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))) .
-
-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))))) .
-
-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))))))) .
-
-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)))) .
-
-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))))) .
-
-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))))))))) .
-
-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))))) .
-
-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))))).
-
-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))))))))) .
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-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))))))) .
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-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))))))))))))) .
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-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))))))))))) .
-
-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)))))))))).
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-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))))))) .
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-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))))))))))) .
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-axiom subst0_refl: \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to (\forall (P: Prop).P)))) .
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-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))))))))) .
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-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))))))))) .
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-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))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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)))))))))))) .
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-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)))))))) .
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-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)))))))))) .
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-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)))))))))) .
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-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))))))) .
-
-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)))))) .
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-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)))))))) .
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-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))))))))) .
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-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)))))))))) .
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-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)))))))))) .
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-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))))))))) .
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-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))))))))) .
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-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)))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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))))))) .
-
-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)))))))))))) .
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-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))))))))) .
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-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 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)))))))))).
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-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)))))))) .
-
-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)))))))) .
-
-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)))))))))) .
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-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))))) .
-
-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)))))))))))) .
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-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))))))))) .
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-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))))))))) .
-
-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)))))))))))))))) .
-
-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)))))))))))))) .
-
-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)))))))))))))) .
-
-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)))))) .
-
-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)))))) .
-
-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)))))))))))))) .
-
-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))))))))) .
-
-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)))))))))))))))) .
-
-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))))) .
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-axiom asucc_inj: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc g a2)) \to (leq g a1 a2)))) .
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-axiom aplus_asort_O_simpl: \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O n) h) (ASort O (next_plus g n h))))) .
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-axiom aplus_asort_le_simpl: \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) .
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-axiom aplus_asort_simpl: \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) .
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-axiom aplus_ahead_simpl: \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) .
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-axiom aplus_asucc_false: \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) h) a) \to (\forall (P: Prop).P)))) .
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-axiom aplus_inj: \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) .
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-axiom ahead_inj_snd: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) .
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-axiom leq_refl: \forall (g: G).(\forall (a: A).(leq g a a)) .
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-axiom leq_eq: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 a2)))) .
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-axiom leq_asucc: \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g a0))))) .
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-axiom leq_sym: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g a2 a1)))) .
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-axiom leq_trans: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) .
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-axiom leq_ahead_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) \to (\forall (P: Prop).P)))) .
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-axiom leq_ahead_asucc_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) (asucc g a1)) \to (\forall (P: Prop).P)))) .
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-axiom leq_asucc_false: \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: Prop).P))) .
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-definition lweight: A \to nat \def let rec lweight (a: A): nat \def (match a with [(ASort _ _) \Rightarrow O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in lweight.
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-definition llt: A \to (A \to Prop) \def \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))).
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-axiom lweight_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat (lweight a1) (lweight a2))))) .
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-axiom llt_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((llt a1 a3) \to (llt a2 a3)))))) .
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-axiom llt_trans: \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 a3) \to (llt a1 a3))))) .
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-axiom llt_head_sx: \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) .
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-axiom llt_head_dx: \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) .
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-axiom llt_wf__q_ind: \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P: ((A \to Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P a))))) P n))) \to (\forall (a: A).(P a))) .
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-axiom llt_wf_ind: \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) .
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-inductive aprem: nat \to (A \to (A \to Prop)) \def
-| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
-| aprem_succ: \forall (a2: A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to (\forall (a1: A).(aprem (S i) (AHead a1 a2) a))))).
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-axiom aprem_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) .
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-axiom aprem_asucc: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i a1 a2) \to (aprem i (asucc g a1) a2))))) .
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-definition gz: G \def Build_G S lt_n_Sn.
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-inductive leqz: A \to (A \to Prop) \def
-| leqz_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus h1 n2) (plus h2 n1)) \to (leqz (ASort h1 n1) (ASort h2 n2))))))
-| leqz_head: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leqz a3 a4) \to (leqz (AHead a1 a3) (AHead a2 a4))))))).
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-axiom aplus_gz_le: \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A (aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) .
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-axiom aplus_gz_ge: \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A (aplus gz (ASort h n) k) (ASort (minus h k) n))))) .
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-axiom next_plus_gz: \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) .
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-axiom leqz_leq: \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) .
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-axiom leq_leqz: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) .
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-inductive arity (g:G): C \to (T \to (A \to Prop)) \def
-| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O n)))
-| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to (arity g c (TLRef i) a)))))))
-| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u (asucc g a)) \to (arity g c (TLRef i) a)))))))
-| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c (THead (Bind b) u t) a2)))))))))
-| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
-| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to (arity g c (THead (Flat Appl) u t) a2)))))))
-| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat Cast) u t) a))))))
-| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).
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-axiom arity_gen_sort: \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c (TSort n) a) \to (leq g a (ASort O n)))))) .
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-axiom arity_gen_lref: \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c (TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))))) .
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-axiom arity_gen_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))))))))) .
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-axiom arity_gen_abst: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c (Bind Abst) u) t a2))))))))) .
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-axiom arity_gen_appl: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2))))))))) .
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-axiom arity_gen_cast: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a: A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a)) (arity g c t a))))))) .
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-axiom arity_gen_appls: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall (a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))))) .
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-axiom node_inh: \forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c: C).(\lambda (t: T).(arity g c t (ASort k n))))))) .
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-axiom arity_gen_cvoid_subst0: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Void) u)) \to (\forall (w: T).(\forall (v: T).((subst0 i w t v) \to (\forall (P: Prop).P)))))))))))) .
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-axiom arity_gen_cvoid: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Void) u)) \to (ex T (\lambda (v: T).(eq T t (lift (S O) i v)))))))))))) .
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-axiom arity_gen_lift: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h: nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t a))))))))) .
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-axiom arity_lift: \forall (g: G).(\forall (c2: C).(\forall (t: T).(\forall (a: A).((arity g c2 t a) \to (\forall (c1: C).(\forall (h: nat).(\forall (d: nat).((drop h d c1 c2) \to (arity g c1 (lift h d t) a))))))))) .
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-axiom arity_lift1: \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 hds t) a)))))))) .
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-axiom arity_mono: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2))))))) .
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-axiom arity_cimp_conf: \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a))))))) .
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-axiom arity_aprem: \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))))) .
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-axiom arity_appls_cast: \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) a)))))))) .
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-axiom arity_appls_abbr: \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (vs: TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs (lift (S i) O v)) a) \to (arity g c (THeads (Flat Appl) vs (TLRef i)) a))))))))) .
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-axiom arity_appls_bind: \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c v a1) \to (\forall (t: T).(\forall (vs: TList).(\forall (a2: A).((arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) a2) \to (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) a2))))))))))) .
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-axiom arity_fsubst0: \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g c1 t1 a) \to (\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u c1 t1 c2 t2) \to (arity g c2 t2 a)))))))))))) .
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-axiom arity_subst0: \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (a: A).((arity g c t1 a) \to (\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t2: T).((subst0 i u t1 t2) \to (arity g c t2 a))))))))))) .
-
-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)))) .
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-axiom pr0_gen_lref: \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) .
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-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))))))) .
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-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)))))))))))) .
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-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)))))) .
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-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)))))) .
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-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))))))) .
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-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))))))))) .
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-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))))))))) .
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-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)))))))))))) .
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-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))))))))))))))) .
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-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))))))))))))))))))) .
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-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)))))))))))))))))))) .
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-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))))))))))))))))) .
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-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)))))))))))))) .
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-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))))))))))))))))))) .
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-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)))))))))))))))) .
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-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))))))))) .
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-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))))))) .
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-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))))))))) .
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-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))))))))) .
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-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))))))))) .
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-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))))))))))) .
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-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)))) .
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-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))))).
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-axiom pr1_pr0: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) .
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-axiom pr1_t: \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 t3) \to (pr1 t1 t3))))) .
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-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)))))) .
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-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)))))) .
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-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))))))) .
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-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))))))) .
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-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)))))))).
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-axiom wcpr0_gen_sort: \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort n)))) .
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-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))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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)))))))))).
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-axiom pr2_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to (eq T x (TSort n))))) .
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-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))))))))) .
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-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)))))))))) .
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-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)))))) .
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-axiom pr2_gen_csort: \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) \to (pr0 t1 t2)))) .
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-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))))))))) .
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-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))))))) .
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-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))))))) .
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-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))))))) .
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-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)))))) .
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-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)))))) .
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-axiom pr2_ctail: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) .
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-axiom pr2_gen_cbind: \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))) .
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-axiom pr2_gen_cflat: \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) .
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-axiom pr2_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) .
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-axiom pr2_gen_appl: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) .
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-axiom pr2_gen_abbr: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) .
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-axiom pr2_gen_void: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) .
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-axiom pr2_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) .
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-axiom pr2_confluence__pr2_free_free: \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))) .
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-axiom pr2_confluence__pr2_free_delta: \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))))))))) .
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-axiom pr2_confluence__pr2_delta_delta: \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))))))))))))))) .
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-axiom pr2_confluence: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)))))))) .
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-axiom pr2_delta1: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) .
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-axiom pr2_subst1: \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) .
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-axiom pr2_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)))))))))))))))) .
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-inductive pr3 (c:C): T \to (T \to Prop) \def
-| pr3_refl: \forall (t: T).(pr3 c t t)
-| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3))))).
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-axiom pr3_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to (eq T x (TSort n))))) .
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-axiom pr3_gen_abst: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) .
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-axiom pr3_gen_cast: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c t1 x)))))) .
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-axiom clear_pr3_trans: \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to (\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2)))))) .
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-axiom pr3_pr2: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c t1 t2)))) .
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-axiom pr3_t: \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall (t3: T).((pr3 c t2 t3) \to (pr3 c t1 t3)))))) .
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-axiom pr3_thin_dx: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) .
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-axiom pr3_head_1: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t))))))) .
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-axiom pr3_head_2: \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u t2))))))) .
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-axiom pr3_head_21: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u1) t1 t2) \to (pr3 c (THead k u1 t1) (THead k u2 t2))))))))) .
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-axiom pr3_head_12: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (k: K).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c k u2) t1 t2) \to (pr3 c (THead k u1 t1) (THead k u2 t2))))))))) .
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-axiom pr3_pr1: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 t2)))) .
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-axiom pr3_cflat: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2)))))) .
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-axiom pr3_pr0_pr2_t: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_pr2_pr2_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_pr2_pr3_t: \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_pr3_pr3_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) .
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-axiom pr3_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift h d t1) (lift h d t2))))))))) .
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-axiom pr3_wcpr0_t: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) .
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-axiom pr3_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 t2)))))))))) .
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-axiom pr3_gen_lref: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to (or (eq T x (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T x (lift (S n) O v)))))))))) .
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-axiom pr3_gen_void: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))))))) .
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-axiom pr3_gen_abbr: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) .
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-axiom pr3_gen_appl: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) .
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-axiom pr3_strip: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)))))))) .
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-axiom pr3_confluence: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall (t2: T).((pr3 c t0 t2) \to (ex2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)))))))) .
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-axiom pr3_subst1: \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) .
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-axiom pr3_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))) .
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-axiom pr3_iso_appls_cast: \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs (THead (Flat Cast) v t)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs t) u2)))))))) .
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-inductive csuba (g:G): C \to (C \to Prop) \def
-| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
-| csuba_head: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (k: K).(\forall (u: T).(csuba g (CHead c1 k u) (CHead c2 k u))))))
-| csuba_abst: \forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (t: T).(\forall (a: A).((arity g c1 t (asucc g a)) \to (\forall (u: T).((arity g c2 u a) \to (csuba g (CHead c1 (Bind Abst) t) (CHead c2 (Bind Abbr) u))))))))).
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-axiom csuba_gen_abbr: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g (CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) .
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-axiom csuba_gen_void: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g (CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) .
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-axiom csuba_gen_abst: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g (CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) .
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-axiom csuba_gen_flat: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall (f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) .
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-axiom csuba_gen_bind: \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) .
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-axiom csuba_refl: \forall (g: G).(\forall (c: C).(csuba g c c)) .
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-axiom csuba_clear_conf: \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c2 e2)))))))) .
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-axiom csuba_drop_abbr: \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) .
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-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))))))))))))) .
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-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)))))))))) .
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-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))))))))))))) .
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-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))))))) .
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-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))))))) .
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-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))))))))))) .
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-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))))))))) .
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-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))))))))) .
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-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))))))) .
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-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))))))) .
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-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)))).
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-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)))) .
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-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)))))) .
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-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))))) .
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-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)))) .
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-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))))).
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-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)))) .
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-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)))))) .
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-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)))) .
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-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)))) .
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-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)))) .
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-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)))))))) .
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-axiom pc3_gen_sort: \forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort n)) \to (eq nat m n)))) .
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-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))))))))) .
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-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)))))))) .
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-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)))))))))) .
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-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))))))))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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)))))))))))))))) .
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-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)))))).
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-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))))) .
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-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)))))))))) .
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-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))))))))))) .
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-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))))))))) .
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-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))))))) .
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-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)))))))))) .
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-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))))))) .
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-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))))))) .
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-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)))))))))))) .
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-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))))))))))) .
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-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))))))))))) .
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-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)))))))))))))))) .
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-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)))))))))))))) .
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-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))))))).
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-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))))))) .
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-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))))))))) .
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-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)))))))))) .
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-axiom csub3_refl: \forall (g: G).(\forall (c: C).(csub3 g c c)) .
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-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)))))))) .
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-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)))))))))))) .
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-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))))))))))) .
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-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)))))))))))) .
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-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))))))))))) .
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-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)))))))))))) .
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-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))))))) .
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-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))))))) .
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-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))))))) .
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-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)))))))) .
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-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))))))))) .
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-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))))))) .
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-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))))))) .
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-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))))))) .
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-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))))))))) .
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-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))))))))) .
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-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)))))) .
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-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))))))))))) .
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-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))))))) .
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-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)))))) .
+include "LambdaDelta/theory.ma".
+
+theorem bind_dec_not:
+ \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
+\def
+ \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2)
+in (let H \def H_x in (or_ind (eq B b1 b2) ((eq B b1 b2) \to (\forall (P:
+Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
+b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0:
+(((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1
+b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
+
+definition TApp:
+ TList \to (T \to TList)
+\def
+ let rec TApp (ts: TList) on ts: (T \to TList) \def (\lambda (v: T).(match ts
+with [TNil \Rightarrow (TCons v TNil) | (TCons t ts0) \Rightarrow (TCons t
+(TApp ts0 v))])) in TApp.
+
+definition tslen:
+ TList \to nat
+\def
+ let rec tslen (ts: TList) on ts: nat \def (match ts with [TNil \Rightarrow O
+| (TCons _ ts0) \Rightarrow (S (tslen ts0))]) in tslen.
+
+definition tslt:
+ TList \to (TList \to Prop)
+\def
+ \lambda (ts1: TList).(\lambda (ts2: TList).(lt (tslen ts1) (tslen ts2))).
+
+theorem tslt_wf__q_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
+\to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
+\to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
+Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
+ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
+(tslen ts)))))).
+
+theorem tslt_wf_ind:
+ \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
+TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
+TList).(P ts)))
+\def
+ let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
+TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
+Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
+(tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
+TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
+nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
+(H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
+m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
+\def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
+(\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
+H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
+ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
+
+theorem theads_tapp:
+ \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T
+(THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
+\def
+ \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
+(v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v
+t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t))))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall
+(t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v
+t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k
+v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k
+v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k
+(TApp t0 v) t1) (H v t1))))))) vs)).
+
+theorem tcons_tapp_ex:
+ \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
+\def
+ \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
+TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
+ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
+ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
+(ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
+TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
+T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
+(\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
+t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
+ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
+(ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
+TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
+(_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
+(x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
+nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
+TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
+t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
+(tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
+nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
+t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
+nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
+TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
+(\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
+(TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
+(tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
+
+theorem tlist_ind_rew:
+ \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
+TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
+TList).(P ts))))
+\def
+ \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
+((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
+t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
+(\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
+TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
+TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
+\to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
+\to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
+(ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
+t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
+(tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
+T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
+(tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
+t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
+(TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
+H4))))) H3))))))) ts2)) ts)))).
+
+theorem iso_gen_sort:
+ \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda
+(n2: nat).(eq T u2 (TSort n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) u2)).(let H0
+\def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
+(iso t t0)).((eq T t (TSort n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2:
+nat).(eq T u2 (TSort n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda
+(H0: (eq T (TSort n0) (TSort n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let
+H2 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_:
+T).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _
+_) \Rightarrow n0])) (TSort n0) (TSort n1) H0) in (eq_ind nat n1 (\lambda (_:
+nat).((eq T (TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TSort
+n3)))))) (\lambda (H3: (eq T (TSort n2) u2)).(eq_ind T (TSort n2) (\lambda
+(t: T).(ex nat (\lambda (n3: nat).(eq T t (TSort n3))))) (ex_intro nat
+(\lambda (n3: nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort
+n2))) u2 H3)) n0 (sym_eq nat n0 n1 H2))) H1))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H0: (eq T (TLRef i1) (TSort n1))).(\lambda (H1: (eq T (TLRef i2)
+u2)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n1) H0) in
+(False_ind ((eq T (TLRef i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
+(TSort n2))))) H2)) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda
+(H0: (eq T (THead k v1 t1) (TSort n1))).(\lambda (H1: (eq T (THead k v2 t2)
+u2)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n1) H0) in
+(False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
+(TSort n2))))) H2)) H1)))]) in (H0 (refl_equal T (TSort n1)) (refl_equal T
+u2))))).
