X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta.ma;h=2c90a295fb0c7842b583e7e066da8a5afe169ad7;hb=2aad3e4b468d3e4199712437e7ef82afbbc9553d;hp=388b054b9e7b2ec0a976d683ed71944da0533c09;hpb=2f5cea9058c4f7c2b323e0a80fd491f69a35c2d8;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma index 388b054b9..2c90a295f 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma @@ -16,1373 +16,52202 @@ 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 +(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t))) +(\lambda (_: (eq T u2 u2)).(or4_intro1 (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 (THead k u2 t) (THead k u1 +x)) (subst0 i0 v (THead k u1 x) (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 u1 +x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 +H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) (\lambda +(t: T).(subst0 i0 v u2 t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v u2 t3)) +(\lambda (t3: T).(subst0 i0 v u2 t3)) (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 (THead k u2 t) (THead k u1 +x)) (subst0 i0 v (THead k u1 x) (THead k u2 t))) (\lambda (x0: T).(\lambda +(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (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 +(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (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 u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 +u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H6)) (\lambda (_: (subst0 i0 v u2 +u2)).(or4_intro1 (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))) 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t3))) (\lambda (x0: T).(\lambda +(_: (subst0 i0 v u2 x0)).(\lambda (_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T +(THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k +u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v +(THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (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 u1 x) t)) (THead k u2 x) (subst0_snd k v x t3 i0 H8 +u2) (subst0_fst v u2 u1 i0 H0 x k)))))) H9)) (\lambda (_: (subst0 i0 v u2 +u2)).(or4_intro1 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 +x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 +x) (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 u1 x) t)) (THead k u2 x) +(subst0_snd k v x t3 i0 H8 u2) (subst0_fst v u2 u1 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t3)) (subst0_both v +u1 u2 i0 H0 k x t3 H8))))) H9)) (\lambda (_: (subst0 i0 v u2 u2)).(or4_intro3 +(eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v +(THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 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))) (\lambda (_: (subst0 i0 v u2 +u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: +T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 +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)) 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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 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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 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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 (_: 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(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) +(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 +(CHead c0 k t) (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 (CHead c0 k t) (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 (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 H19 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H16 e H2) in +(or4_ind (drop n0 O c0 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 n0 O c0 (CHead e0 k +w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: +T).(subst0 (minus i0 (s k 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 n0 O c0 (CHead e2 k u0)))))) (\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 (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 +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 n0)) v u0 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) 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 (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 (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 (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))))))) (\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: 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(\lambda (n: nat).(subst0 (minus +(s (Bind b) i0) n) v x2 x3)) H23 (s x0 (S n0)) (s_S x0 n0)))) 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 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 (u0: T).(eq C e (CHead e1 k +u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: +T).(drop n0 O c0 (CHead e2 k u0)))))) (\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) 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: 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(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 (_: +T).(\lambda (w: T).(drop 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 n0)) v u0 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) 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 (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 +(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 (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))))))) +(\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 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(Bind b) i0) n) v x1 x2)) H24 (s x0 (S +n0)) (s_S x0 n0)))) e H21)))))))))) H20)) H19)))))) (\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 H19 \def (H0 c0 v H16 e H2) in (or4_ind (drop (S n0) O c0 +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 c0 (CHead e0 k w)))))) (\lambda (k: +K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus 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 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)))))) (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 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 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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))))))) (drop_drop (Flat f) n0 c0 +e H20 t))) (\lambda (H20: (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 (u0: T).(\lambda (_: T).(eq C e 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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: 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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 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(_: 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 (_: 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(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 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+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 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(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 (_: 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+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 +(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead +(Bind b) u t) t2) (csubst0 i u0 c c2))).(and_ind (eq T (THead (Bind b) u t) +t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind +b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u +t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 +i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) 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 c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c +(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u)))) +t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2) +(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 +i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead (Bind b) u t) +t2)).