ind (ex T (\lambda (t: T).(eq T x (THead (Bind Abst) x0
+t)))) (\forall (t: T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P:
+Prop).P))) (or (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead
+(Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind
+Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda
+(_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1 (THead (Bind
+Abst) u2 u)) \to (\forall (P: Prop).P)))) (\lambda (H10: (ex T (\lambda (t:
+T).(eq T x (THead (Bind Abst) x0 t))))).(ex_ind T (\lambda (t: T).(eq T x
+(THead (Bind Abst) x0 t))) (or (ex4_2 T T (\lambda (u: T).(\lambda (_:
+T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2:
+T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2:
+T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall
+(u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P))))
+(\lambda (x1: T).(\lambda (H11: (eq T x (THead (Bind Abst) x0 x1))).(let H12
+\def (eq_ind T x (\lambda (t: T).(nf2 c t)) H5 (THead (Bind Abst) x0 x1) H11)
+in (let H13 \def (eq_ind T x (\lambda (t: T).(pr3 c u1 t)) H4 (THead (Bind
+Abst) x0 x1) H11) in (or_introl (ex4_2 T T (\lambda (u: T).(\lambda (_:
+T).(pc3 c u1 (THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2:
+T).(ty3 g c (THead (Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2:
+T).(pr3 c u2 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall
+(u: T).((pc3 c u1 (THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P)))
+(ex4_2_intro T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1 (THead (Bind Abst)
+u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead (Bind Abst) v2 u)
+t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2))) (\lambda (_:
+T).(\lambda (v2: T).(nf2 c v2))) x1 x0 (pc3_pr3_t c u1 (THead (Bind Abst) x0
+x1) H13 (THead (Bind Abst) u2 x1) (pr3_head_12 c u2 x0 H7 (Bind Abst) x1 x1
+(pr3_refl (CHead c (Bind Abst) x0) x1))) (ty3_sred_pr3 c u1 (THead (Bind
+Abst) x0 x1) H13 g t1 H) H7 H8)))))) H10)) (\lambda (H10: ((\forall (t:
+T).((eq T x (THead (Bind Abst) x0 t)) \to (\forall (P:
+Prop).P))))).(or_intror (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c u1
+(THead (Bind Abst) u2 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c (THead
+(Bind Abst) v2 u) t1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c u2 v2)))
+(\lambda (_: T).(\lambda (v2: T).(nf2 c v2)))) (\forall (u: T).((pc3 c u1
+(THead (Bind Abst) u2 u)) \to (\forall (P: Prop).P))) (\lambda (u:
+T).(\lambda (H11: (pc3 c u1 (THead (Bind Abst) u2 u))).(\lambda (P:
+Prop).(let H12 \def H11 in (ex2_ind T (\lambda (t: T).(pr3 c u1 t)) (\lambda
+(t: T).(pr3 c (THead (Bind Abst) u2 u) t)) P (\lambda (x1: T).(\lambda (H13:
+(pr3 c u1 x1)).(\lambda (H14: (pr3 c (THead (Bind Abst) u2 u) x1)).(ex2_ind T
+(\lambda (t: T).(pr3 c x1 t)) (\lambda (t: T).(pr3 c x t)) P (\lambda (x2:
+T).(\lambda (H15: (pr3 c x1 x2)).(\lambda (H16: (pr3 c x x2)).(let H_y \def
+(nf2_pr3_unfold c x x2 H16 H5) in (let H17 \def (eq_ind_r T x2 (\lambda (t:
+T).(pr3 c x1 t)) H15 x H_y) in (let H18 \def (pr3_gen_abst c u2 u x1 H14) in
+(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T x1 (THead (Bind Abst)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c u2 u3))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
+u0) u t3))))) P (\lambda (x3: T).(\lambda (x4: T).(\lambda (H19: (eq T x1
+(THead (Bind Abst) x3 x4))).(\lambda (H20: (pr3 c u2 x3)).(\lambda (_:
+((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) u x4))))).(let
+H22 \def (eq_ind T x1 (\lambda (t: T).(pr3 c t x)) H17 (THead (Bind Abst) x3
+x4) H19) in (let H23 \def (pr3_gen_abst c x3 x4 x H22) in (ex3_2_ind T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c x3 u3))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) x4
+t3))))) P (\lambda (x5: T).(\lambda (x6: T).(\lambda (H24: (eq T x (THead
+(Bind Abst) x5 x6))).(\lambda (H25: (pr3 c x3 x5)).(\lambda (_: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x4 x6))))).(let H27 \def (eq_ind
+T x (\lambda (t: T).(\forall (t0: T).((eq T t (THead (Bind Abst) x0 t0)) \to
+(\forall (P: Prop).P)))) H10 (THead (Bind Abst) x5 x6) H24) in (let H28 \def
+(eq_ind T x (\lambda (t: T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in
+(let H29 \def (nf2_gen_abst c x5 x6 H28) in (and_ind (nf2 c x5) (nf2 (CHead c
+(Bind Abst) x5) x6) P (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c
+(Bind Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7)
+in (H27 x6 (sym_equal T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6)
+(f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K
+(Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) P))))
+H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x H4)))))
+H12))))))) H9)))))) H6)))))) H3)))))))))))).