+
+theorem iso_gen_lref:
+ \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda
+(n2: nat).(eq T u2 (TLRef n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) u2)).(let H0
+\def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
+(iso t t0)).((eq T t (TLRef n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2:
+nat).(eq T u2 (TLRef n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda
+(H0: (eq T (TSort n0) (TLRef n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let
+H2 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef n1) H0) in (False_ind ((eq T
+(TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TLRef n3))))) H2))
+H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef
+n1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1]))
+(TLRef i1) (TLRef n1) H0) in (eq_ind nat n1 (\lambda (_: nat).((eq T (TLRef
+i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 (TLRef n2)))))) (\lambda (H3:
+(eq T (TLRef i2) u2)).(eq_ind T (TLRef i2) (\lambda (t: T).(ex nat (\lambda
+(n2: nat).(eq T t (TLRef n2))))) (ex_intro nat (\lambda (n2: nat).(eq T
+(TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))) u2 H3)) i1 (sym_eq nat
+i1 n1 H2))) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda (H0: (eq T
+(THead k v1 t1) (TLRef n1))).(\lambda (H1: (eq T (THead k v2 t2) u2)).((let
+H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n1) H0) in
+(False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
+(TLRef n2))))) H2)) H1)))]) in (H0 (refl_equal T (TLRef n1)) (refl_equal T
+u2))))).
+
+theorem iso_gen_head:
+ \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso
+(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))))))
+\def
+ \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda
+(H: (iso (THead k v1 t1) u2)).(let H0 \def (match H in iso return (\lambda
+(t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (THead k v1 t1))
+\to ((eq T t0 u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq
+T (TSort n1) (THead k v1 t1))).(\lambda (H1: (eq T (TSort n2) u2)).((let H2
+\def (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H0) in (False_ind ((eq T
+(TSort n2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0:
+(eq T (TLRef i1) (THead k v1 t1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let
+H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H0) in (False_ind ((eq T
+(TLRef i2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))) H2)) H1))) | (iso_head v0 v2 t0 t2 k0) \Rightarrow
+(\lambda (H0: (eq T (THead k0 v0 t0) (THead k v1 t1))).(\lambda (H1: (eq T
+(THead k0 v2 t2) u2)).((let H2 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k v1
+t1) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0
+| (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H0) in
+((let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_:
+T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _
+_) \Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H0) in (eq_ind K k
+(\lambda (k1: K).((eq T v0 v1) \to ((eq T t0 t1) \to ((eq T (THead k1 v2 t2)
+u2) \to (ex_2 T T (\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead k v3
+t3))))))))) (\lambda (H5: (eq T v0 v1)).(eq_ind T v1 (\lambda (_: T).((eq T
+t0 t1) \to ((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3: T).(\lambda
+(t3: T).(eq T u2 (THead k v3 t3)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T
+t1 (\lambda (_: T).((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3:
+T).(\lambda (t3: T).(eq T u2 (THead k v3 t3))))))) (\lambda (H7: (eq T (THead
+k v2 t2) u2)).(eq_ind T (THead k v2 t2) (\lambda (t: T).(ex_2 T T (\lambda
+(v3: T).(\lambda (t3: T).(eq T t (THead k v3 t3)))))) (ex_2_intro T T
+(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2
+t2 (refl_equal T (THead k v2 t2))) u2 H7)) t0 (sym_eq T t0 t1 H6))) v0
+(sym_eq T v0 v1 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H0
+(refl_equal T (THead k v1 t1)) (refl_equal T u2))))))).
+
+theorem iso_refl:
+ \forall (t: T).(iso t t)
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n:
+nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_:
+(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
+
+theorem lifts_tapp:
+ \forall (h: nat).(\forall (d: nat).(\forall (v: T).(\forall (vs: TList).(eq
+TList (lifts h d (TApp vs v)) (TApp (lifts h d vs) (lift h d v))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (v: T).(\lambda (vs:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts h d (TApp t v)) (TApp
+(lifts h d t) (lift h d v)))) (refl_equal TList (TCons (lift h d v) TNil))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq TList (lifts h d (TApp
+t0 v)) (TApp (lifts h d t0) (lift h d v)))).(eq_ind_r TList (TApp (lifts h d
+t0) (lift h d v)) (\lambda (t1: TList).(eq TList (TCons (lift h d t) t1)
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v))))) (refl_equal TList
+(TCons (lift h d t) (TApp (lifts h d t0) (lift h d v)))) (lifts h d (TApp t0
+v)) H)))) vs)))).
+
+theorem dnf_dec2:
+ \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
+O) d v))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda
+(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d:
+nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n)
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d
+v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort
+n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (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 (or (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T
+(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v:
+T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef n) t0)) (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).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0
+w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift
+(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w
+(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S
+O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (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).(subst0 n w (TLRef n) t0)) (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)).(or_intror (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n)
+(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq T (TLRef n) t0)) (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).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
+t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda
+(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
+in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0
+d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1)
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O)
+d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0
+(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
+(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S
+O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))
+(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift
+(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d
+v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w
+t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
+T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w)
+in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift
+(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S
+O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s
+k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda
+(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w
+(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d
+w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0
+t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2))
+(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6)
+(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5))))))
+(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T
+(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex
+T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x))
+(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def
+H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T
+(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d
+v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d)
+x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O)
+(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d)
+x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1
+H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex
+T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x:
+T).(\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 (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s
+k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S
+O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
+k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v:
+T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O)
+d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
+(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2
+t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T
+(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v))))
+(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T
+(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda
+(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda
+(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (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).(subst0 d w (THead k (lift (S O) d x) t1)
+t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift
+(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3))
+(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x0: T).(\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).(or (\forall (w: T).(ex T (\lambda
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x)
+(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
+k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
+T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v))))
+(ex_intro T (\lambda (v: T).(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).(eq T (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) t2)) (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).
+
+theorem pr2_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
+T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v1) t1 t2) \to
+(\forall (v2: T).(pr2 (CHead c (Bind b) v2) t1 t2))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
+(v1: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind
+b) v1) t1 t2)).(\lambda (v2: T).(insert_eq C (CHead c (Bind b) v1) (\lambda
+(c0: C).(pr2 c0 t1 t2)) (pr2 (CHead c (Bind b) v2) t1 t2) (\lambda (y:
+C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v1)) \to (pr2 (CHead c (Bind
+b) v2) t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H2: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v1))).(pr2_free
+(CHead c (Bind b) v2) t3 t4 H2)))))) (\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0
+(CHead c (Bind b) v1))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1
+(CHead d (Bind Abbr) u))) H2 (CHead c (Bind b) v1) H5) in (nat_ind (\lambda
+(n: nat).((getl n (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)) \to ((subst0
+n u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t)))) (\lambda (H7: (getl O
+(CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 O u t4
+t)).(let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d
+(Bind Abbr) u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u)
+H7))) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr) u) v1
+(getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in ((let H11
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d
+(Bind Abbr) u) (CHead c (Bind b) v1) (clear_gen_bind b c (CHead d (Bind Abbr)
+u) v1 (getl_gen_O (CHead c (Bind b) v1) (CHead d (Bind Abbr) u) H7))) in
+(\lambda (H12: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H14 \def (eq_ind
+T u (\lambda (t0: T).(subst0 O t0 t4 t)) H8 v1 H11) in (let H15 \def
+(eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abbr))) H Abbr H12) in (eq_ind B
+Abbr (\lambda (b0: B).(pr2 (CHead c (Bind b0) v2) t3 t)) (let H16 \def (match
+(H15 (refl_equal B Abbr)) in False return (\lambda (_: False).(pr2 (CHead c
+(Bind Abbr) v2) t3 t)) with []) in H16) b H12)))))) H10)) H9)))) (\lambda
+(i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) v1) (CHead d (Bind Abbr)
+u)) \to ((subst0 i0 u t4 t) \to (pr2 (CHead c (Bind b) v2) t3 t))))).(\lambda
+(H7: (getl (S i0) (CHead c (Bind b) v1) (CHead d (Bind Abbr) u))).(\lambda
+(H8: (subst0 (S i0) u t4 t)).(pr2_delta (CHead c (Bind b) v2) 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) v1 i0 H7) v2) t3 t4 H3 t H8))))) i H6 H4)))))))))))))
+y t1 t2 H1))) H0)))))))).
+
+theorem pr3_flat:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall (f: F).(pr3 c (THead
+(Flat f) u1 t1) (THead (Flat f) u2 t2)))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
+(f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f
+u2))))))))).
+
+theorem pr3_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1:
+T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind
+b) u1) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall
+(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind
+b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
+(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3
+(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B
+Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x:
+T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def
+(pr3_gen_abbr c u1 t1 x H0) in (or_ind (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)) (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 (H2: (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 (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))) (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 (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x
+(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3
+(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (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)) (ex3_2_intro 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))) x0 x1
+H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S
+O) O x))).(or_intror (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)) H2)) H1)))))))) (\lambda (H:
+(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1
+\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or
+(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).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c
+(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_:
+(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1:
+T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1
+\def (pr3_gen_void c u1 t1 x H0) in (or_ind (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
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c
+(Bind Void) u1) t1 (lift (S O) O 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).(pr3 (CHead c (Bind Void)
+u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda
+(H2: (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 (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0)
+u) t1 t2))))))).(ex3_2_ind 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 (b0: B).(\forall (u: T).(pr3 (CHead
+c (Bind b0) u) t1 t2))))) (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).(pr3 (CHead c (Bind Void) u1) t1
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0
+x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall
+(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (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).(pr3
+(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S
+O) O x)) (ex3_2_intro 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).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1
+H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1
+(lift (S O) O x))).(or_intror (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).(pr3 (CHead c (Bind Void) u1) t1
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).
+
+theorem pr3_iso_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).(let u1 \def (THeads (Flat
+Appl) vs (TLRef i)) in (\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) vs (lift (S i) O w))
+u2))))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
+(\lambda (t: TList).(let u1 \def (THeads (Flat Appl) t (TLRef i)) in (\forall
+(u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to
+(pr3 c (THeads (Flat Appl) t (lift (S i) O w)) u2)))))) (\lambda (u2:
+T).(\lambda (H0: (pr3 c (TLRef i) u2)).(\lambda (H1: (((iso (TLRef i) u2) \to
+(\forall (P: Prop).P)))).(let H2 \def (pr3_gen_lref c u2 i H0) in (or_ind (eq
+T u2 (TLRef i)) (ex3_3 C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T u2 (lift (S i) O v)))))) (pr3 c (lift (S i) O w) u2) (\lambda
+(H3: (eq T u2 (TLRef i))).(let H4 \def (eq_ind T u2 (\lambda (t: T).((iso
+(TLRef i) t) \to (\forall (P: Prop).P))) H1 (TLRef i) H3) in (eq_ind_r T
+(TLRef i) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (H4 (iso_refl (TLRef
+i)) (pr3 c (lift (S i) O w) (TLRef i))) u2 H3))) (\lambda (H3: (ex3_3 C T T
+(\lambda (d0: C).(\lambda (u: T).(\lambda (_: T).(getl i c (CHead d0 (Bind
+Abbr) u))))) (\lambda (d0: C).(\lambda (u: T).(\lambda (v: T).(pr3 d0 u v))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T u2 (lift (S i) O
+v))))))).(ex3_3_ind C T T (\lambda (d0: C).(\lambda (u: T).(\lambda (_:
+T).(getl i c (CHead d0 (Bind Abbr) u))))) (\lambda (d0: C).(\lambda (u:
+T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(v: T).(eq T u2 (lift (S i) O v))))) (pr3 c (lift (S i) O w) u2) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H4: (getl i c (CHead x0
+(Bind Abbr) x1))).(\lambda (H5: (pr3 x0 x1 x2)).(\lambda (H6: (eq T u2 (lift
+(S i) O x2))).(let H7 \def (eq_ind T u2 (\lambda (t: T).((iso (TLRef i) t)
+\to (\forall (P: Prop).P))) H1 (lift (S i) O x2) H6) in (eq_ind_r T (lift (S
+i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C
+(CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind
+Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1)
+H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow
+c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d
+(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal
+C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead
+x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind
+Abbr) x1) H4)) 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 w H10) in (let H13
+\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def
+(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d
+H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d
+H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15)))))))
+H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef
+i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P:
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w))
+u2)))))).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (TLRef i))) u2)).(\lambda (H2: (((iso (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (TLRef i))) u2) \to (\forall (P: Prop).P)))).(let H3
+\def (pr3_gen_appl c t (THeads (Flat Appl) t0 (TLRef i)) u2 H1) 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 t u3))) (\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 (u3: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\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)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\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 (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 t
+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) t
+(THeads (Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (H4: (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 t u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) 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 t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THeads (Flat Appl) t0 (TLRef i)) t2))) (pr3 c (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t
+x0)).(\lambda (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let H8 \def
+(eq_ind T u2 (\lambda (t1: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
+t0 (TLRef i))) t1) \to (\forall (P: Prop).P))) H2 (THead (Flat Appl) x0 x1)
+H5) in (eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t1: T).(pr3 c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) t1)) (H8 (iso_head t
+x0 (THeads (Flat Appl) t0 (TLRef i)) x1 (Flat Appl)) (pr3 c (THead (Flat
+Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) (THead (Flat Appl) x0 x1)))
+u2 H5))))))) H4)) (\lambda (H4: (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 t
+u3))))) (\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 (u3: T).(\lambda (t2: T).(pr3
+c (THead (Bind Abbr) u3 t2) u2))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (_: T).(pr3 c t u3))))) (\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))))))) (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O
+w))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H6: (pr3 c t
+x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
+Abst) x0 x1))).(\lambda (H8: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c
+(Bind b) u) x1 x3))))).(pr3_t (THead (Bind Abbr) t x1) (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t
+(THead (Bind Abst) x0 x1)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
+(S i) O w))) c (pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead
+(Bind Abst) x0 x1) (H0 (THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso
+(THeads (Flat Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (P:
+Prop).(iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H9 P)))) t Appl) (THead
+(Bind Abbr) t x1) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1))
+(THead (Bind Abbr) t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst)
+x0 x1)) (THead (Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1
+(pr0_refl x1))))) u2 (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t
+x1) c (pr3_head_12 c t x2 H6 (Bind Abbr) x1 x3 (H8 Abbr x2)) u2 H5))))))))))
+H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\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 (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 t 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))))))))).(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
+(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 t 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) t (THeads (Flat
+Appl) t0 (lift (S i) O w))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0 (TLRef i))
+(THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H8: (pr3 c t x4)).(\lambda
+(H9: (pr3 c x1 x5)).(\lambda (H10: (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) t (THeads (Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Bind x0)
+x1 (THead (Flat Appl) (lift (S O) O t) x2)) (THead (Flat Appl) t (THeads
+(Flat Appl) t0 (lift (S i) O w))) c (pr3_t (THead (Flat Appl) t (THead (Bind
+x0) x1 x2)) (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O w))) c
+(pr3_thin_dx c (THeads (Flat Appl) t0 (lift (S i) O w)) (THead (Bind x0) x1
+x2) (H0 (THead (Bind x0) x1 x2) H6 (\lambda (H11: (iso (THeads (Flat Appl) t0
+(TLRef i)) (THead (Bind x0) x1 x2))).(\lambda (P:
+Prop).(iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H11 P)))) t Appl) (THead
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr3_pr2 c (THead (Flat
+Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl) (lift
+(S O) O t) x2)) (pr2_free c (THead (Flat Appl) t (THead (Bind x0) x1 x2))
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr0_upsilon x0
+H5 t t (pr0_refl t) 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 t) x2) (THead
+(Flat Appl) (lift (S O) O x4) x2) (pr3_head_12 (CHead c (Bind x0) x1) (lift
+(S O) O t) (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) t x4 H8) (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 H9 (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 H10
+(lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))).