(\lambda (H10: (csubst0 i u0 c c2)).(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 c2 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 c2 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 c2 t0 a2)) (arity_bind g b H0 c2 x a1 (H2 d1 u0 i +H5 c2 x (fsubst0_both i u0 c u x H13 c2 H10)) 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 c2 (Bind b) x) t (fsubst0_fst (S i) u0 (CHead c +(Bind b) u) t (CHead c2 (Bind b) x) (csubst0_both_bind b i u0 u x H13 c c2 +H10)))) 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 c2 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 c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 i H5 c2 u (fsubst0_fst i u0 +c u c2 H10)) 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 c2 (Bind b) u) x +(fsubst0_both (S i) u0 (CHead c (Bind b) u) t x H13 (CHead c2 (Bind b) u) +(csubst0_fst_bind b i c c2 u0 H10 u)))) 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 c2 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 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) +x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 H12)))))) H11)) +(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 +(s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 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 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: +(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(and_ind (eq T +(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2)) +(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0 +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) +(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u +t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(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 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))) +(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 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 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))) +(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c2 t2 (AHead a1 +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)))) +(\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 c2 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 c2 t0 (AHead a1 a2))) (arity_head 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 (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: +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 (H2: (arity g c t (AHead a1 a2))).(\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 (AHead a1 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 +(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 +c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)) +(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) +t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 +(THead (Flat Appl) 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 (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 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 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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: 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+(\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: 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+(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 +w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 +O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def +(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in +(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w +x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x +H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) +(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 +H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: +T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 +w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: +(subst0 (s (Bind Abbr) 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 (Bind +Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead +(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 +w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 +x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x +x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 +(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead +(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) +(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd +(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind +Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: +(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 (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 +x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O +x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 +x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal +nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 +\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S +i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) +(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead +(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) +w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda +(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: +(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 +H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 +(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) +(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S +i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i +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)))))) 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+(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 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(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 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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 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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 (_: 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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 (_: 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(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 (_: 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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 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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 (_: 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(\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))) 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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: 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(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: 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(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 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(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 (x3: +T).(\lambda (H17: (eq T x (THead (Bind Abbr) x0 x3))).(\lambda (H18: (subst0 +(s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t1: T).(subst0 O u1 x2 t1)) +(\lambda (t1: T).(pr0 t1 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 +(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 (x4: +T).(\lambda (H19: (subst0 O u1 x2 x4)).(\lambda (H20: (pr0 x4 x1)).