+
+theorem ty3_inference:
+ \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2:
+T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to (\forall (P:
+Prop).P))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0:
+C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2:
+T).((ty3 g c0 t t2) \to (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda
+(t2: T).(match t2 return (\lambda (t: T).(((\forall (c1: C).(\forall (t1:
+T).((flt c1 t1 c2 t) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2)))
+(\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or
+(ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3)
+\to (\forall (P: Prop).P)))))) with [(TSort n) \Rightarrow (\lambda (_:
+((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (TSort n)) \to (or (ex T
+(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to
+(\forall (P: Prop).P))))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2
+(TSort n) t3))) (\forall (t3: T).((ty3 g c2 (TSort n) t3) \to (\forall (P:
+Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next
+g n)) (ty3_sort g c2 n)))) | (TLRef n) \Rightarrow (\lambda (H: ((\forall
+(c1: C).(\forall (t1: T).((flt c1 t1 c2 (TLRef n)) \to (or (ex T (\lambda
+(t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall
+(P: Prop).P))))))))).(let H_x \def (getl_dec c2 n) in (let H0 \def H_x in
+(or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n
+c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) \to (\forall (P:
+Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3:
+T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3
+C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e
+(Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda
+(v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g
+c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P:
+Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2:
+(getl n c2 (CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0
+x2 n H2)) in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3:
+T).((ty3 g x0 x2 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3:
+T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to
+(\forall (P: Prop).P)))) (\lambda (H4: (ex T (\lambda (t2: T).(ty3 g x0 x2
+t2)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 t3)) (or (ex T (\lambda (t3:
+T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to
+(\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2
+x)).((match x1 return (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to
+(or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g
+c2 (TLRef n) t3) \to (\forall (P: Prop).P)))))) with [Abbr \Rightarrow
+(\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) x2))).(or_introl (ex T
+(\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef
+n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2
+(TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x H5)))) | Abst
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2
+(TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g
+c2 (TLRef n) t3)) (lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) | Void
+\Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Void) x2))).(or_intror
+(ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2
+(TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3
+g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t))))) P (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda
+(t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3
+c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abbr)
+x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind
+Void) x2) (\lambda (c: C).(getl n c2 c)) H6 (CHead x3 (Bind Abbr) x4)
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10))
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
+ee return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _
+k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _)
+\Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0
+(Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind P
+H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3
+c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abst)
+x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind
+Void) x2) (\lambda (c: C).(getl n c2 c)) H6 (CHead x3 (Bind Abst) x4)
+(getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10))
+in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match
+ee return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _
+k _) \Rightarrow (match k return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _)
+\Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0
+(Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind P
+H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))]) H2))) H4)) (\lambda
+(H4: ((\forall (t2: T).((ty3 g x0 x2 t2) \to (\forall (P:
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))
+(\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(\lambda (P:
+Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda
+(H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2
+(lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3:
+C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x5)
+t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) x4))).(\lambda (H9: (ty3
+g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c:
+C).(getl n c2 c)) H2 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind
+x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (let H11 \def (f_equal C C
+(\lambda (e: C).(match e return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow x0 | (CHead c _ _) \Rightarrow c])) (CHead x0 (Bind x1) x2)
+(CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead
+x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match
+e return (\lambda (_: C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _)
+\Rightarrow (match k return (\lambda (_: K).B) with [(Bind b) \Rightarrow b |
+(Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr)
+x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8))
+in ((let H13 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) (CHead
+x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1)
+x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda (_: (eq B x1
+Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t:
+T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17 \def
+(eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def
+(eq_ind_r C x3 (\lambda (c: C).(getl n c2 (CHead c (Bind Abbr) x2))) H16 x0
+H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0
+H15) in (H4 x5 H19 P)))))))) H12)) H11))))))))) H6)) (\lambda (H6: (ex3_3 C T
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u)
+t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u
+t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3
+c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_:
+T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4:
+T).(\lambda (x5: T).(\lambda (H7: (pc3 c2 (lift (S n) O x4) t3)).(\lambda
+(H8: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (H9: (ty3 g x3 x4
+x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c: C).(getl n
+c2 c)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n
+H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e:
+C).(match e return (\lambda (_: C).C) with [(CSort _) \Rightarrow x0 | (CHead
+c _ _) \Rightarrow c])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4)