+
+theorem pr3_iso_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2:
+T).(\forall (t: T).(let u1 \def (THead (Flat Appl) v1 (THead (Bind b) v2 t))
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O v1) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v1: T).(\lambda
+(v2: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H0: (pr3 c
+(THead (Flat Appl) v1 (THead (Bind b) v2 t)) u2)).(\lambda (H1: (((iso (THead
+(Flat Appl) v1 (THead (Bind b) v2 t)) u2) \to (\forall (P: Prop).P)))).(let
+H2 \def (pr3_gen_appl c v1 (THead (Bind b) v2 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 v1 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c (THead (Bind b) v2 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 v1 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) z1
+t2)))))))) (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).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1
+z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
+T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 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 v1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (H3: (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 v1 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 c (THead (Bind b) v2 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 v1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THead (Bind b) v2 t) t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O v1) t)) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq
+T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v1 x0)).(\lambda (_:
+(pr3 c (THead (Bind b) v2 t) x1)).(let H7 \def (eq_ind T u2 (\lambda (t0:
+T).((iso (THead (Flat Appl) v1 (THead (Bind b) v2 t)) t0) \to (\forall (P:
+Prop).P))) H1 (THead (Flat Appl) x0 x1) H4) in (eq_ind_r T (THead (Flat Appl)
+x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S
+O) O v1) t)) t0)) (H7 (iso_head v1 x0 (THead (Bind b) v2 t) x1 (Flat Appl))
+(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) 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 (u3:
+T).(\lambda (_: T).(pr3 c v1 u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 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 v1 u3)))))
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c
+(THead (Bind b) v2 t) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u:
+T).(pr3 (CHead c (Bind b0) u) z1 t2))))))) (pr3 c (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) 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 v1 x2)).(\lambda (H6: (pr3 c (THead (Bind b)
+v2 t) (THead (Bind Abst) x0 x1))).(\lambda (H7: ((\forall (b0: B).(\forall
+(u: T).(pr3 (CHead c (Bind b0) u) x1 x3))))).(pr3_t (THead (Bind Abbr) x2 x3)
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let H_x \def
+(pr3_gen_bind b H c v2 t (THead (Bind Abst) x0 x1) H6) in (let H8 \def H_x in
+(or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))))
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1))) (pr3 c
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind
+Abbr) x2 x3)) (\lambda (H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
+T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3
+(CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda
+(x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4
+x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2)
+t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return
+(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow
+Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst)
+x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda
+(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0
+| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind
+Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0]))
+(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T
+x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda
+(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def
+(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def
+(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst
+H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H
+Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22
+\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_:
+False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t))
+(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13)))))))
+H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind
+Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
+x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift
+(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead
+(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O
+v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop
+(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst)
+x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead
+(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead
+(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2
+x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst)
+x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2
+(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead
+(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta
+x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3)
+(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead
+(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1)))
+(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4)))))))))
+H3)) (\lambda (H3: (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).(pr3 c (THead (Bind b) v2 t) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
+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 v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2)))))))
+(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
+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).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda
+(b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u3:
+T).(\lambda (y2: T).(pr3 c (THead (Bind b0) 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 v1 u3))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0)
+y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O
+v1) 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 (THead (Bind b) v2 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 v1 x4)).(\lambda (H8: (pr3 c x1
+x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t
+(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_:
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind
+b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda
+(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl)
+(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq
+T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2
+x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal
+T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match
+k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
+((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0
+_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
+((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in
+(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def
+(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17)
+in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18)
+in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2
+x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0
+Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat
+Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8)
+(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S
+O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O
+x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c
+(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3
+(pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3
+(Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11:
+(pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
+(Bind x0) x1 x2)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1)
+t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat
+Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (Bind b)
+(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x4) (pr3_lift
+(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2)
+v1 x4 H7) t (lift (S O) O (THead (Bind x0) x1 x2)) H11 Appl)) (THead (Bind
+x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (eq_ind T (lift (S O) O
+(THead (Flat Appl) x4 (THead (Bind x0) x1 x2))) (\lambda (t0: T).(pr3 c
+(THead (Bind b) v2 t0) (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O
+x4) x3)))) (pr3_sing c (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O
+x4) x2)) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x4 (THead (Bind
+x0) x1 x2)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl)
+x4 (THead (Bind x0) x1 x2)))) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S
+O) O x4) x2)) (pr0_zeta b H (THead (Flat Appl) x4 (THead (Bind x0) x1 x2))
+(THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) (pr0_upsilon x0
+H4 x4 x4 (pr0_refl x4) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)) v2)) (THead
+(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (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))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead
+(Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O))))
+H10))) u2 H6))))))))))))) H3)) H2)))))))))).
+
+theorem pr3_iso_appls_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u:
+T).(\forall (t: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) vs
+(THead (Flat Appl) v (THead (Bind b) u t))) in (\forall (c: C).(\forall (u2:
+T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c
+(THeads (Flat Appl) vs (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
+t))) u2)))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (v: T).(\lambda
+(u: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0:
+TList).(let u1 \def (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
+b) u t))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead
+(Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))) (\lambda (c:
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) v (THead (Bind b)
+u t)) u2)).(\lambda (H1: (((iso (THead (Flat Appl) v (THead (Bind b) u t))
+u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b H v u t c u2 H0 H1)))))
+(\lambda (t0: T).(\lambda (t1: TList).(\lambda (H0: ((\forall (c: C).(\forall
+(u2: T).((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
+t))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+b) u t))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t1
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))) u2))))))).(\lambda
+(c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t)))) u2)).(\lambda
+(H2: (((iso (THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v
+(THead (Bind b) u t)))) u2) \to (\forall (P: Prop).P)))).(let H3 \def
+(pr3_gen_appl c t0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind
+b) u t))) u2 H1) 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 Appl) v (THead (Bind b) u 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
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2)))))))) (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).(pr3
+c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
+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 (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2)))))))) (pr3 c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
+Appl) (lift (S O) O v) t)))) u2) (\lambda (H4: (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 Appl) v (THead (Bind b) u 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
+Appl) v (THead (Bind b) u t))) t2))) (pr3 c (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t))))
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T u2 (THead (Flat
+Appl) x0 x1))).(\lambda (_: (pr3 c t0 x0)).(\lambda (_: (pr3 c (THeads (Flat
+Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) x1)).(let H8 \def
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t0 (THeads (Flat Appl)
+t1 (THead (Flat Appl) v (THead (Bind b) u t)))) t2) \to (\forall (P:
+Prop).P))) H2 (THead (Flat Appl) x0 x1) H5) in (eq_ind_r T (THead (Flat Appl)
+x0 x1) (\lambda (t2: T).(pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) t2)) (H8
+(iso_head t0 x0 (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
+t))) x1 (Flat Appl)) (pr3 c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) (THead (Flat
+Appl) x0 x1))) u2 H5))))))) H4)) (\lambda (H4: (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
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) 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
+Appl) v (THead (Bind b) u t))) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b0:
+B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) z1 t2))))))) (pr3 c (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O v) t)))) u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H5: (pr3 c (THead (Bind Abbr) x2 x3)
+u2)).(\lambda (H6: (pr3 c t0 x2)).(\lambda (H7: (pr3 c (THeads (Flat Appl) t1
+(THead (Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0
+x1))).(\lambda (H8: ((\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind
+b0) u0) x1 x3))))).(pr3_t (THead (Bind Abbr) t0 x1) (THead (Flat Appl) t0
+(THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v)
+t)))) c (pr3_t (THead (Flat Appl) t0 (THead (Bind Abst) x0 x1)) (THead (Flat
+Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S
+O) O v) t)))) c (pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u
+(THead (Flat Appl) (lift (S O) O v) t))) (THead (Bind Abst) x0 x1) (H0 c
+(THead (Bind Abst) x0 x1) H7 (\lambda (H9: (iso (THeads (Flat Appl) t1 (THead
+(Flat Appl) v (THead (Bind b) u t))) (THead (Bind Abst) x0 x1))).(\lambda (P:
+Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1 (THead (Bind b) u t)
+t1 H9 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 H6 (Bind Abbr) x1 x3
+(H8 Abbr x2)) u2 H5)))))))))) H4)) (\lambda (H4: (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).(pr3 c
+(THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u t))) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0)
+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 (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind
+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).(pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead (Bind b) u
+t))) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind
+b0) 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 (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c
+(THead (Flat Appl) t0 (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat
+Appl) (lift (S O) O v) t)))) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not
+(eq B x0 Abst))).(\lambda (H6: (pr3 c (THeads (Flat Appl) t1 (THead (Flat
+Appl) v (THead (Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (H7: (pr3 c
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda
+(H8: (pr3 c t0 x4)).(\lambda (H9: (pr3 c x1 x5)).(\lambda (H10: (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 (THead (Bind b) u
+(THead (Flat Appl) (lift (S O) O v) 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 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c (pr3_t
+(THead (Flat Appl) t0 (THead (Bind x0) x1 x2)) (THead (Flat Appl) t0 (THeads
+(Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl) (lift (S O) O v) t)))) c
+(pr3_thin_dx c (THeads (Flat Appl) t1 (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O v) t))) (THead (Bind x0) x1 x2) (H0 c (THead (Bind x0) x1 x2)
+H6 (\lambda (H11: (iso (THeads (Flat Appl) t1 (THead (Flat Appl) v (THead
+(Bind b) u t))) (THead (Bind x0) x1 x2))).(\lambda (P:
+Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind b) u t) t1
+H11 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 H5 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 H8) (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 H9 (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 H10 (lift (S O) O x4) Appl)) u2
+H7)))))))))))))) H4)) H3))))))))) vs)))))).
+
+theorem pr3_iso_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u:
+T).(\forall (t: T).(let u1 \def (THeads (Flat Appl) vs (THead (Bind b) u t))
+in (\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl)
+(lifts (S O) O vs) t)) u2))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (vs:
+TList).(tlist_ind_rew (\lambda (t: TList).(\forall (u: T).(\forall (t0:
+T).(let u1 \def (THeads (Flat Appl) t (THead (Bind b) u t0)) in (\forall (c:
+C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P:
+Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t)
+t0)) u2))))))))) (\lambda (u: T).(\lambda (t: T).(\lambda (c: C).(\lambda
+(u2: T).(\lambda (H0: (pr3 c (THead (Bind b) u t) u2)).(\lambda (_: (((iso
+(THead (Bind b) u t) u2) \to (\forall (P: Prop).P)))).H0)))))) (\lambda (ts:
+TList).(\lambda (t: T).(\lambda (H0: ((\forall (u: T).(\forall (t0:
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) ts (THead
+(Bind b) u t0)) u2) \to ((((iso (THeads (Flat Appl) ts (THead (Bind b) u t0))
+u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind b) u (THeads (Flat
+Appl) (lifts (S O) O ts) t0)) u2))))))))).(\lambda (u: T).(\lambda (t0:
+T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H1: (pr3 c (THeads (Flat Appl)
+(TApp ts t) (THead (Bind b) u t0)) u2)).(\lambda (H2: (((iso (THeads (Flat
+Appl) (TApp ts t) (THead (Bind b) u t0)) u2) \to (\forall (P:
+Prop).P)))).(eq_ind_r TList (TApp (lifts (S O) O ts) (lift (S O) O t))
+(\lambda (t1: TList).(pr3 c (THead (Bind b) u (THeads (Flat Appl) t1 t0))
+u2)) (eq_ind_r T (THeads (Flat Appl) (lifts (S O) O ts) (THead (Flat Appl)
+(lift (S O) O t) t0)) (\lambda (t1: T).(pr3 c (THead (Bind b) u t1) u2)) (let
+H3 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind b) u t0))
+(\lambda (t1: T).(pr3 c t1 u2)) H1 (THeads (Flat Appl) ts (THead (Flat Appl)
+t (THead (Bind b) u t0))) (theads_tapp (Flat Appl) ts t (THead (Bind b) u
+t0))) in (let H4 \def (eq_ind T (THeads (Flat Appl) (TApp ts t) (THead (Bind
+b) u t0)) (\lambda (t1: T).((iso t1 u2) \to (\forall (P: Prop).P))) H2
+(THeads (Flat Appl) ts (THead (Flat Appl) t (THead (Bind b) u t0)))
+(theads_tapp (Flat Appl) ts t (THead (Bind b) u t0))) in (TList_ind (\lambda
+(t1: TList).(((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall
+(u3: T).((pr3 c0 (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to
+((((iso (THeads (Flat Appl) t1 (THead (Bind b) u0 t2)) u3) \to (\forall (P:
+Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O
+t1) t2)) u3)))))))) \to ((pr3 c (THeads (Flat Appl) t1 (THead (Flat Appl) t
+(THead (Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) t1 (THead (Flat
+Appl) t (THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c
+(THead (Bind b) u (THeads (Flat Appl) (lifts (S O) O t1) (THead (Flat Appl)
+(lift (S O) O t) t0))) u2))))) (\lambda (_: ((\forall (u0: T).(\forall (t1:
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) TNil (THead
+(Bind b) u0 t1)) u3) \to ((((iso (THeads (Flat Appl) TNil (THead (Bind b) u0
+t1)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0 (THead (Bind b) u0 (THeads
+(Flat Appl) (lifts (S O) O TNil) t1)) u3))))))))).(\lambda (H6: (pr3 c
+(THeads (Flat Appl) TNil (THead (Flat Appl) t (THead (Bind b) u t0)))
+u2)).(\lambda (H7: (((iso (THeads (Flat Appl) TNil (THead (Flat Appl) t
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P)))).(pr3_iso_appl_bind b
+H t u t0 c u2 H6 H7)))) (\lambda (t1: T).(\lambda (ts0: TList).(\lambda (_:
+((((\forall (u0: T).(\forall (t2: T).(\forall (c0: C).(\forall (u3: T).((pr3
+c0 (THeads (Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads
+(Flat Appl) ts0 (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to
+(pr3 c0 (THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O ts0) t2))
+u3)))))))) \to ((pr3 c (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
+(Bind b) u t0))) u2) \to ((((iso (THeads (Flat Appl) ts0 (THead (Flat Appl) t
+(THead (Bind b) u t0))) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead
+(Bind b) u (THeads (Flat Appl) (lifts (S O) O ts0) (THead (Flat Appl) (lift
+(S O) O t) t0))) u2)))))).(\lambda (H5: ((\forall (u0: T).(\forall (t2:
+T).(\forall (c0: C).(\forall (u3: T).((pr3 c0 (THeads (Flat Appl) (TCons t1
+ts0) (THead (Bind b) u0 t2)) u3) \to ((((iso (THeads (Flat Appl) (TCons t1
+ts0) (THead (Bind b) u0 t2)) u3) \to (\forall (P: Prop).P))) \to (pr3 c0
+(THead (Bind b) u0 (THeads (Flat Appl) (lifts (S O) O (TCons t1 ts0)) t2))
+u3))))))))).(\lambda (H6: (pr3 c (THeads (Flat Appl) (TCons t1 ts0) (THead
+(Flat Appl) t (THead (Bind b) u t0))) u2)).(\lambda (H7: (((iso (THeads (Flat
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) u2) \to
+(\forall (P: Prop).P)))).(H5 u (THead (Flat Appl) (lift (S O) O t) t0) c u2
+(pr3_iso_appls_appl_bind b H t u t0 (TCons t1 ts0) c u2 H6 H7) (\lambda (H8:
+(iso (THeads (Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O t) t0))) u2)).(\lambda (P: Prop).(H7 (iso_trans (THeads (Flat
+Appl) (TCons t1 ts0) (THead (Flat Appl) t (THead (Bind b) u t0))) (THeads
+(Flat Appl) (TCons t1 ts0) (THead (Bind b) u (THead (Flat Appl) (lift (S O) O
+t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead
+(Bind b) u t0))) (THeads (Flat Appl) ts0 (THead (Bind b) u (THead (Flat Appl)
+(lift (S O) O t) t0))) (Flat Appl)) u2 H8) P)))))))))) ts H0 H3 H4))) (THeads
+(Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat
+Appl) (lifts (S O) O ts) (lift (S O) O t) t0)) (lifts (S O) O (TApp ts t))
+(lifts_tapp (S O) O t ts))))))))))) vs))).
+
+theorem pr3_iso_beta:
+ \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat
+Appl) v (THead (Bind Abst) w t)) in (\forall (c: C).(\forall (u2: T).((pr3 c
+u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THead (Bind
+Abbr) v t) u2))))))))
+\def
+ \lambda (v: T).(\lambda (w: T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2:
+T).(\lambda (H: (pr3 c (THead (Flat Appl) v (THead (Bind Abst) w t))
+u2)).(\lambda (H0: (((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) u2)
+\to (\forall (P: Prop).P)))).(let H1 \def (pr3_gen_appl c v (THead (Bind
+Abst) w t) u2 H) 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 v
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst) w 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 v u3))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
+w 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 (THead (Bind Abst) w 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 v 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 (Bind Abbr) v t) u2) (\lambda (H2: (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 v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THead (Bind Abst) w 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 v u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THead (Bind Abst)
+w t) t2))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H3: (eq T u2 (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c v
+x0)).(\lambda (_: (pr3 c (THead (Bind Abst) w t) x1)).(let H6 \def (eq_ind T
+u2 (\lambda (t0: T).((iso (THead (Flat Appl) v (THead (Bind Abst) w t)) t0)
+\to (\forall (P: Prop).P))) H0 (THead (Flat Appl) x0 x1) H3) in (eq_ind_r T
+(THead (Flat Appl) x0 x1) (\lambda (t0: T).(pr3 c (THead (Bind Abbr) v t)
+t0)) (H6 (iso_head v x0 (THead (Bind Abst) w t) x1 (Flat Appl)) (pr3 c (THead
+(Bind Abbr) v t) (THead (Flat Appl) x0 x1))) u2 H3))))))) H2)) (\lambda (H2:
+(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 v u3))))) (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst)
+w 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 v
+u3))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(pr3 c (THead (Bind Abst) w 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 (Bind Abbr) v t)
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H3: (pr3 c (THead (Bind Abbr) x2 x3) u2)).(\lambda (H4: (pr3 c v
+x2)).(\lambda (H5: (pr3 c (THead (Bind Abst) w t) (THead (Bind Abst) x0
+x1))).(\lambda (H6: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b)
+u) x1 x3))))).(let H7 \def (pr3_gen_abst c w t (THead (Bind Abst) x0 x1) H5)
+in (ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
+(x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b:
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0]))
+(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0)
+\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in
+(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1
+H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0
+H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c
+(pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2)
+(H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2))
+(\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\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) w 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 v
+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))))))))).(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
+(THead (Bind Abst) w 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 v 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 (Bind Abbr) v t) u2) (\lambda (x0: B).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
+T).(\lambda (H3: (not (eq B x0 Abst))).(\lambda (H4: (pr3 c (THead (Bind
+Abst) w t) (THead (Bind x0) x1 x2))).(\lambda (H5: (pr3 c (THead (Bind x0) x5
+(THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (_: (pr3 c v
+x4)).(\lambda (_: (pr3 c x1 x5)).(\lambda (H8: (pr3 (CHead c (Bind x0) x5) x2
+x3)).(let H9 \def (pr3_gen_abst c w t (THead (Bind x0) x1 x2) H4) in
+(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c w
+u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3
+(CHead c (Bind b) u) t t2))))) (pr3 c (THead (Bind Abbr) v t) u2) (\lambda
+(x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead
+(Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b)
+\Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead
+(Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _)
+\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2)
+(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 |
+(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0)
+x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1
+x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda
+(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0))))
+H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0))
+H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind
+b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b:
+B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2))
+H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b
+Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in
+False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with [])
+in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))).