(or_introl +(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)))) (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))))))) x0 x3 H17 (pr2_free c u1 x0 H12) (or3_intro2 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 x3))) (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 x3)))) (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 x3))) x4 x1 (pr2_delta (CHead c (Bind +Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H13 x4 H19) H20 (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) x3 H18)))))))) (pr0_subst0_back x0 x2 x1 +O H14 u1 H12))))) 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 x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind +Abbr) i) u x1 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 +x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 +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 (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 (x3: T).(\lambda (x4: T).(\lambda (H17: (eq T x +(THead (Bind Abbr) x3 x4))).(\lambda (H18: (subst0 i u x0 x3)).(\lambda (H19: +(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t1: T).(subst0 O u1 +x2 t1)) (\lambda (t1: T).(pr0 t1 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 (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 (x5: T).(\lambda (H20: (subst0 O u1 x2 x5)).(\lambda (H21: (pr0 x5 +x1)).(or_introl (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)))) (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 (_: 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(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: 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(\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 (_: 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(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)))). - -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))))))) . - -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)))))))) . - -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))))))) . - -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)))))))))) . - -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))))))) . - -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))))))))) . - -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))))))))))) . - -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))))))) . - -inductive A: Set \def -| ASort: nat \to (nat \to A) -| AHead: A \to (A \to A). - -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. - -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. - -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))))))). - -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)))))))))) . - -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)))))))) . - -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))))))))) . - -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)))))))) . - -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))))))))) . - -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)))))) . - -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))))) . - -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)))) . - -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))))) . - -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))))) . - -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)))) . - -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))))) . - -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)))))) . - -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))))))) . - -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)))))) . - -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)))) . - -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))))) . - -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)))))) . - -axiom leq_refl: \forall (g: G).(\forall (a: A).(leq g a a)) . - -axiom leq_eq: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 a2)))) . - -axiom leq_asucc: \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g a0))))) . - -axiom leq_sym: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g a2 a1)))) . - -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)))))) . - -axiom leq_ahead_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) \to (\forall (P: Prop).P)))) . - -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)))) . - -axiom leq_asucc_false: \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: Prop).P))) . - -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. - -definition llt: A \to (A \to Prop) \def \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))). - -axiom lweight_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat (lweight a1) (lweight a2))))) . - -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)))))) . - -axiom llt_trans: \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 a3) \to (llt a1 a3))))) . - -axiom llt_head_sx: \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) . - -axiom llt_head_dx: \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) . - -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))) . - -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))) . - -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))))). - -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))))))))) . - -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))))) . - -definition gz: G \def Build_G S lt_n_Sn. - -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))))))). - -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)))))) . - -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))))) . - -axiom next_plus_gz: \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) . - -axiom leqz_leq: \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) . - -axiom leq_leqz: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) . - -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)))))). - -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)))))) . - -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)))))))))) . - -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)))))))))) . - -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))))))))) . - -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))))))))) . - -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))))))) . - -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)))))))) . - -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))))))) . - -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)))))))))))) . - -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)))))))))))) . - -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))))))))) . - -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))))))))) . - -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)))))))) . - -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))))))) . - -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))))))) . - -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))))))))))))) . - -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)))))))) . - -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))))))))) . - -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))))))))))) . - -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)))))))))))) . - -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))))))))))) . - -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)))))) . - -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))))))) . - -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)))))) . - -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))))))))) . - -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))))))))))))) . - -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))))))))) . - -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))))))))) . - -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)))))))))) . - -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)))))))) . - -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)))))))))))))) . - -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)))))))))))))))))))) . - -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)))))))) . - -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)))))))))) . - -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)))))))))))) . - -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)))))))))))))))) . - -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))))). - -axiom