+(getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in
+((let H12 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k
+return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2
+(CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def
+(f_equal C T (\lambda (e: C).(match e return (\lambda (_: C).T) with [(CSort
+_) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2)
+(CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead
+x3 (Bind Abst) x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C
+x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3
+(Bind Abst) t))) H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t:
+T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t:
+T).(pc3 c2 (lift (S n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3
+(\lambda (c: C).(getl n c2 (CHead c (Bind Abst) x2))) H16 x0 H15) in (let H20
+\def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 H15) in (H4 x5
+H20 P))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5)))))))
+H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P:
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)))
+(\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))
+(\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(\lambda (P:
+Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t:
+T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda
+(_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u:
+T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda
+(u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u)))))
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda
+(H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2
+(lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl
+n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_:
+T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift (S n) O x2)
+t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (ty3 g
+x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 P))))))) H3)) (\lambda (H3:
+(ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S
+n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2
+(CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t:
+T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u:
+T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda
+(u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e:
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift (S n) O x1)
+t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g
+x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 P))))))) H3)) (ty3_gen_lref g c2
+t3 n H2))))))) H0)))) | (THead k t t0) \Rightarrow (\lambda (H: ((\forall
+(c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k t t0)) \to (or (ex T
+(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to
+(\forall (P: Prop).P))))))))).((match k return (\lambda (k0: K).(((\forall
+(c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k0 t t0)) \to (or (ex T
+(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to
+(\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead
+k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead k0 t t0) t3) \to (\forall
+(P: Prop).P)))))) with [(Bind b) \Rightarrow (\lambda (H0: ((\forall (c1:
+C).(\forall (t1: T).((flt c1 t1 c2 (THead (Bind b) t t0)) \to (or (ex T
+(\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to
+(\forall (P: Prop).P))))))))).(let H1 \def (H0 c2 t (flt_thead_sx (Bind b) c2
+t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3:
+T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead
+(Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H2: (ex T (\lambda
+(t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda
+(x: T).(\lambda (H3: (ty3 g c2 t x)).(let H4 \def (H0 (CHead c2 (Bind b) t)
+t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g
+(CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 g (CHead c2 (Bind b) t)
+t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0)
+t3) \to (\forall (P: Prop).P)))) (\lambda (H5: (ex T (\lambda (t2: T).(ty3 g
+(CHead c2 (Bind b) t) t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2
+(Bind b) t) t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall
+(P: Prop).P)))) (\lambda (x0: T).(\lambda (H6: (ty3 g (CHead c2 (Bind b) t)
+t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) x0 t3)) (or
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda
+(x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) x0 x1)).(or_introl (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3
+g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T
+(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0)
+(ty3_bind g c2 t x H3 b t0 x0 H6 x1 H7))))) (ty3_correct g (CHead c2 (Bind b)
+t) t0 x0 H6)))) H5)) (\lambda (H5: ((\forall (t2: T).((ty3 g (CHead c2 (Bind
+b) t) t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead
+(Bind b) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6:
+(ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: Prop).(ex4_3_ind T T T
+(\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t
+t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t
+t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2
+(Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3
+g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda
+(_: (ty3 g c2 t x1)).(\lambda (H9: (ty3 g (CHead c2 (Bind b) t) t0
+x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H5 x0 H9 P))))))))
+(ty3_gen_bind g b c2 t t0 t3 H6))))))) H4)))) H2)) (\lambda (H2: ((\forall
+(t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3
+g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3:
+T).(\lambda (H3: (ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P:
+Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3
+c2 (THead (Bind b) t t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_:
+T).(ty3 g c2 t t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
+(CHead c2 (Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda
+(t6: T).(ty3 g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0)
+t3)).(\lambda (H5: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t)
+t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P))))))))
+(ty3_gen_bind g b c2 t t0 t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda
+(H0: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f) t t0))
+\to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (P: Prop).P))))))))).((match f return (\lambda (f0:
+F).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f0) t t0))
+\to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g
+c2 (THead (Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0)
+t t0) t3) \to (\forall (P: Prop).P)))))) with [Appl \Rightarrow (\lambda (H1:
+((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat Appl) t t0))
+\to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1
+t1 t2) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx
+(Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3)))
+(\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))))
+(\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda
+(t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat
+Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3)
+\to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t
+x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex
+T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to
+(\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat
+Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3)
+\to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t2: T).(ty3 g c2
+t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0:
+T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P:
+Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(ex_ind T
+(\lambda (t3: T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead
+(Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0)
+t3) \to (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x
+x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10)
+in (let H11 \def H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u:
+T).(nf2 c2 u)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0)
+t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall
+(P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13:
+(nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0
+\def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \def H_x0 in (or_ind
+(ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3
+u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u)
+x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_:
+T).(\lambda (v2: T).(nf2 c2 v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind
+Abst) x3 u)) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl)
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H16: (ex4_2 T T (\lambda (u:
+T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u:
+T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_:
+T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2
+v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead
+(Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind
+Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda
+(_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl)
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3
+g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda
+(_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21
+\def (eq_ind_r T x5 (\lambda (t: T).(pr3 c2 x3 t)) H19 x3 H_y) in (let H22
+\def (eq_ind_r T x5 (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) t x4) x1))
+H18 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl)
+t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
+(\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat
+Appl) t t0) t3)) (THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g
+c2 t x3 (ty3_tred g c2 t x H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst)
+x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u:
+T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P:
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead
+(Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u:
+T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4))
+t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u
+t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x4:
+T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind
+Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4
+x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20
+x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0
+H7) in (H16 x5 (pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead
+(Bind Abst) x4 x5) H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3
+(pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H12)) (Bind Abst) x5)) P))))))))
+(ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2
+t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t2:
+T).((ty3 g c2 t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3:
+T).(\lambda (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P:
+Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat
+Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3
+g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2
+t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat
+Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead
+(Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0
+x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3:
+((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror
+(ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))
+(\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Appl) t t0)
+t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3
+c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u:
+T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u:
+T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1))
+t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (H7: (ty3
+g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) |
+Cast \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1
+c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1
+t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P:
+Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in
+(or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2
+t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead
+(Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0)
+t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g
+c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x:
+T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat
+Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall
+(t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3:
+T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T
+(\lambda (t2: T).(ty3 g c2 t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0
+t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)))
+(\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P:
+Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T
+(\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast)
+t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g
+c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x
+in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (or (ex T
+(\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3:
+T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))))
+(\lambda (H10: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast)
+t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2
+(THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0
+x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P:
+Prop).P)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead
+(Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0
+t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y
+\def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P))))
+(ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7))))
+H6)) (\lambda (H6: ((\forall (t2: T).((ty3 g c2 t0 t2) \to (\forall (P:
+Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t
+t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to
+(\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat
+Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P
+(\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P)))
+(ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t2:
+T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda
+(t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2
+(THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3:
+T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P:
+Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t
+t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t
+t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P)))
+(ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))])
+H0))]) H))]))) c t1))).
projects / helm.git / blobdiff