+
+theorem pr3_iso_appls_beta:
+ \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1
+\def (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst) w t))) in
+(\forall (c: C).(\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to
+(\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) us (THead (Bind Abbr)
+v t)) u2)))))))))
+\def
+ \lambda (us: TList).(TList_ind (\lambda (t: TList).(\forall (v: T).(\forall
+(w: T).(\forall (t0: T).(let u1 \def (THeads (Flat Appl) t (THead (Flat Appl)
+v (THead (Bind Abst) w t0))) in (\forall (c: C).(\forall (u2: T).((pr3 c u1
+u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat
+Appl) t (THead (Bind Abbr) v t0)) u2)))))))))) (\lambda (v: T).(\lambda (w:
+T).(\lambda (t: T).(\lambda (c: C).(\lambda (u2: T).(\lambda (H: (pr3 c
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2)).(\lambda (H0: (((iso
+(THead (Flat Appl) v (THead (Bind Abst) w t)) u2) \to (\forall (P:
+Prop).P)))).(pr3_iso_beta v w t c u2 H H0)))))))) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: ((\forall (v: T).(\forall (w: T).(\forall (t1:
+T).(\forall (c: C).(\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (THead
+(Flat Appl) v (THead (Bind Abst) w t1))) u2) \to ((((iso (THeads (Flat Appl)
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) u2) \to (\forall (P:
+Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))
+u2)))))))))).(\lambda (v: T).(\lambda (w: T).(\lambda (t1: T).(\lambda (c:
+C).(\lambda (u2: T).(\lambda (H0: (pr3 c (THead (Flat Appl) t (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) u2)).(\lambda (H1:
+(((iso (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Flat Appl) v
+(THead (Bind Abst) w t1)))) u2) \to (\forall (P: Prop).P)))).(let H2 \def
+(pr3_gen_appl c t (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind
+Abst) w t1))) 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 t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl)
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) 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 t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w 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
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) (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 t 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) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) u2)
+(\lambda (H3: (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 t u3)))
+(\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) 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 t u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 c
+(THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) t2)))
+(pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1)))
+u2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T u2 (THead (Flat
+Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda (_: (pr3 c (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1))) x1)).(let H7 \def
+(eq_ind T u2 (\lambda (t2: T).((iso (THead (Flat Appl) t (THeads (Flat Appl)
+t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))) 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) t (THeads (Flat Appl) t0
+(THead (Bind Abbr) v t1))) t2)) (H7 (iso_head t x0 (THeads (Flat Appl) t0
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) x1 (Flat Appl)) (pr3 c (THead
+(Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) (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 (u3:
+T).(\lambda (_: T).(pr3 c t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w 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))))))))).(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 t u3))))) (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (THead (Flat
+Appl) v (THead (Bind Abst) w 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))))))) (pr3 c (THead (Flat
+Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) 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 t x2)).(\lambda (H6: (pr3
+c (THeads (Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t1)))
+(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) t x1)
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
+(pr3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind Abst) x0 x1) (H v w t1
+c (THead (Bind Abst) x0 x1) H6 (\lambda (H8: (iso (THeads (Flat Appl) t0
+(THead (Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind Abst) x0
+x1))).(\lambda (P: Prop).(iso_flats_flat_bind_false Appl Appl Abst x0 v x1
+(THead (Bind Abst) w t1) t0 H8 P)))) t Appl) (THead (Bind Abbr) t x1)
+(pr3_pr2 c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr)
+t x1) (pr2_free c (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead
+(Bind Abbr) t x1) (pr0_beta x0 t t (pr0_refl t) x1 x1 (pr0_refl x1))))) u2
+(pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) t x1) c (pr3_head_12 c t
+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) t0 (THead (Flat Appl) v (THead (Bind Abst)
+w t1))) (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 t 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))))))))).(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 (THead (Flat
+Appl) v (THead (Bind Abst) w t1))) (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 t 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) t (THeads (Flat Appl) t0 (THead
+(Bind Abbr) v t1))) 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) t0 (THead (Flat Appl) v
+(THead (Bind Abst) w t1))) (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 t 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) t (THeads (Flat Appl) t0 (THead (Bind Abbr)
+v t1))) c (pr3_t (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2))
+(THead (Flat Appl) t (THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c
+(pr3_t (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Flat Appl) t
+(THeads (Flat Appl) t0 (THead (Bind Abbr) v t1))) c (pr3_thin_dx c (THeads
+(Flat Appl) t0 (THead (Bind Abbr) v t1)) (THead (Bind x0) x1 x2) (H v w t1 c
+(THead (Bind x0) x1 x2) H5 (\lambda (H10: (iso (THeads (Flat Appl) t0 (THead
+(Flat Appl) v (THead (Bind Abst) w t1))) (THead (Bind x0) x1 x2))).(\lambda
+(P: Prop).(iso_flats_flat_bind_false Appl Appl x0 x1 v x2 (THead (Bind Abst)
+w t1) t0 H10 P)))) t Appl) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O)
+O t) x2)) (pr3_pr2 c (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead
+(Bind x0) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (pr2_free c (THead
+(Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x1 (THead (Flat Appl)
+(lift (S O) O t) x2)) (pr0_upsilon x0 H4 t t (pr0_refl t) 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 t) x2) (THead (Flat Appl) (lift (S O) O x4) x2) (pr3_head_12
+(CHead c (Bind x0) x1) (lift (S O) O t) (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) t 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)))))))))))) us).
+
+theorem csuba_gen_abst_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
+(CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
+\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
+[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
+(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0:
+C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq
+C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
+(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort
+n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst)
+u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
+c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
+(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k
+u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to
+((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
+u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
+(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to
+((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
+k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0:
+K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
+c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))
+(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
+(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
+d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
+(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7)))
+c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
+H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
+c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst)
+u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
+(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \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 c0
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
+(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def
+(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind ((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 (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
+(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))).
+
+theorem csuba_gen_void_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
+(CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind
+Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
+(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return
+(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
+\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c
+(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
+[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
+(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0:
+C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq
+C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
+(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort
+n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
+u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
+Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
+c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
+(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k
+u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to
+((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void)
+u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
+u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3)
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
+(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to
+((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
+k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
+(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0:
+K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
+c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))
+(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
+(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void)
+u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
+d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
+(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7)))
+c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
+H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
+c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void)
+u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
+(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \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 c0
+(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
+(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def
+(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
+\Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind ((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 (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u)))
+(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
+(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))).
+
+theorem csuba_gen_abbr_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c
+(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a))))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba
+return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
+c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2:
+C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda
+(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr)
+u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind
+Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))
+(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def
+(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda
+(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
+d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
+(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head
+c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda
+(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k
+u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to
+((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))
+(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u)
+(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
+in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0:
+C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C
+(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))
+(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0:
+K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C
+(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
+(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u
+u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda
+(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2:
+C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C
+(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1))
+H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2
+d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2)
+\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4:
+(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead
+c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1
+(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to
+((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))))))))))
+(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let
+H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
+(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u)
+(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1)
+\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a)
+\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1
+(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity
+g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst)
+t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda
+(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1
+a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
+A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
+a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
+a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u
+(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in
+(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))).
+
+theorem csuba_gen_flat_rev:
+ \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
+(f: F).((csuba g c (CHead d1 (Flat f) u1)) \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 d2 d1)))))))))
+\def
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H
+in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0
+c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \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 d2 d1))))))))) with [(csuba_sort n)
+\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n)
+(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort
+n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
+g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3
+\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0
+k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C
+(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda
+(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2)
+\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3:
+(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat
+f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
+k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1
+(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq
+T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
+T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K
+(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8:
+(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat
+f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9:
+(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C
+(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
+(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u
+(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6)))
+H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow
+(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead
+c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t)
+(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \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 c0 (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5:
+(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind
+C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((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 (CHead c1 (Bind Abst) t) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6)))
+c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat
+f) u1))))))))).
+
+theorem csuba_gen_bind_rev:
+ \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
+(v1: T).((csuba g c2 (CHead e1 (Bind b1) v1)) \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 e2 e1))))))))))
+\def
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def
+(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba
+? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \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 e2
+e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
+c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort
+n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) 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).(csuba g e2
+e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3
+\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda
+(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2)
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))
+H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C
+(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1)
+v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead
+e1 (Bind b1) v1)) \to ((csuba g c1 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 e2 e1))))))))
+(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
+(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c:
+C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind
+b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda
+(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))
+(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq
+C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0:
+K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C
+c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1)
+v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda
+(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1
+(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k
+(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2
+H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C
+(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u)
+(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c:
+C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1
+c0) \to ((arity g c1 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 c (CHead e2 (Bind
+b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
+e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1)
+v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
+(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8
+\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
+with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
+(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c:
+C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t
+(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda
+(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
+v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
+e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_:
+B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to
+((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda
+(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda
+(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity
+g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
+e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t
+(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2
+(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst)
+t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1
+(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind
+Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind
+Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8)))
+H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C
+(CHead e1 (Bind b1) v1))))))))).
+
+theorem csuba_clear_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c2 c1) \to
+(\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1))
+(\lambda (e2: C).(clear c2 e2))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c2
+c1)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear
+c0 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c
+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 e2 e1))
+(\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 c4
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear c3
+e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2:
+(clear (CHead c4 k u) e1)).(K_ind (\lambda (k0: K).((clear (CHead c4 k0 u)
+e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear
+(CHead c3 k0 u) e2))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind
+b) u) e1)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind b) u) e2))))
+(ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind b) u))) (\lambda
+(e2: C).(clear (CHead c3 (Bind b) u) e2)) (CHead c3 (Bind b) u) (csuba_head g
+c3 c4 H0 (Bind b) u) (clear_bind b c3 u)) e1 (clear_gen_bind b c4 e1 u H3))))
+(\lambda (f: F).(\lambda (H3: (clear (CHead c4 (Flat f) u) e1)).(let H4 \def
+(H1 e1 (clear_gen_flat f c4 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g
+e2 e1)) (\lambda (e2: C).(clear c3 e2)) (ex2 C (\lambda (e2: C).(csuba g e2
+e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2))) (\lambda (x:
+C).(\lambda (H5: (csuba g x e1)).(\lambda (H6: (clear c3 x)).(ex_intro2 C
+(\lambda (e2: C).(csuba g e2 e1)) (\lambda (e2: C).(clear (CHead c3 (Flat f)
+u) e2)) x H5 (clear_flat c3 x H6 f u))))) H4)))) k H2))))))))) (\lambda (c3:
+C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall
+(e1: C).((clear c4 e1) \to (ex2 C (\lambda (e2: C).(csuba g e2 e1)) (\lambda
+(e2: C).(clear c3 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 c4 (Bind Abbr) u)
+e1)).(eq_ind_r C (CHead c4 (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda
+(e2: C).(csuba g e2 c)) (\lambda (e2: C).(clear (CHead c3 (Bind Abst) t)
+e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g e2 (CHead c4 (Bind Abbr) u)))
+(\lambda (e2: C).(clear (CHead c3 (Bind Abst) t) e2)) (CHead c3 (Bind Abst)
+t) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abst c3 t)) e1
+(clear_gen_bind Abbr c4 e1 u H4))))))))))))) c2 c1 H)))).
+
+theorem csuba_drop_abst_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
+O c1 (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g
+c2 c1) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))))))))))
+\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 Abst) u)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(drop n O c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))))))
+(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1
+(CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0:
+(csuba g c2 c1)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c2 c)) H0
+(CHead d1 (Bind Abst) u) (drop_gen_refl c1 (CHead d1 (Bind Abst) u) H)) in
+(let H_x \def (csuba_gen_abst_rev g d1 c2 u H1) in (let H2 \def H_x in
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H3:
+(eq C c2 (CHead x (Bind Abst) u))).(\lambda (H4: (csuba g x d1)).(eq_ind_r C
+(CHead x (Bind Abst) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (ex_intro2 C
+(\lambda (d2: C).(drop O O (CHead x (Bind Abst) u) (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abst) 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 Abst) u))
+\to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda
+(d2: C).(drop n O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S
+n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))
+(\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n)
+O (CSort n0) (CHead d1 (Bind Abst) u))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (_: (csuba g c2 (CSort n0))).(and3_ind (eq C (CHead d1 (Bind
+Abst) 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 Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u) (CSort n0))).(\lambda (_:
+(eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq
+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 Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)))))) with [refl_equal \Rightarrow (\lambda (H5: (eq C
+(CHead d1 (Bind Abst) u) (CSort n0))).(let H6 \def (eq_ind C (CHead d1 (Bind
+Abst) u) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with
+[(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0)
+H5) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H6)))]) in (H5 (refl_equal C
+(CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u)
+H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u:
+T).((drop (S n) O c (CHead d1 (Bind Abst) u)) \to (\forall (g: G).(\forall
+(c2: C).((csuba g c2 c) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead
+d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))))))).(\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 Abst) u))).(\lambda (g: G).(\lambda (c2:
+C).(\lambda (H2: (csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba
+g c2 (CHead c k0 t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u)) \to
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1)))))) (\lambda (b: B).(\lambda (H3: (csuba g c2 (CHead
+c (Bind b) t))).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst)
+u))).(B_ind (\lambda (b0: B).((csuba g c2 (CHead c (Bind b0) t)) \to ((drop
+(r (Bind b0) n) O c (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2:
+C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6:
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def
+(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
+g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g c t a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (H8: (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
+g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t)))
+(\lambda (d2: C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x:
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x
+c)).(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 Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)))) (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 Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H12:
+(drop n O x (CHead x0 (Bind Abst) u))).(\lambda (H13: (csuba g x0 d1)).(let
+H14 \def (refl_equal nat (r (Bind Abst) n)) in (let H15 \def (eq_ind nat n
+(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H12 (r (Bind Abst)
+n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr)
+t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop
+(Bind Abbr) n x (CHead x0 (Bind Abst) 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 Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))) (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2:
+A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g
+x0 c)).(\lambda (_: (arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t
+x2)).(eq_ind_r C (CHead x0 (Bind Abst) x1) (\lambda (c0: C).(ex2 C (\lambda
+(d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1)))) (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 Abst) u))) (\lambda (d2: C).(csuba g d2 d1))
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda
+(H14: (drop n O x0 (CHead x (Bind Abst) u))).(\lambda (H15: (csuba g x
+d1)).(let H16 \def (refl_equal nat (r (Bind Abst) n)) in (let H17 \def
+(eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abst) u))) H14
+(r (Bind Abst) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead
+x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)) x (drop_drop (Bind Abst) n x0 (CHead x (Bind Abst) u) H17 x1) H15))))))
+H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g c2 (CHead c (Bind
+Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst)
+u))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let H7 \def H_x in
+(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2:
+C).(csuba g d2 c)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8:
+(eq C c2 (CHead x (Bind Abst) t))).(\lambda (H9: (csuba g x c)).(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 Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (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 Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead
+x0 (Bind Abst) u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal
+nat (r (Bind Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0:
+nat).(drop n0 O x (CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abst)
+n x (CHead x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))))
+(\lambda (H5: (csuba g c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r
+(Bind Void) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def
+(csuba_gen_void_rev g c c2 t H5) in (let H7 \def H_x in (ex2_ind C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g d2 c))
+(ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x
+(Bind Void) t))).(\lambda (H9: (csuba g x c)).(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 Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (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 Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Void) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abst)
+u))).(\lambda (H12: (csuba g x0 d1)).(let H13 \def (refl_equal nat (r (Bind
+Abst) n)) in (let H14 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
+(CHead x0 (Bind Abst) u))) H11 (r (Bind Abst) n) H13) in (ex_intro2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Void) n x (CHead
+x0 (Bind Abst) u) H14 t) H12)))))) H10)) c2 H8)))) H7))))) b H3 H4))))
+(\lambda (f: F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda
+(H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u))).(let H_x \def
+(csuba_gen_flat_rev g c c2 t f H3) in (let H5 \def H_x 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 d2 c))) (ex2 C (\lambda (d2: C).(drop (S n)
+O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda
+(x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f)
+x1))).(\lambda (H7: (csuba g x0 c)).(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
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (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 Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(drop (S n) O
+(CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abst)
+u))).(\lambda (H10: (csuba g x d1)).(ex_intro2 C (\lambda (d2: C).(drop (S n)
+O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g
+d2 d1)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u) H9 x1) H10))))
+H8)) c2 H6))))) H5)))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u) t n
+H1)))))))))))) c1)))) i).