pr3_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to (eq T x (TSort n))))) . - -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)))))))))) . - -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)))))) . - -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)))))) . - -axiom pr3_pr2: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c t1 t2)))) . - -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)))))) . - -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))))))) . - -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))))))) . - -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))))))) . - -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))))))))) . - -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))))))))) . - -axiom pr3_pr1: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 t2)))) . - -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)))))) . - -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)))))))) . - -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)))))))) . - -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)))))))) . - -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)))))))) . - -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))))))))) . - -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)))))) . - -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)))))))))) . - -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)))))))))) . - -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))))))) . - -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))))))) . - -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))))))))))))) . - -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)))))))) . - -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)))))))) . - -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)))))))))))) . - -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)))))))))))))))) . - -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)))))))) . - -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))))))))). - -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))))))) . - -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))))))) . - -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)))))))))) . - -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))))))))) . - -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)))))))))) . - -axiom csuba_refl: \forall (g: G).(\forall (c: C).(csuba g c c)) . - -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)))))))) . - -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: +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) x0 x1) (THead (Flat Appl) w (THead +(Bind Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (_: (pc3 c (THead (Bind Abst) u x2) x)).(\lambda (_: (ty3 g c u +x3)).(\lambda (H14: (ty3 g (CHead c (Bind Abst) u) t0 x2)).(\lambda (H15: +(ty3 g (CHead c (Bind Abst) u) x2 x4)).(ex_ind T (\lambda (t4: T).(ty3 g c u +t4)) (ty3 g c (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead (Bind +Abst) u t0))) (\lambda (x5: T).(\lambda (H16: (ty3 g c u x5)).(ty3_conv g c +(THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead +(Bind Abst) u x2)) (ty3_appl g c w u H0 (THead (Bind Abst) u t0) x2 (ty3_bind +g c u x5 H16 Abst t0 x2 H14 x4 H15)) (THead (Flat Appl) x0 x1) (THead (Flat +Appl) x0 (THead (Bind Abst) u t0)) (ty3_appl g c x0 u (H1 i u0 c x0 +(fsubst0_snd i u0 c w x0 H9) e H6) x1 t0 (H3 (s (Flat Appl) i) u0 c x1 +(fsubst0_snd (s (Flat Appl) i) u0 c v x1 H10) e H6)) (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) x0 (THead (Bind Abst) u t0)) (fsubst0_snd i u0 c (THead +(Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) x0 (THead (Bind +Abst) u t0)) (subst0_fst u0 x0 w i H9 (THead (Bind Abst) u t0) (Flat Appl))) +e H6)))) (ty3_correct g c w u H0))))))))) (ty3_gen_bind g Abst c u t0 x +H11)))) (ty3_correct g c v (THead (Bind Abst) u t0) H2)) t3 H8)))))) H7)) +(subst0_gen_head (Flat Appl) u0 w v t3 i H5)))))) (\lambda (c3: C).(\lambda +(H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e +(Bind Abbr) u0))).(ty3_appl g c3 w u (H1 i u0 c3 w (fsubst0_fst i u0 c w c3 +H5) e H6) v t0 (H3 i u0 c3 v (fsubst0_fst i u0 c v c3 H5) e H6)))))) (\lambda +(t3: T).(\lambda (H5: (subst0 i u0 (THead (Flat Appl) w v) t3)).(\lambda (c3: +C).(\lambda (H6: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H7: (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 c3 t3 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H8: +(ex2 T (\lambda (u2: T).(eq T t3 (THead (Flat Appl) u2 v))) (\lambda (u2: +T).(subst0 i u0 w u2)))).(ex2_ind T 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H6) e H7)) t3 H9)))) H8)) (\lambda (H8: (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 c3 t3 (THead (Flat Appl) w +(THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: +(eq T t3 (THead (Flat Appl) x0 x1))).(\lambda (H10: (subst0 i u0 w +x0)).(\lambda (H11: (subst0 (s (Flat Appl) i) u0 v x1)).(eq_ind_r T (THead +(Flat Appl) x0 x1) (\lambda (t4: T).(ty3 g c3 t4 (THead (Flat Appl) w (THead +(Bind Abst) u t0)))) (ex_ind T (\lambda (t4: T).(ty3 g c3 (THead (Bind Abst) +u t0) t4)) (ty3 g c3 (THead (Flat Appl) x0 x1) (THead (Flat Appl) w (THead +(Bind Abst) u t0))) (\lambda (x: T).(\lambda 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(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c x0 (fsubst0_snd i u +c t3 x0 H9) e H6)) t5 H8)))))) H7)) (subst0_gen_head (Flat Cast) u t3 t2 t5 i +H5)))))) (\lambda (c3: C).(\lambda (H5: (csubst0 i u c c3)).(\lambda (e: +C).(\lambda (H6: (getl i c (CHead e (Bind Abbr) u))).(ty3_cast g c3 t2 t3 (H1 +i u c3 t2 (fsubst0_fst i u c t2 c3 H5) e H6) t0 (H3 i u c3 t3 (fsubst0_fst i +u c t3 c3 H5) e H6)))))) (\lambda (t5: T).(\lambda (H5: (subst0 i u (THead +(Flat Cast) t3 t2) t5)).(\lambda (c3: C).(\lambda (H6: (csubst0 i u c +c3)).(\lambda (e: C).(\lambda (H7: (getl i c (CHead e (Bind Abbr) +u))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) +(\lambda (u2: T).(subst0 i u t3 u2))) (ex2 T (\lambda (t6: T).(eq T t5 (THead +(Flat Cast) t3 t6))) (\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6))) +(ex3_2 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 (H8: (ex2 T (\lambda (u2: T).(eq T t5 (THead (Flat Cast) u2 t2))) +(\lambda (u2: T).(subst0 i u t3 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t5 +(THead (Flat Cast) u2 t2))) (\lambda (u2: T).(subst0 i u t3 u2)) (ty3 g c3 t5 +t3) (\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) x +t2))).(\lambda (H10: (subst0 i u t3 x)).(eq_ind_r T (THead (Flat Cast) x t2) +(\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) x t2) x (ty3_cast g c3 +t2 x (ty3_conv g c3 x t0 (H3 i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e +H7) t2 t3 (H1 i u c3 t2 (fsubst0_fst i u c t2 c3 H6) e H7) (pc3_s c3 t3 x +(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 +H6) e H7))) t0 (H3 i u c3 x (fsubst0_both i u c t3 x H10 c3 H6) e H7)) +(pc3_fsubst0 c t3 t3 (pc3_refl c t3) i u c3 x (fsubst0_both i u c t3 x H10 c3 +H6) e H7)) t5 H9)))) H8)) (\lambda (H8: (ex2 T (\lambda (t2: T).(eq T t5 +(THead (Flat Cast) t3 t2))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u t2 +t3)))).(ex2_ind T (\lambda (t6: T).(eq T t5 (THead (Flat Cast) t3 t6))) +(\lambda (t6: T).(subst0 (s (Flat Cast) i) u t2 t6)) (ty3 g c3 t5 t3) +(\lambda (x: T).(\lambda (H9: (eq T t5 (THead (Flat Cast) t3 x))).(\lambda +(H10: (subst0 (s (Flat Cast) i) u t2 x)).(eq_ind_r T (THead (Flat Cast) t3 x) +(\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))) 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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))).