+
+theorem csuba_drop_abbr_rev:
+ \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i
+O c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba
+g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+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 Abbr) u1)) \to (\forall (g:
+G).(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(drop n
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1:
+T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u1))).(\lambda (g:
+G).(\lambda (c2: C).(\lambda (H0: (csuba g c2 c1)).(let H1 \def (eq_ind C c1
+(\lambda (c: C).(csuba g c2 c)) H0 (CHead d1 (Bind Abbr) u1) (drop_gen_refl
+c1 (CHead d1 (Bind Abbr) u1) H)) in (let H_x \def (csuba_gen_abbr_rev g d1 c2
+u1 H1) in (let H2 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H3:
+(ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(drop O O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O
+O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x
+(Bind Abbr) u1))).(\lambda (H5: (csuba g x d1)).(eq_ind_r C (CHead x (Bind
+Abbr) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr)
+u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind
+Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2:
+C).(drop O O (CHead x (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1)) x (drop_refl (CHead x (Bind Abbr) 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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind
+Abst) x1))).(\lambda (H5: (csuba g x0 d1)).(\lambda (H6: (arity g x0 x1
+(asucc g x2))).(\lambda (H7: (arity g d1 u1 x2)).(eq_ind_r C (CHead x0 (Bind
+Abst) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abst)
+x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind
+Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abst) x1)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abst) 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 Abbr) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c2 c1) \to
+(or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1:
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop
+(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))))))))))) (\lambda (n0: nat).(\lambda (d1:
+C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind
+Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g c2 (CSort
+n0))).(and3_ind (eq C (CHead d1 (Bind Abbr) 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
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (H2: (eq C (CHead d1 (Bind Abbr) u1) (CSort n0))).(\lambda (_: (eq
+nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 in eq 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 Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) with
+[refl_equal \Rightarrow (\lambda (H5: (eq C (CHead d1 (Bind Abbr) u1) (CSort
+n0))).(let H6 \def (eq_ind C (CHead d1 (Bind Abbr) u1) (\lambda (e: C).(match
+e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False |
+(CHead _ _ _) \Rightarrow True])) I (CSort n0) H5) in (False_ind (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H6)))]) in (H5
+(refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind
+Abbr) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1:
+C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abbr) u1)) \to (\forall
+(g: G).(\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda (d2: C).(drop
+(S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 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 Abbr) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2:
+(csuba g c2 (CHead c k t))).(K_ind (\lambda (k0: K).((csuba g c2 (CHead c k0
+t)) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (b:
+B).(\lambda (H3: (csuba g c2 (CHead c (Bind b) t))).(\lambda (H4: (drop (r
+(Bind b) n) O c (CHead d1 (Bind Abbr) u1))).(B_ind (\lambda (b0: B).((csuba g
+c2 (CHead c (Bind b0) t)) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind
+Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a))))))))) (\lambda (H5: (csuba g c2 (CHead c (Bind Abbr) t))).(\lambda (H6:
+(drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def
+(csuba_gen_abbr_rev g c c2 t H5) in (let H7 \def H_x in (or_ind (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba
+g d2 c))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g c t a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abbr) t))) (\lambda (d2: C).(csuba g d2 c)))).(ex2_ind C (\lambda (d2: C).(eq
+C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x:
+C).(\lambda (H9: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H10: (csuba g x
+c)).(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 Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 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 Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or
+(ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n
+O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind
+C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind
+Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0:
+C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) u1))).(\lambda (H14:
+(csuba g x0 d1)).(let H15 \def (refl_equal nat (r (Bind Abst) n)) in (let H16
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abbr)
+u1))) H13 (r (Bind Abst) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (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 Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x0 (drop_drop (Bind Abbr)
+n x (CHead x0 (Bind Abbr) 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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop n O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (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 Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x
+(CHead x0 (Bind Abst) x1))).(\lambda (H14: (csuba g x0 d1)).(\lambda (H15:
+(arity g x0 x1 (asucc g x2))).(\lambda (H16: (arity g d1 u1 x2)).(let H17
+\def (refl_equal nat (r (Bind Abst) n)) in (let H18 \def (eq_ind nat n
+(\lambda (n0: nat).(drop n0 O x (CHead x0 (Bind Abst) x1))) H13 (r (Bind
+Abst) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x
+(Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 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 Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n
+x (CHead x0 (Bind Abst) 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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 c)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a)))))).(ex4_3_ind C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+c)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc
+g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t a))))
+(or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2
+(CHead x0 (Bind Abst) x1))).(\lambda (H10: (csuba g x0 c)).(\lambda (_:
+(arity g x0 x1 (asucc g x2))).(\lambda (_: (arity g c t x2)).(eq_ind_r C
+(CHead x0 (Bind Abst) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop
+(S n) O c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 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 Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n
+O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))).(ex2_ind
+C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind
+Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O
+(CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x:
+C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abbr) u1))).(\lambda (H16:
+(csuba g x d1)).(let H17 \def (refl_equal nat (r (Bind Abst) n)) in (let H18
+\def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x (Bind Abbr)
+u1))) H15 (r (Bind Abst) n) H17) in (or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Bind
+Abst) n x0 (CHead x (Bind Abbr) 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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abst) x1)
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5:
+A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abst) x4))).(\lambda (H16:
+(csuba g x3 d1)).(\lambda (H17: (arity g x3 x4 (asucc g x5))).(\lambda (H18:
+(arity g d1 u1 x5)).(let H19 \def (refl_equal nat (r (Bind Abst) n)) in (let
+H20 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x0 (CHead x3 (Bind Abst)
+x4))) H15 (r (Bind Abst) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(drop (S n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S
+n) O (CHead x0 (Bind Abst) x1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x3 x4 x5
+(drop_drop (Bind Abst) n x0 (CHead x3 (Bind Abst) x4) H20 x1) H16 H17
+H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7))))) (\lambda (H5: (csuba g
+c2 (CHead c (Bind Abst) t))).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead
+d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_abst_rev g c c2 t H5) in (let
+H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst)
+t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n)
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x
+(Bind Abst) t))).(\lambda (H9: (csuba g x c)).(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 Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 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 Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11:
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (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
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr)
+u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind
+Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
+(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
+d2 d1)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) 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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst)
+x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g
+x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r
+(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O
+x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 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
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) x0 x1 x2 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) x1) H17 t)
+H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) (\lambda (H5: (csuba g
+c2 (CHead c (Bind Void) t))).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead
+d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_void_rev g c c2 t H5) in (let
+H7 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void)
+t))) (\lambda (d2: C).(csuba g d2 c)) (or (ex2 C (\lambda (d2: C).(drop (S n)
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x
+(Bind Void) t))).(\lambda (H9: (csuba g x c)).(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 Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 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 Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n
+O x (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead
+x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H11:
+(ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (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
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr)
+u1))).(\lambda (H13: (csuba g x0 d1)).(let H14 \def (refl_equal nat (r (Bind
+Abst) n)) in (let H15 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O x
+(CHead x0 (Bind Abbr) u1))) H12 (r (Bind Abst) n) H14) in (or_introl (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O
+(CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
+d2 d1)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) 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 Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst)
+x1))).(\lambda (H13: (csuba g x0 d1)).(\lambda (H14: (arity g x0 x1 (asucc g
+x2))).(\lambda (H15: (arity g d1 u1 x2)).(let H16 \def (refl_equal nat (r
+(Bind Abst) n)) in (let H17 \def (eq_ind nat n (\lambda (n0: nat).(drop n0 O
+x (CHead x0 (Bind Abst) x1))) H12 (r (Bind Abst) n) H16) in (or_intror (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 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
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) x0 x1 x2 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) x1) H17 t)
+H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7))))) b H3 H4)))) (\lambda (f:
+F).(\lambda (H3: (csuba g c2 (CHead c (Flat f) t))).(\lambda (H4: (drop (r
+(Flat f) n) O c (CHead d1 (Bind Abbr) u1))).(let H_x \def (csuba_gen_flat_rev
+g c c2 t f H3) in (let H5 \def H_x 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 d2 c))) (or (ex2 C (\lambda (d2: C).(drop (S n) O
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead
+d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
+(eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g x0 c)).(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 Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3
+C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 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 Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C
+(\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S
+n) O x0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(drop (S
+n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (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 Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abbr)
+u1))).(\lambda (H11: (csuba g x d1)).(or_introl (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (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 Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x (drop_drop (Flat f) n
+x0 (CHead x (Bind Abbr) 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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(drop (S n) O x0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(drop
+(S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (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 Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n)
+O x0 (CHead x2 (Bind Abst) x3))).(\lambda (H11: (csuba g x2 d1)).(\lambda
+(H12: (arity g x2 x3 (asucc g x4))).(\lambda (H13: (arity g d1 u1
+x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1)
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (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 Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 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 Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind
+Abst) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5)))))) k H2
+(drop_gen_drop k c (CHead d1 (Bind Abbr) u1) t n H1)))))))))))) c1)))) i).
+
+theorem csuba_getl_abst_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g
+c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abst) u) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u)))
+(\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (x:
+C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind
+Abst) u))).(C_ind (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1
+(Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)))))))) (\lambda (n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda
+(H4: (clear (CSort n) (CHead d1 (Bind Abst) u))).(clear_gen_sort (CHead d1
+(Bind Abst) u) n H4 (\forall (c2: C).((csuba g c2 c1) \to (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear x0
+(CHead d1 (Bind Abst) u)) \to (\forall (c2: C).((csuba g c2 c1) \to (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (H3:
+(drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear (CHead x0 k t) (CHead d1
+(Bind Abst) u))).(K_ind (\lambda (k0: K).((drop i O c1 (CHead x0 k0 t)) \to
+((clear (CHead x0 k0 t) (CHead d1 (Bind Abst) u)) \to (\forall (c2:
+C).((csuba g c2 c1) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda (b: B).(\lambda
+(H5: (drop i O c1 (CHead x0 (Bind b) t))).(\lambda (H6: (clear (CHead x0
+(Bind b) t) (CHead d1 (Bind Abst) u))).(let H7 \def (f_equal C C (\lambda (e:
+C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 |
+(CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H8 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abst | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abst])])) (CHead d1 (Bind Abst) u) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abst) u) t H6)) in ((let H9 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d1 (Bind
+Abst) u) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abst) u)
+t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 x0)).(\lambda
+(c2: C).(\lambda (H12: (csuba g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda
+(t0: T).(drop i O c1 (CHead x0 (Bind b) t0))) H5 u H9) in (let H14 \def
+(eq_ind_r B b (\lambda (b0: B).(drop i O c1 (CHead x0 (Bind b0) u))) H13 Abst
+H10) in (let H15 \def (eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c
+(Bind Abst) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abst_rev i c1 d1 u
+H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl i
+c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda
+(x1: C).(\lambda (H17: (drop i O c2 (CHead x1 (Bind Abst) u))).(\lambda (H18:
+(csuba g x1 d1)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2 (CHead x1
+(Bind Abst) u) (CHead x1 (Bind Abst) u) H17 (clear_bind Abst x1 u)) H18))))
+H16)))))))))) H8)) H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead
+x0 (Flat f) t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind
+Abst) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c
+(CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1))))))) (nat_ind (\lambda (n: nat).(\forall (x1: C).((drop n
+O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (ex2 C
+(\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(csuba g d2 d1)))))))) (\lambda (x1: C).(\lambda (H8: (drop O O x1 (CHead
+x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g c2 x1)).(let H10
+\def (eq_ind C x1 (\lambda (c: C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t)
+(drop_gen_refl x1 (CHead x0 (Flat f) t) H8)) in (let H_y \def (clear_flat x0
+(CHead d1 (Bind Abst) u) (clear_gen_flat f x0 (CHead d1 (Bind Abst) u) t H6)
+f t) in (let H11 \def (csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10
+(CHead d1 (Bind Abst) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2
+(CHead d1 (Bind Abst) u))) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda
+(d2: C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) (\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abst)
+u))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abst_rev g d1 x2
+u H12) in (let H14 \def H_x in (ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+(\lambda (x3: C).(\lambda (H15: (eq C x2 (CHead x3 (Bind Abst) u))).(\lambda
+(H16: (csuba g x3 d1)).(let H17 \def (eq_ind C x2 (\lambda (c: C).(clear c2
+c)) H13 (CHead x3 (Bind Abst) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl
+O c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) x3
+(getl_intro O c2 (CHead x3 (Bind Abst) u) c2 (drop_refl c2) H17) H16)))))
+H14)))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x1:
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1)
+\to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda
+(d2: C).(csuba g d2 d1))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S n) O
+x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2
+x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind
+b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead
+x0 (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x2: B).(\lambda (x3:
+C).(\lambda (x4: T).(\lambda (H12: (clear x1 (CHead x3 (Bind x2)
+x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f) t))).(let H14 \def
+(csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4) H12) in (ex2_ind C
+(\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4))) (\lambda (e2:
+C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x5: C).(\lambda (H15:
+(csuba g x5 (CHead x3 (Bind x2) x4))).(\lambda (H16: (clear c2 x5)).(let H_x
+\def (csuba_gen_bind_rev g x2 x3 x5 x4 H15) in (let H17 \def H_x in
+(ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x5
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 x3)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x6: B).(\lambda (x7:
+C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6)
+x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c:
+C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13
+x7 H19) in (ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abst) u)))
+(\lambda (d2: C).(csuba g d2 d1)) (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) (\lambda (x9:
+C).(\lambda (H22: (getl n x7 (CHead x9 (Bind Abst) u))).(\lambda (H23: (csuba
+g x9 d1)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst)
+u))) (\lambda (d2: C).(csuba g d2 d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20
+(CHead x9 (Bind Abst) u) n H22) H23)))) H21)))))))) H17)))))) H14)))))))
+H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2)))) H0))))))).
+
+theorem csuba_getl_abbr_rev:
+ \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall
+(i: nat).((getl i c1 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba
+g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 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 Abbr) u1))).(let H0 \def
+(getl_gen_all c1 (CHead d1 (Bind Abbr) u1) i H) in (ex2_ind C (\lambda (e:
+C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u1)))
+(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2:
+(clear x (CHead d1 (Bind Abbr) u1))).(C_ind (\lambda (c: C).((drop i O c1 c)
+\to ((clear c (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2
+c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda
+(n: nat).(\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n)
+(CHead d1 (Bind Abbr) u1))).(clear_gen_sort (CHead d1 (Bind Abbr) u1) n H4
+(\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))))))))) (\lambda (x0: C).(\lambda (_: (((drop i O c1 x0) \to ((clear
+x0 (CHead d1 (Bind Abbr) u1)) \to (\forall (c2: C).((csuba g c2 c1) \to (or
+(ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))))).(\lambda (k: K).(\lambda
+(t: T).(\lambda (H3: (drop i O c1 (CHead x0 k t))).(\lambda (H4: (clear
+(CHead x0 k t) (CHead d1 (Bind Abbr) u1))).(K_ind (\lambda (k0: K).((drop i O
+c1 (CHead x0 k0 t)) \to ((clear (CHead x0 k0 t) (CHead d1 (Bind Abbr) u1))
+\to (\forall (c2: C).((csuba g c2 c1) \to (or (ex2 C (\lambda (d2: C).(getl i
+c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))))))))) (\lambda (b: B).(\lambda (H5: (drop i O c1 (CHead x0 (Bind b)
+t))).(\lambda (H6: (clear (CHead x0 (Bind b) t) (CHead d1 (Bind Abbr)
+u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c]))
+(CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead
+d1 (Bind Abbr) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e:
+C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
+(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
+[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind
+Abbr) u1) (CHead x0 (Bind b) t) (clear_gen_bind b x0 (CHead d1 (Bind Abbr)
+u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0)
+\Rightarrow t0])) (CHead d1 (Bind Abbr) u1) (CHead x0 (Bind b) t)
+(clear_gen_bind b x0 (CHead d1 (Bind Abbr) u1) t H6)) in (\lambda (H10: (eq B
+Abbr b)).(\lambda (H11: (eq C d1 x0)).(\lambda (c2: C).(\lambda (H12: (csuba
+g c2 c1)).(let H13 \def (eq_ind_r T t (\lambda (t0: T).(drop i O c1 (CHead x0
+(Bind b) t0))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b0:
+B).(drop i O c1 (CHead x0 (Bind b0) u1))) H13 Abbr H10) in (let H15 \def
+(eq_ind_r C x0 (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u1))) H14 d1
+H11) in (let H16 \def (csuba_drop_abbr_rev i c1 d1 u1 H15 g c2 H12) in
+(or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H17: (ex2 C (\lambda
+(d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))) (\lambda (x1: C).(\lambda (H18: (drop i O c2 (CHead x1 (Bind Abbr)
+u1))).(\lambda (H19: (csuba g x1 d1)).(or_introl (ex2 C (\lambda (d2:
+C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) x1 (getl_intro i c2
+(CHead x1 (Bind Abbr) u1) (CHead x1 (Bind Abbr) u1) H18 (clear_bind Abbr x1
+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 Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x1:
+C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H18: (drop i O c2 (CHead x1
+(Bind Abst) x2))).(\lambda (H19: (csuba g x1 d1)).(\lambda (H20: (arity g x1
+x2 (asucc g x3))).(\lambda (H21: (arity g d1 u1 x3)).(or_intror (ex2 C
+(\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex4_3_intro C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))) x1 x2 x3
+(getl_intro i c2 (CHead x1 (Bind Abst) x2) (CHead x1 (Bind Abst) x2) H18
+(clear_bind Abst x1 x2)) H19 H20 H21))))))))) H17)) H16)))))))))) H8))
+H7))))) (\lambda (f: F).(\lambda (H5: (drop i O c1 (CHead x0 (Flat f)
+t))).(\lambda (H6: (clear (CHead x0 (Flat f) t) (CHead d1 (Bind Abbr)
+u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c: C).((drop i O c (CHead x0
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 c) \to (or (ex2 C (\lambda
+(d2: C).(getl i c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x1:
+C).((drop n O x1 (CHead x0 (Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1)
+\to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))) (\lambda
+(x1: C).(\lambda (H8: (drop O O x1 (CHead x0 (Flat f) t))).(\lambda (c2:
+C).(\lambda (H9: (csuba g c2 x1)).(let H10 \def (eq_ind C x1 (\lambda (c:
+C).(csuba g c2 c)) H9 (CHead x0 (Flat f) t) (drop_gen_refl x1 (CHead x0 (Flat
+f) t) H8)) in (let H_y \def (clear_flat x0 (CHead d1 (Bind Abbr) u1)
+(clear_gen_flat f x0 (CHead d1 (Bind Abbr) u1) t H6) f t) in (let H11 \def
+(csuba_clear_trans g (CHead x0 (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u1)
+H_y) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead d1 (Bind Abbr) u1)))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x2: C).(\lambda (H12: (csuba g x2 (CHead d1 (Bind Abbr)
+u1))).(\lambda (H13: (clear c2 x2)).(let H_x \def (csuba_gen_abbr_rev g d1 x2
+u1 H12) in (let H14 \def H_x in (or_ind (ex2 C (\lambda (d2: C).(eq C x2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x2 (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H15:
+(ex2 C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(eq C x2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x3: C).(\lambda (H16: (eq C x2 (CHead
+x3 (Bind Abbr) u1))).(\lambda (H17: (csuba g x3 d1)).(let H18 \def (eq_ind C
+x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abbr) u1) H16) in
+(or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1)))
+(\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2))))) (\lambda
+(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C
+(\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)) x3 (getl_intro O c2 (CHead x3 (Bind Abbr) 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 x2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C x2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))) (or (ex2 C (\lambda (d2: C).(getl
+O c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C
+T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H16:
+(eq C x2 (CHead x3 (Bind Abst) x4))).(\lambda (H17: (csuba g x3 d1)).(\lambda
+(H18: (arity g x3 x4 (asucc g x5))).(\lambda (H19: (arity g d1 u1 x5)).(let
+H20 \def (eq_ind C x2 (\lambda (c: C).(clear c2 c)) H13 (CHead x3 (Bind Abst)
+x4) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))) x3 x4 x5 (getl_intro O c2 (CHead x3 (Bind Abst)
+x4) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14)))))) H11))))))))
+(\lambda (n: nat).(\lambda (H8: ((\forall (x1: C).((drop n O x1 (CHead x0
+(Flat f) t)) \to (\forall (c2: C).((csuba g c2 x1) \to (or (ex2 C (\lambda
+(d2: C).(getl n c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n
+c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))))))))).(\lambda (x1: C).(\lambda (H9: (drop (S
+n) O x1 (CHead x0 (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g c2
+x1)).(let H11 \def (drop_clear x1 (CHead x0 (Flat f) t) n H9) in (ex2_3_ind B
+C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x1 (CHead e (Bind
+b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead
+x0 (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (H12: (clear x1
+(CHead x3 (Bind x2) x4))).(\lambda (H13: (drop n O x3 (CHead x0 (Flat f)
+t))).(let H14 \def (csuba_clear_trans g x1 c2 H10 (CHead x3 (Bind x2) x4)
+H12) in (ex2_ind C (\lambda (e2: C).(csuba g e2 (CHead x3 (Bind x2) x4)))
+(\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2
+(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))))) (\lambda (x5: C).(\lambda (H15: (csuba g x5 (CHead x3 (Bind x2)
+x4))).(\lambda (H16: (clear c2 x5)).(let H_x \def (csuba_gen_bind_rev g x2 x3
+x5 x4 H15) in (let H17 \def H_x in (ex2_3_ind B C T (\lambda (b2: B).(\lambda
+(e2: C).(\lambda (v2: T).(eq C x5 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 x3)))) (or (ex2 C (\lambda
+(d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
+d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl
+(S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (x6: B).(\lambda (x7:
+C).(\lambda (x8: T).(\lambda (H18: (eq C x5 (CHead x7 (Bind x6)
+x8))).(\lambda (H19: (csuba g x7 x3)).(let H20 \def (eq_ind C x5 (\lambda (c:
+C).(clear c2 c)) H16 (CHead x7 (Bind x6) x8) H18) in (let H21 \def (H8 x3 H13
+x7 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (or
+(ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) (\lambda (H22:
+(ex2 C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1)))).(ex2_ind C (\lambda (d2: C).(getl n x7 (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) (or (ex2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))) (\lambda (x9: C).(\lambda (H23: (getl n x7
+(CHead x9 (Bind Abbr) u1))).(\lambda (H24: (csuba g x9 d1)).(or_introl (ex2 C
+(\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(getl (S n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C (\lambda (d2:
+C).(getl (S n) c2 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1)) x9 (getl_clear_bind x6 c2 x7 x8 H20 (CHead x9 (Bind Abbr) u1) n H23)
+H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(getl n x7 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))).(ex4_3_ind C T
+A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x7 (CHead d2
+(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))
+(\lambda (x9: C).(\lambda (x10: T).(\lambda (x11: A).(\lambda (H23: (getl n
+x7 (CHead x9 (Bind Abst) x10))).(\lambda (H24: (csuba g x9 d1)).(\lambda
+(H25: (arity g x9 x10 (asucc g x11))).(\lambda (H26: (arity g d1 u1
+x11)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr)
+u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S
+n) c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))) x9 x10 x11 (getl_clear_bind x6 c2 x7 x8 H20
+(CHead x9 (Bind Abst) x10) n H23) H24 H25 H26))))))))) H22)) H21))))))))
+H17)))))) H14))))))) H11)))))))) i) H7))))) k H3 H4))))))) x H1 H2))))
+H0))))))).
+
+theorem sn3_gen_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
+(THead (Bind b) u t)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Bind b) u t))).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
+T).(sn3 c t0)) (land (sn3 c u) (sn3 (CHead c (Bind b) u) t)) (\lambda (y:
+T).(\lambda (H0: (sn3 c y)).(unintro T t (\lambda (t0: T).((eq T y (THead
+(Bind b) u t0)) \to (land (sn3 c u) (sn3 (CHead c (Bind b) u) t0)))) (unintro
+T u (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind b) t0 x)) \to
+(land (sn3 c t0) (sn3 (CHead c (Bind b) t0) x))))) (sn3_ind c (\lambda (t0:
+T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Bind b) x x0)) \to
+(land (sn3 c x) (sn3 (CHead c (Bind b) x) 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 (Bind b) x x0)) \to (land (sn3 c x) (sn3 (CHead c
+(Bind b) x) x0)))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T
+t1 (THead (Bind b) x x0))).(let H4 \def (eq_ind T t1 (\lambda (t0:
+T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c
+t0 t2) \to (\forall (x1: T).(\forall (x2: T).((eq T t2 (THead (Bind b) x1
+x2)) \to (land (sn3 c x1) (sn3 (CHead c (Bind b) x1) x2))))))))) H2 (THead
+(Bind b) x x0) H3) in (let H5 \def (eq_ind T t1 (\lambda (t0: T).(\forall
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
+(sn3 c t2))))) H1 (THead (Bind b) x x0) H3) in (conj (sn3 c x) (sn3 (CHead c
+(Bind b) x) 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 (Bind b) t2 x0) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
+b) t2 x0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x |
+(TLRef _) \Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind b) x
+x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
+T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T
+x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b)
+t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (and_ind (sn3 c t2)
+(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda
+(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b)
+x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H7: (pr3 (CHead c (Bind b) x) x0 t2)).(let H8 \def (H4
+(THead (Bind b) x t2) (\lambda (H8: (eq T (THead (Bind b) x x0) (THead (Bind
+b) x t2))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind b) x
+x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0:
+T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T
+t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in
+(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0
+t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (and_ind (sn3 c x) (sn3
+(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c
+x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y
+H0))))) H))))).
+
+theorem sn3_gen_head:
+ \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
+(THead k u t)) \to (sn3 c u)))))
+\def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (c: C).(\forall (u:
+T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
+B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
+(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
+(and_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
+c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
+F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
+(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in
+(and_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
+(sn3 c t)).H1)) H0)))))))) k).
+
+theorem sn3_gen_cflat:
+ \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 (CHead
+c (Flat f) u) t) \to (sn3 c t)))))
+\def
+ \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
+(sn3 (CHead c (Flat f) u) t)).(sn3_ind (CHead c (Flat f) u) (\lambda (t0:
+T).(sn3 c t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
+(sn3 (CHead c (Flat f) u) t2)))))).(\lambda (H1: ((\forall (t2: T).((((eq T
+t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Flat f) u) t1 t2) \to
+(sn3 c t2)))))).(sn3_sing c t1 (\lambda (t2: T).(\lambda (H2: (((eq T t1 t2)
+\to (\forall (P: Prop).P)))).(\lambda (H3: (pr3 c t1 t2)).(H1 t2 H2
+(pr3_cflat c t1 t2 H3 f u))))))))) t H))))).
+
+theorem sn3_cflat:
+ \forall (c: C).(\forall (t: T).((sn3 c t) \to (\forall (f: F).(\forall (u:
+T).(sn3 (CHead c (Flat f) u) t)))))
+\def
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(\lambda (f:
+F).(\lambda (u: T).(sn3_ind c (\lambda (t0: T).(sn3 (CHead c (Flat f) u) 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
+(sn3 (CHead c (Flat f) u) t2)))))).(sn3_pr2_intro (CHead c (Flat f) u) t1
+(\lambda (t2: T).(\lambda (H2: (((eq T t1 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H3: (pr2 (CHead c (Flat f) u) t1 t2)).(H1 t2 H2
+(pr3_pr2 c t1 t2 (pr2_gen_cflat f c u t1 t2 H3)))))))))) t H))))).
+
+theorem sn3_shift:
+ \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c
+(THead (Bind b) v t)) \to (sn3 (CHead c (Bind b) v) t)))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
+H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
+(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
+v) t)).H2)) H0))))))).
+
+theorem sn3_change:
+ \forall (b: B).((not (eq B b Abbr)) \to (\forall (c: C).(\forall (v1:
+T).(\forall (t: T).((sn3 (CHead c (Bind b) v1) t) \to (\forall (v2: T).(sn3
+(CHead c (Bind b) v2) t)))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abbr))).(\lambda (c: C).(\lambda
+(v1: T).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c (Bind b) v1) t)).(\lambda
+(v2: T).(sn3_ind (CHead c (Bind b) v1) (\lambda (t0: T).(sn3 (CHead c (Bind
+b) v2) t0)) (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2)
+\to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to (sn3
+(CHead c (Bind b) v1) t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) v1) t1 t2) \to
+(sn3 (CHead c (Bind b) v2) t2)))))).(sn3_pr2_intro (CHead c (Bind b) v2) t1
+(\lambda (t2: T).(\lambda (H3: (((eq T t1 t2) \to (\forall (P:
+Prop).P)))).(\lambda (H4: (pr2 (CHead c (Bind b) v2) t1 t2)).(H2 t2 H3
+(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
+v1)))))))))) t H0))))))).
+
+theorem sn3_cpr3_trans:
+ \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
+(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
+t)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr3 c u1
+u2)).(\lambda (k: K).(\lambda (t: T).(\lambda (H0: (sn3 (CHead c k u1)
+t)).(sn3_ind (CHead c k u1) (\lambda (t0: T).(sn3 (CHead c k u2) t0))
+(\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall
+(P: Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u1)
+t2)))))).(\lambda (H2: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 (CHead c k u1) t1 t2) \to (sn3 (CHead c k u2)
+t2)))))).(sn3_sing (CHead c k u2) t1 (\lambda (t2: T).(\lambda (H3: (((eq T
+t1 t2) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr3 (CHead c k u2) t1
+t2)).(H2 t2 H3 (pr3_pr3_pr3_t c u1 u2 H t1 t2 k H4))))))))) t H0))))))).
+
+theorem sn3_bind:
+ \forall (b: B).(\forall (c: C).(\forall (u: T).((sn3 c u) \to (\forall (t:
+T).((sn3 (CHead c (Bind b) u) t) \to (sn3 c (THead (Bind b) u t)))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sn3 c
+u)).(sn3_ind c (\lambda (t: T).(\forall (t0: T).((sn3 (CHead c (Bind b) t)
+t0) \to (sn3 c (THead (Bind b) 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 (CHead c
+(Bind b) t2) t) \to (sn3 c (THead (Bind b) t2 t))))))))).(\lambda (t:
+T).(\lambda (H2: (sn3 (CHead c (Bind b) t1) t)).(sn3_ind (CHead c (Bind b)
+t1) (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (\lambda (t2:
+T).(\lambda (H3: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind b)
+t1) t3)))))).(\lambda (H4: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 c (THead (Bind b)
+t1 t3))))))).(sn3_sing c (THead (Bind b) t1 t2) (\lambda (t3: T).(\lambda
+(H5: (((eq T (THead (Bind b) t1 t2) t3) \to (\forall (P: Prop).P)))).(\lambda
+(H6: (pr3 c (THead (Bind b) t1 t2) t3)).(let H_x \def (bind_dec_not b Abst)
+in (let H7 \def H_x in (or_ind (eq B b Abst) (not (eq B b Abst)) (sn3 c t3)
+(\lambda (H8: (eq B b Abst)).(let H9 \def (eq_ind B b (\lambda (b0: B).(pr3 c
+(THead (Bind b0) t1 t2) t3)) H6 Abst H8) in (let H10 \def (eq_ind B b
+(\lambda (b0: B).((eq T (THead (Bind b0) t1 t2) t3) \to (\forall (P:
+Prop).P))) H5 Abst H8) in (let H11 \def (eq_ind B b (\lambda (b0: B).(\forall
+(t4: T).((((eq T t2 t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
+b0) t1) t2 t4) \to (sn3 c (THead (Bind b0) t1 t4)))))) H4 Abst H8) in (let
+H12 \def (eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T t2 t4) \to
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) t2 t4) \to (sn3
+(CHead c (Bind b0) t1) t4))))) H3 Abst H8) in (let H13 \def (eq_ind B b
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
+\to ((pr3 c t1 t4) \to (\forall (t0: T).((sn3 (CHead c (Bind b0) t4) t0) \to
+(sn3 c (THead (Bind b0) t4 t0)))))))) H1 Abst H8) in (let H14 \def
+(pr3_gen_abst c t1 t2 t3 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 t4))))) (sn3 c t3) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H15: (eq T t3 (THead (Bind Abst) x0
+x1))).(\lambda (H16: (pr3 c t1 x0)).(\lambda (H17: ((\forall (b0: B).(\forall
+(u0: T).(pr3 (CHead c (Bind b0) u0) t2 x1))))).(let H18 \def (eq_ind T t3
+(\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2) t0) \to (\forall (P:
+Prop).P))) H10 (THead (Bind Abst) x0 x1) H15) in (eq_ind_r T (THead (Bind
+Abst) x0 x1) (\lambda (t0: T).(sn3 c t0)) (let H_x0 \def (term_dec t1 x0) in
+(let H19 \def H_x0 in (or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P:
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H20: (eq T t1 x0)).(let
+H21 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T (THead (Bind Abst) t1 t2)
+(THead (Bind Abst) t0 x1)) \to (\forall (P: Prop).P))) H18 t1 H20) in (let
+H22 \def (eq_ind_r T x0 (\lambda (t0: T).(pr3 c t1 t0)) H16 t1 H20) in
+(eq_ind T t1 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t0 x1))) (let H_x1
+\def (term_dec t2 x1) in (let H23 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abst) t1 x1)) (\lambda
+(H24: (eq T t2 x1)).(let H25 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T
+(THead (Bind Abst) t1 t2) (THead (Bind Abst) t1 t0)) \to (\forall (P:
+Prop).P))) H21 t2 H24) in (let H26 \def (eq_ind_r T x1 (\lambda (t0:
+T).(\forall (b0: B).(\forall (u0: T).(pr3 (CHead c (Bind b0) u0) t2 t0))))
+H17 t2 H24) in (eq_ind T t2 (\lambda (t0: T).(sn3 c (THead (Bind Abst) t1
+t0))) (H25 (refl_equal T (THead (Bind Abst) t1 t2)) (sn3 c (THead (Bind Abst)
+t1 t2))) x1 H24)))) (\lambda (H24: (((eq T t2 x1) \to (\forall (P:
+Prop).P)))).(H11 x1 H24 (H17 Abst t1))) H23))) x0 H20)))) (\lambda (H20:
+(((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x1 \def (term_dec t2 x1)
+in (let H21 \def H_x1 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P:
+Prop).P)) (sn3 c (THead (Bind Abst) x0 x1)) (\lambda (H22: (eq T t2 x1)).(let
+H23 \def (eq_ind_r T x1 (\lambda (t0: T).(\forall (b0: B).(\forall (u0:
+T).(pr3 (CHead c (Bind b0) u0) t2 t0)))) H17 t2 H22) in (eq_ind T t2 (\lambda
+(t0: T).(sn3 c (THead (Bind Abst) x0 t0))) (H13 x0 H20 H16 t2 (sn3_cpr3_trans
+c t1 x0 H16 (Bind Abst) t2 (sn3_sing (CHead c (Bind Abst) t1) t2 H12))) x1
+H22))) (\lambda (H22: (((eq T t2 x1) \to (\forall (P: Prop).P)))).(H13 x0 H20
+H16 x1 (sn3_cpr3_trans c t1 x0 H16 (Bind Abst) x1 (H12 x1 H22 (H17 Abst
+t1))))) H21)))) H19))) t3 H15))))))) H14)))))))) (\lambda (H8: (not (eq B b
+Abst))).(let H_x0 \def (pr3_gen_bind b H8 c t1 t2 t3 H6) in (let H9 \def H_x0
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
+b) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) t2 t4)))) (pr3 (CHead c (Bind
+b) t1) t2 (lift (S O) O t3)) (sn3 c t3) (\lambda (H10: (ex3_2 T T (\lambda
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
+(CHead c (Bind b) t1) t2 t4))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3 (CHead c (Bind b)
+t1) t2 t4))) (sn3 c t3) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq
+T t3 (THead (Bind b) x0 x1))).(\lambda (H12: (pr3 c t1 x0)).(\lambda (H13:
+(pr3 (CHead c (Bind b) t1) t2 x1)).(let H14 \def (eq_ind T t3 (\lambda (t0:
+T).((eq T (THead (Bind b) t1 t2) t0) \to (\forall (P: Prop).P))) H5 (THead
+(Bind b) x0 x1) H11) in (eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0:
+T).(sn3 c t0)) (let H_x1 \def (term_dec t1 x0) in (let H15 \def H_x1 in
+(or_ind (eq T t1 x0) ((eq T t1 x0) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Bind b) x0 x1)) (\lambda (H16: (eq T t1 x0)).(let H17 \def (eq_ind_r T x0
+(\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t0 x1)) \to
+(\forall (P: Prop).P))) H14 t1 H16) in (let H18 \def (eq_ind_r T x0 (\lambda
+(t0: T).(pr3 c t1 t0)) H12 t1 H16) in (eq_ind T t1 (\lambda (t0: T).(sn3 c
+(THead (Bind b) t0 x1))) (let H_x2 \def (term_dec t2 x1) in (let H19 \def
+H_x2 in (or_ind (eq T t2 x1) ((eq T t2 x1) \to (\forall (P: Prop).P)) (sn3 c
+(THead (Bind b) t1 x1)) (\lambda (H20: (eq T t2 x1)).(let H21 \def (eq_ind_r
+T x1 (\lambda (t0: T).((eq T (THead (Bind b) t1 t2) (THead (Bind b) t1 t0))
+\to (\forall (P: Prop).P))) H17 t2 H20) in (let H22 \def (eq_ind_r T x1
+(\lambda (t0: T).(pr3 (CHead c (Bind b) t1) t2 t0)) H13 t2 H20) in (eq_ind T
+t2 (\lambda (t0: T).(sn3 c (THead (Bind b) t1 t0))) (H21 (refl_equal T (THead
+(Bind b) t1 t2)) (sn3 c (THead (Bind b) t1 t2))) x1 H20)))) (\lambda (H20:
+(((eq T t2 x1) \to (\forall (P: Prop).P)))).(H4 x1 H20 H13)) H19))) x0
+H16)))) (\lambda (H16: (((eq T t1 x0) \to (\forall (P: Prop).P)))).(let H_x2
+\def (term_dec t2 x1) in (let H17 \def H_x2 in (or_ind (eq T t2 x1) ((eq T t2
+x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind b) x0 x1)) (\lambda (H18:
+(eq T t2 x1)).(let H19 \def (eq_ind_r T x1 (\lambda (t0: T).(pr3 (CHead c
+(Bind b) t1) t2 t0)) H13 t2 H18) in (eq_ind T t2 (\lambda (t0: T).(sn3 c
+(THead (Bind b) x0 t0))) (H1 x0 H16 H12 t2 (sn3_cpr3_trans c t1 x0 H12 (Bind
+b) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3))) x1 H18))) (\lambda (H18: (((eq
+T t2 x1) \to (\forall (P: Prop).P)))).(H1 x0 H16 H12 x1 (sn3_cpr3_trans c t1
+x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
+(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
+t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
+(Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0:
+T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12:
+(pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1)
+t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl
+c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
+
+theorem sn3_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
+t)) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
+(Bind Abst) w t))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
+(Bind Abbr) v t))).(insert_eq T (THead (Bind Abbr) v t) (\lambda (t0: T).(sn3
+c t0)) (\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) v (THead
+(Bind Abst) w t))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t
+(\lambda (t0: T).((eq T y (THead (Bind Abbr) v t0)) \to (\forall (w: T).((sn3
+c w) \to (sn3 c (THead (Flat Appl) v (THead (Bind Abst) w t0))))))) (unintro
+T v (\lambda (t0: T).(\forall (x: T).((eq T y (THead (Bind Abbr) t0 x)) \to
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) t0 (THead (Bind
+Abst) w x)))))))) (sn3_ind c (\lambda (t0: T).(\forall (x: T).(\forall (x0:
+T).((eq T t0 (THead (Bind Abbr) x x0)) \to (\forall (w: T).((sn3 c w) \to
+(sn3 c (THead (Flat Appl) x (THead (Bind Abst) w 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 (Bind Abbr) x x0)) \to
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) x (THead (Bind Abst)
+w x0))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t1
+(THead (Bind Abbr) x x0))).(\lambda (w: T).(\lambda (H4: (sn3 c w)).(let H5
+\def (eq_ind T t1 (\lambda (t0: T).(\forall (t2: T).((((eq T t0 t2) \to
+(\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (\forall (x1: T).(\forall (x2:
+T).((eq T t2 (THead (Bind Abbr) x1 x2)) \to (\forall (w0: T).((sn3 c w0) \to
+(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) w0 x2)))))))))))) H2 (THead
+(Bind Abbr) x x0) H3) in (let H6 \def (eq_ind T t1 (\lambda (t0: T).(\forall
+(t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to
+(sn3 c t2))))) H1 (THead (Bind Abbr) x x0) H3) in (sn3_ind c (\lambda (t0:
+T).(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t0 x0)))) (\lambda (t2:
+T).(\lambda (H7: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P:
+Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H8: ((\forall
+(t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to
+(sn3 c (THead (Flat Appl) x (THead (Bind Abst) t3 x0)))))))).(sn3_pr2_intro c
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (\lambda (t3: T).(\lambda
+(H9: (((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t3) \to (\forall
+(P: Prop).P)))).(\lambda (H10: (pr2 c (THead (Flat Appl) x (THead (Bind Abst)
+t2 x0)) t3)).(let H11 \def (pr2_gen_appl c x (THead (Bind Abst) t2 x0) t3
+H10) 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 x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind Abst) t2 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 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 (THead (Bind Abst) t2 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 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 t3)
+(\lambda (H12: (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 x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0)
+t4))))).(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 x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) t2 x0) t4))) (sn3
+c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H13: (eq T t3 (THead (Flat
+Appl) x1 x2))).(\lambda (H14: (pr2 c x x1)).(\lambda (H15: (pr2 c (THead
+(Bind Abst) t2 x0) x2)).(let H16 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
+Prop).P))) H9 (THead (Flat Appl) x1 x2) H13) in (eq_ind_r T (THead (Flat
+Appl) x1 x2) (\lambda (t0: T).(sn3 c t0)) (let H17 \def (pr2_gen_abst c t2 x0
+x2 H15) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead
+(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t2 u2)))
+(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c (Bind b) u) x0 t4))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3:
+T).(\lambda (x4: T).(\lambda (H18: (eq T x2 (THead (Bind Abst) x3
+x4))).(\lambda (H19: (pr2 c t2 x3)).(\lambda (H20: ((\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 x4))))).(let H21 \def (eq_ind T x2
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0))
+(THead (Flat Appl) x1 t0)) \to (\forall (P: Prop).P))) H16 (THead (Bind Abst)
+x3 x4) H18) in (eq_ind_r T (THead (Bind Abst) x3 x4) (\lambda (t0: T).(sn3 c
+(THead (Flat Appl) x1 t0))) (let H_x \def (term_dec t2 x3) in (let H22 \def
+H_x in (or_ind (eq T t2 x3) ((eq T t2 x3) \to (\forall (P: Prop).P)) (sn3 c
+(THead (Flat Appl) x1 (THead (Bind Abst) x3 x4))) (\lambda (H23: (eq T t2
+x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0: T).((eq T (THead (Flat Appl)
+x (THead (Bind Abst) t2 x0)) (THead (Flat Appl) x1 (THead (Bind Abst) t0
+x4))) \to (\forall (P: Prop).P))) H21 t2 H23) in (let H25 \def (eq_ind_r T x3
+(\lambda (t0: T).(pr2 c t2 t0)) H19 t2 H23) in (eq_ind T t2 (\lambda (t0:
+T).(sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t0 x4)))) (let H_x0 \def
+(term_dec x x1) in (let H26 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
+(\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind Abst) t2
+x4))) (\lambda (H27: (eq T x x1)).(let H28 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) (THead (Flat Appl)
+t0 (THead (Bind Abst) t2 x4))) \to (\forall (P: Prop).P))) H24 x H27) in (let
+H29 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H27) in (eq_ind
+T x (\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) t2
+x4)))) (let H_x1 \def (term_dec x0 x4) in (let H30 \def H_x1 in (or_ind (eq T
+x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x
+(THead (Bind Abst) t2 x4))) (\lambda (H31: (eq T x0 x4)).(let H32 \def
+(eq_ind_r T x4 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind
+Abst) t2 x0)) (THead (Flat Appl) x (THead (Bind Abst) t2 t0))) \to (\forall
+(P: Prop).P))) H28 x0 H31) in (let H33 \def (eq_ind_r T x4 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0
+H31) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x (THead
+(Bind Abst) t2 t0)))) (H32 (refl_equal T (THead (Flat Appl) x (THead (Bind
+Abst) t2 x0))) (sn3 c (THead (Flat Appl) x (THead (Bind Abst) t2 x0)))) x4
+H31)))) (\lambda (H31: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
+(Bind Abbr) x x4) (\lambda (H32: (eq T (THead (Bind Abbr) x x0) (THead (Bind
+Abbr) x x4))).(\lambda (P: Prop).(let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x x4) H32) in (let H34 \def (eq_ind_r T x4
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H31 x0 H33) in
+(let H35 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H33) in (H34 (refl_equal T x0)
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
+(Bind Abbr) x x4)) t2 (sn3_sing c t2 H7))) H30))) x1 H27)))) (\lambda (H27:
+(((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind Abbr) x1 x4)
+(\lambda (H28: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x1
+x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
+\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x1 x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x1 x4) H28) in (\lambda (H31: (eq T x
+x1)).(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (let H33 \def
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
+H27 x H31) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
+x H31) in (H33 (refl_equal T x) P)))))) H29)))) (pr3_head_12 c x x1 (pr3_pr2
+c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
+Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) t2 (sn3_sing c t2
+H7))) H26))) x3 H23)))) (\lambda (H23: (((eq T t2 x3) \to (\forall (P:
+Prop).P)))).(let H_x0 \def (term_dec x x1) in (let H24 \def H_x0 in (or_ind
+(eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl)
+x1 (THead (Bind Abst) x3 x4))) (\lambda (H25: (eq T x x1)).(let H26 \def
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14 x H25) in (eq_ind T x
+(\lambda (t0: T).(sn3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))))
+(let H_x1 \def (term_dec x0 x4) in (let H27 \def H_x1 in (or_ind (eq T x0 x4)
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x (THead
+(Bind Abst) x3 x4))) (\lambda (H28: (eq T x0 x4)).(let H29 \def (eq_ind_r T
+x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+x0 t0)))) H20 x0 H28) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
+Appl) x (THead (Bind Abst) x3 t0)))) (H8 x3 H23 (pr3_pr2 c t2 x3 H19)) x4
+H28))) (\lambda (H28: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H5 (THead
+(Bind Abbr) x x4) (\lambda (H29: (eq T (THead (Bind Abbr) x x0) (THead (Bind
+Abbr) x x4))).(\lambda (P: Prop).(let H30 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x x4) H29) in (let H31 \def (eq_ind_r T x4
+(\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H28 x0 H30) in
+(let H32 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u:
+T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H30) in (H31 (refl_equal T x0)
+P)))))) (pr3_pr2 c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4)
+(pr2_head_2 c x x0 x4 (Bind Abbr) (H20 Abbr x))) x x4 (refl_equal T (THead
+(Bind Abbr) x x4)) x3 (H7 x3 H23 (pr3_pr2 c t2 x3 H19)))) H27))) x1 H25)))
+(\lambda (H25: (((eq T x x1) \to (\forall (P: Prop).P)))).(H5 (THead (Bind
+Abbr) x1 x4) (\lambda (H26: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr)
+x1 x4))).(\lambda (P: Prop).(let H27 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _)
+\Rightarrow x | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x1 x4) H26) in ((let H28 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind
+Abbr) x x0) (THead (Bind Abbr) x1 x4) H26) in (\lambda (H29: (eq T x
+x1)).(let H30 \def (eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x0 t0)))) H20 x0 H28) in (let H31 \def
+(eq_ind_r T x1 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
+H25 x H29) in (let H32 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H14
+x H29) in (H31 (refl_equal T x) P)))))) H27)))) (pr3_head_12 c x x1 (pr3_pr2
+c x x1 H14) (Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x1) x0 x4 (H20
+Abbr x1))) x1 x4 (refl_equal T (THead (Bind Abbr) x1 x4)) x3 (H7 x3 H23
+(pr3_pr2 c t2 x3 H19)))) H24)))) H22))) x2 H18))))))) H17)) t3 H13)))))))
+H12)) (\lambda (H12: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) t2 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 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))))))))).(ex4_4_ind T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) t2 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 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 t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (H13: (eq T (THead (Bind Abst) t2 x0) (THead
+(Bind Abst) x1 x2))).(\lambda (H14: (eq T t3 (THead (Bind Abbr) x3
+x4))).(\lambda (H15: (pr2 c x x3)).(\lambda (H16: ((\forall (b: B).(\forall
+(u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H17 \def (eq_ind T t3
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0)
+\to (\forall (P: Prop).P))) H9 (THead (Bind Abbr) x3 x4) H14) in (eq_ind_r T
+(THead (Bind Abbr) x3 x4) (\lambda (t0: T).(sn3 c t0)) (let H18 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _) \Rightarrow t0]))
+(THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t0)
+\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind Abst) x1 x2) H13) in
+(\lambda (_: (eq T t2 x1)).(let H21 \def (eq_ind_r T x2 (\lambda (t0:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t0 x4)))) H16 x0
+H19) in (let H_x \def (term_dec x x3) in (let H22 \def H_x in (or_ind (eq T x
+x3) ((eq T x x3) \to (\forall (P: Prop).P)) (sn3 c (THead (Bind Abbr) x3 x4))
+(\lambda (H23: (eq T x x3)).(let H24 \def (eq_ind_r T x3 (\lambda (t0:
+T).(pr2 c x t0)) H15 x H23) in (eq_ind T x (\lambda (t0: T).(sn3 c (THead
+(Bind Abbr) t0 x4))) (let H_x0 \def (term_dec x0 x4) in (let H25 \def H_x0 in
+(or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Bind Abbr) x x4)) (\lambda (H26: (eq T x0 x4)).(let H27 \def (eq_ind_r T x4
+(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x0
+t0)))) H21 x0 H26) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Bind Abbr)
+x t0))) (sn3_sing c (THead (Bind Abbr) x x0) H6) x4 H26))) (\lambda (H26:
+(((eq T x0 x4) \to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x x4)
+(\lambda (H27: (eq T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x
+x4))).(\lambda (P: Prop).(let H28 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x x4) H27) in (let H29 \def (eq_ind_r T x4 (\lambda (t0:
+T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H26 x0 H28) in (let H30 \def
+(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) x0 t0)))) H21 x0 H28) in (H29 (refl_equal T x0) P)))))) (pr3_pr2
+c (THead (Bind Abbr) x x0) (THead (Bind Abbr) x x4) (pr2_head_2 c x x0 x4
+(Bind Abbr) (H21 Abbr x))))) H25))) x3 H23))) (\lambda (H23: (((eq T x x3)
+\to (\forall (P: Prop).P)))).(H6 (THead (Bind Abbr) x3 x4) (\lambda (H24: (eq
+T (THead (Bind Abbr) x x0) (THead (Bind Abbr) x3 x4))).(\lambda (P:
+Prop).(let H25 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
+(THead _ t0 _) \Rightarrow t0])) (THead (Bind Abbr) x x0) (THead (Bind Abbr)
+x3 x4) H24) in ((let H26 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _)
+\Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind Abbr) x x0)
+(THead (Bind Abbr) x3 x4) H24) in (\lambda (H27: (eq T x x3)).(let H28 \def
+(eq_ind_r T x4 (\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c
+(Bind b) u) x0 t0)))) H21 x0 H26) in (let H29 \def (eq_ind_r T x3 (\lambda
+(t0: T).((eq T x t0) \to (\forall (P0: Prop).P0))) H23 x H27) in (let H30
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr2 c x t0)) H15 x H27) in (H29
+(refl_equal T x) P)))))) H25)))) (pr3_head_12 c x x3 (pr3_pr2 c x x3 H15)
+(Bind Abbr) x0 x4 (pr3_pr2 (CHead c (Bind Abbr) x3) x0 x4 (H21 Abbr x3)))))
+H22)))))) H18)) t3 H14)))))))))) H12)) (\lambda (H12: (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\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) t2 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 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 (Bind Abst) t2 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 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 t3)
+(\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
+(x5: T).(\lambda (x6: T).(\lambda (H13: (not (eq B x1 Abst))).(\lambda (H14:
+(eq T (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3))).(\lambda (H15: (eq
+T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda
+(_: (pr2 c x x5)).(\lambda (H17: (pr2 c x2 x6)).(\lambda (H18: (pr2 (CHead c
+(Bind x1) x6) x3 x4)).(let H19 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind Abst) t2 x0)) t0) \to (\forall (P:
+Prop).P))) H9 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))
+H15) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)) (\lambda (t0: T).(sn3 c t0)) (let H20 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abst |
+(TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in ((let H21
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ t0 _)
+\Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14) in
+((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind Abst) t2 x0) (THead (Bind x1) x2 x3) H14)
+in (\lambda (H23: (eq T t2 x2)).(\lambda (H24: (eq B Abst x1)).(let H25 \def
+(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H18 x0
+H22) in (let H26 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H17 t2
+H23) in (let H27 \def (eq_ind_r B x1 (\lambda (b: B).(pr2 (CHead c (Bind b)
+x6) x0 x4)) H25 Abst H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b:
+B).(not (eq B b Abst))) H13 Abst H24) in (eq_ind B Abst (\lambda (b: B).(sn3
+c (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (let H29
+\def (match (H28 (refl_equal B Abst)) in False return (\lambda (_:
+False).(sn3 c (THead (Bind Abst) x6 (THead (Flat Appl) (lift (S O) O x5)
+x4)))) with []) in H29) x1 H24)))))))) H21)) H20)) t3 H15)))))))))))))) H12))
+H11))))))))) w H4))))))))))) y H0))))) H)))).
+
+theorem nf3_appl_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) w)) \to (\forall (v: T).((sn3 c (THead (Flat Appl) v
+(lift (S i) O w))) \to (sn3 c (THead (Flat Appl) v (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (v: T).(\lambda (H0: (sn3 c
+(THead (Flat Appl) v (lift (S i) O w)))).(insert_eq T (THead (Flat Appl) v
+(lift (S i) O w)) (\lambda (t: T).(sn3 c t)) (sn3 c (THead (Flat Appl) v
+(TLRef i))) (\lambda (y: T).(\lambda (H1: (sn3 c y)).(unintro T v (\lambda
+(t: T).((eq T y (THead (Flat Appl) t (lift (S i) O w))) \to (sn3 c (THead
+(Flat Appl) t (TLRef i))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).((eq T
+t (THead (Flat Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x
+(TLRef i)))))) (\lambda (t1: T).(\lambda (H2: ((\forall (t2: T).((((eq T t1
+t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c
+t2)))))).(\lambda (H3: ((\forall (t2: T).((((eq T t1 t2) \to (\forall (P:
+Prop).P))) \to ((pr3 c t1 t2) \to (\forall (x: T).((eq T t2 (THead (Flat
+Appl) x (lift (S i) O w))) \to (sn3 c (THead (Flat Appl) x (TLRef
+i)))))))))).(\lambda (x: T).(\lambda (H4: (eq T t1 (THead (Flat Appl) x (lift
+(S i) O w)))).(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 (\forall
+(x0: T).((eq T t2 (THead (Flat Appl) x0 (lift (S i) O w))) \to (sn3 c (THead
+(Flat Appl) x0 (TLRef i))))))))) H3 (THead (Flat Appl) x (lift (S i) O w))
+H4) 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 c t t2) \to (sn3 c t2))))) H2
+(THead (Flat Appl) x (lift (S i) O w)) H4) in (sn3_pr2_intro c (THead (Flat
+Appl) x (TLRef i)) (\lambda (t2: T).(\lambda (H7: (((eq T (THead (Flat Appl)
+x (TLRef i)) t2) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr2 c (THead
+(Flat Appl) x (TLRef i)) t2)).(let H9 \def (pr2_gen_appl c x (TLRef i) t2 H8)
+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 (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 x
+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 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 (H10: (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 (TLRef i) t3))))).(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 (TLRef i) t3))) (sn3 c t2) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H11: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr2 c
+x x0)).(\lambda (H13: (pr2 c (TLRef i) x1)).(let H14 \def (eq_ind T t2
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P:
+Prop).P))) H7 (THead (Flat Appl) x0 x1) H11) in (eq_ind_r T (THead (Flat
+Appl) x0 x1) (\lambda (t: T).(sn3 c t)) (let H15 \def (pr2_gen_lref c x1 i
+H13) in (or_ind (eq T x1 (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 x1 (lift (S i) O u))))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda (H16:
+(eq T x1 (TLRef i))).(let H17 \def (eq_ind T x1 (\lambda (t: T).((eq T (THead
+(Flat Appl) x (TLRef i)) (THead (Flat Appl) x0 t)) \to (\forall (P:
+Prop).P))) H14 (TLRef i) H16) in (eq_ind_r T (TLRef i) (\lambda (t: T).(sn3 c
+(THead (Flat Appl) x0 t))) (let H_x \def (term_dec x x0) in (let H18 \def H_x
+in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c (THead
+(Flat Appl) x0 (TLRef i))) (\lambda (H19: (eq T x x0)).(let H20 \def
+(eq_ind_r T x0 (\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead
+(Flat Appl) t (TLRef i))) \to (\forall (P: Prop).P))) H17 x H19) in (let H21
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H19) in (eq_ind T x
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (TLRef i)))) (H20 (refl_equal T
+(THead (Flat Appl) x (TLRef i))) (sn3 c (THead (Flat Appl) x (TLRef i)))) x0
+H19)))) (\lambda (H19: (((eq T x x0) \to (\forall (P: Prop).P)))).(H5 (THead
+(Flat Appl) x0 (lift (S i) O w)) (\lambda (H20: (eq T (THead (Flat Appl) x
+(lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O w)))).(\lambda (P:
+Prop).(let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x |
+(THead _ t _) \Rightarrow t])) (THead (Flat Appl) x (lift (S i) O w)) (THead
+(Flat Appl) x0 (lift (S i) O w)) H20) in (let H22 \def (eq_ind_r T x0
+(\lambda (t: T).((eq T x t) \to (\forall (P0: Prop).P0))) H19 x H21) in (let
+H23 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H21) in (H22
+(refl_equal T x) P)))))) (pr3_pr2 c (THead (Flat Appl) x (lift (S i) O w))
+(THead (Flat Appl) x0 (lift (S i) O w)) (pr2_head_1 c x x0 H12 (Flat Appl)
+(lift (S i) O w))) x0 (refl_equal T (THead (Flat Appl) x0 (lift (S i) O
+w))))) H18))) x1 H16))) (\lambda (H16: (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 x1 (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 x1 (lift (S i) O u)))) (sn3 c (THead (Flat Appl) x0 x1)) (\lambda
+(x2: C).(\lambda (x3: T).(\lambda (H17: (getl i c (CHead x2 (Bind Abbr)
+x3))).(\lambda (H18: (eq T x1 (lift (S i) O x3))).(let H19 \def (eq_ind T x1
+(\lambda (t: T).((eq T (THead (Flat Appl) x (TLRef i)) (THead (Flat Appl) x0
+t)) \to (\forall (P: Prop).P))) H14 (lift (S i) O x3) H18) in (eq_ind_r T
+(lift (S i) O x3) (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 t))) (let H20
+\def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H
+(CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2
+(Bind Abbr) x3) H17)) in (let H21 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _)
+\Rightarrow c0])) (CHead d (Bind Abbr) w) (CHead x2 (Bind Abbr) x3)
+(getl_mono c (CHead d (Bind Abbr) w) i H (CHead x2 (Bind Abbr) x3) H17)) in
+((let H22 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d
+(Bind Abbr) w) (CHead x2 (Bind Abbr) x3) (getl_mono c (CHead d (Bind Abbr) w)
+i H (CHead x2 (Bind Abbr) x3) H17)) in (\lambda (H23: (eq C d x2)).(let H24
+\def (eq_ind_r T x3 (\lambda (t: T).(getl i c (CHead x2 (Bind Abbr) t))) H20
+w H22) in (eq_ind T w (\lambda (t: T).(sn3 c (THead (Flat Appl) x0 (lift (S
+i) O t)))) (let H25 \def (eq_ind_r C x2 (\lambda (c0: C).(getl i c (CHead c0
+(Bind Abbr) w))) H24 d H23) in (let H_x \def (term_dec x x0) in (let H26 \def
+H_x in (or_ind (eq T x x0) ((eq T x x0) \to (\forall (P: Prop).P)) (sn3 c
+(THead (Flat Appl) x0 (lift (S i) O w))) (\lambda (H27: (eq T x x0)).(let H28
+\def (eq_ind_r T x0 (\lambda (t: T).(pr2 c x t)) H12 x H27) in (eq_ind T x
+(\lambda (t: T).(sn3 c (THead (Flat Appl) t (lift (S i) O w)))) (sn3_sing c
+(THead (Flat Appl) x (lift (S i) O w)) H6) x0 H27))) (\lambda (H27: (((eq T x
+x0) \to (\forall (P: Prop).P)))).(H6 (THead (Flat Appl) x0 (lift (S i) O w))
+(\lambda (H28: (eq T (THead (Flat Appl) x (lift (S i) O w)) (THead (Flat
+Appl) x0 (lift (S i) O w)))).(\lambda (P: Prop).(let H29 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t]))
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
+w)) H28) in (let H30 \def (eq_ind_r T x0 (\lambda (t: T).((eq T x t) \to
+(\forall (P0: Prop).P0))) H27 x H29) in (let H31 \def (eq_ind_r T x0 (\lambda
+(t: T).(pr2 c x t)) H12 x H29) in (H30 (refl_equal T x) P)))))) (pr3_pr2 c
+(THead (Flat Appl) x (lift (S i) O w)) (THead (Flat Appl) x0 (lift (S i) O
+w)) (pr2_head_1 c x x0 H12 (Flat Appl) (lift (S i) O w))))) H26)))) x3
+H22)))) H21))) x1 H18)))))) H16)) H15)) t2 H11))))))) H10)) (\lambda (H10:
+(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 x 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))))))))).(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 x 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 (H11: (eq T (TLRef i) (THead (Bind Abst) x0
+x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr2 c
+x x2)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b)
+u) x1 x3))))).(let H15 \def (eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat
+Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7 (THead (Bind Abbr) x2
+x3) H12) in (eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(sn3 c t))
+(let H16 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
+x1) H11) in (False_ind (sn3 c (THead (Bind Abbr) x2 x3)) H16)) t2
+H12)))))))))) H10)) (\lambda (H10: (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\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 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 (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 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 (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0
+Abst))).(\lambda (H12: (eq T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda
+(H13: (eq T t2 (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
+x3)))).(\lambda (_: (pr2 c x x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_:
+(pr2 (CHead c (Bind x0) x5) x2 x3)).(let H17 \def (eq_ind T t2 (\lambda (t:
+T).((eq T (THead (Flat Appl) x (TLRef i)) t) \to (\forall (P: Prop).P))) H7
+(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) H13) 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 H18 \def (eq_ind T (TLRef i) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind x0) x1 x2) H12) in (False_ind (sn3 c (THead (Bind x0) x5 (THead
+(Flat Appl) (lift (S O) O x4) x3))) H18)) t2 H13)))))))))))))) H10))
+H9))))))))))))) y H1)))) H0))))))).
+
+theorem sn3_appl_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
+T).((sn3 c u) \to (\forall (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) u)
+(THead (Flat Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v
+(THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (H0: (sn3 c u)).(sn3_ind c (\lambda (t: T).(\forall (t0:
+T).(\forall (v: T).((sn3 (CHead c (Bind b) t) (THead (Flat Appl) (lift (S O)
+O v) t0)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t 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 (t: T).(\forall (v: T).((sn3 (CHead c (Bind b) t2) (THead (Flat
+Appl) (lift (S O) O v) t)) \to (sn3 c (THead (Flat Appl) v (THead (Bind b) t2
+t))))))))))).(\lambda (t: T).(\lambda (v: T).(\lambda (H3: (sn3 (CHead c
+(Bind b) t1) (THead (Flat Appl) (lift (S O) O v) t))).(insert_eq T (THead
+(Flat Appl) (lift (S O) O v) t) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1)
+t0)) (sn3 c (THead (Flat Appl) v (THead (Bind b) t1 t))) (\lambda (y:
+T).(\lambda (H4: (sn3 (CHead c (Bind b) t1) y)).(unintro T t (\lambda (t0:
+T).((eq T y (THead (Flat Appl) (lift (S O) O v) t0)) \to (sn3 c (THead (Flat
+Appl) v (THead (Bind b) t1 t0))))) (unintro T v (\lambda (t0: T).(\forall (x:
+T).((eq T y (THead (Flat Appl) (lift (S O) O t0) x)) \to (sn3 c (THead (Flat
+Appl) t0 (THead (Bind b) t1 x)))))) (sn3_ind (CHead c (Bind b) t1) (\lambda
+(t0: T).(\forall (x: T).(\forall (x0: T).((eq T t0 (THead (Flat Appl) (lift
+(S O) O x) x0)) \to (sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0)))))))
+(\lambda (t2: T).(\lambda (H5: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
+(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (sn3 (CHead c (Bind
+b) t1) t3)))))).(\lambda (H6: ((\forall (t3: T).((((eq T t2 t3) \to (\forall
+(P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t2 t3) \to (\forall (x:
+T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) (lift (S O) O x) x0)) \to
+(sn3 c (THead (Flat Appl) x (THead (Bind b) t1 x0))))))))))).(\lambda (x:
+T).(\lambda (x0: T).(\lambda (H7: (eq T t2 (THead (Flat Appl) (lift (S O) O
+x) x0))).(let H8 \def (eq_ind T t2 (\lambda (t0: T).(\forall (t3: T).((((eq T
+t0 t3) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b) t1) t0 t3) \to
+(\forall (x1: T).(\forall (x2: T).((eq T t3 (THead (Flat Appl) (lift (S O) O
+x1) x2)) \to (sn3 c (THead (Flat Appl) x1 (THead (Bind b) t1 x2)))))))))) H6
+(THead (Flat Appl) (lift (S O) O x) x0) H7) in (let H9 \def (eq_ind T t2
+(\lambda (t0: T).(\forall (t3: T).((((eq T t0 t3) \to (\forall (P: Prop).P)))
+\to ((pr3 (CHead c (Bind b) t1) t0 t3) \to (sn3 (CHead c (Bind b) t1) t3)))))
+H5 (THead (Flat Appl) (lift (S O) O x) x0) H7) in (sn3_pr2_intro c (THead
+(Flat Appl) x (THead (Bind b) t1 x0)) (\lambda (t3: T).(\lambda (H10: (((eq T
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t3) \to (\forall (P:
+Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) x (THead (Bind b) t1
+x0)) t3)).(let H12 \def (pr2_gen_appl c x (THead (Bind b) t1 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 x u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0) t4)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind b) t1 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 x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0)
+u0) z1 t4)))))))) (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) t1 x0) (THead
+(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
+b0) 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 (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1 z2)))))))) (sn3 c t3)
+(\lambda (H13: (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 x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 x0)
+t4))))).(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 x u2)))
+(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind b) t1 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 x x1)).(\lambda (H16: (pr2 c (THead
+(Bind b) t1 x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P)))
+H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2)
+(\lambda (t0: T).(sn3 c t0)) (let H_x \def (pr3_gen_bind b H c t1 x0 x2) in
+(let H18 \def (H_x (pr3_pr2 c (THead (Bind b) t1 x0) x2 H16)) in (or_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4)))) (pr3 (CHead c (Bind
+b) t1) x0 (lift (S O) O x2)) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr3 (CHead c (Bind b) t1) x0 t4))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind b) u2 t4)))) (\lambda
+(u2: T).(\lambda (_: T).(pr3 c t1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr3
+(CHead c (Bind b) t1) x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda
+(x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Bind b) x3
+x4))).(\lambda (H21: (pr3 c t1 x3)).(\lambda (H22: (pr3 (CHead c (Bind b) t1)
+x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t0: T).((eq T (THead (Flat Appl)
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 t0)) \to (\forall (P:
+Prop).P))) H17 (THead (Bind b) x3 x4) H20) in (eq_ind_r T (THead (Bind b) x3
+x4) (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 t0))) (let H_x0 \def
+(term_dec t1 x3) in (let H24 \def H_x0 in (or_ind (eq T t1 x3) ((eq T t1 x3)
+\to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Bind b) x3
+x4))) (\lambda (H25: (eq T t1 x3)).(let H26 \def (eq_ind_r T x3 (\lambda (t0:
+T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1
+(THead (Bind b) t0 x4))) \to (\forall (P: Prop).P))) H23 t1 H25) in (let H27
+\def (eq_ind_r T x3 (\lambda (t0: T).(pr3 c t1 t0)) H21 t1 H25) in (eq_ind T
+t1 (\lambda (t0: T).(sn3 c (THead (Flat Appl) x1 (THead (Bind b) t0 x4))))
+(let H_x1 \def (term_dec x0 x4) in (let H28 \def H_x1 in (or_ind (eq T x0 x4)
+((eq T x0 x4) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead
+(Bind b) t1 x4))) (\lambda (H29: (eq T x0 x4)).(let H30 \def (eq_ind_r T x4
+(\lambda (t0: T).((eq T (THead (Flat Appl) x (THead (Bind b) t1 x0)) (THead
+(Flat Appl) x1 (THead (Bind b) t1 t0))) \to (\forall (P: Prop).P))) H26 x0
+H29) in (let H31 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b)
+t1) x0 t0)) H22 x0 H29) in (eq_ind T x0 (\lambda (t0: T).(sn3 c (THead (Flat
+Appl) x1 (THead (Bind b) t1 t0)))) (let H_x2 \def (term_dec x x1) in (let H32
+\def H_x2 in (or_ind (eq T x x1) ((eq T x x1) \to (\forall (P: Prop).P)) (sn3
+c (THead (Flat Appl) x1 (THead (Bind b) t1 x0))) (\lambda (H33: (eq T x
+x1)).(let H34 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl)
+x (THead (Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to
+(\forall (P: Prop).P))) H30 x H33) in (let H35 \def (eq_ind_r T x1 (\lambda
+(t0: T).(pr2 c x t0)) H15 x H33) in (eq_ind T x (\lambda (t0: T).(sn3 c
+(THead (Flat Appl) t0 (THead (Bind b) t1 x0)))) (H34 (refl_equal T (THead
+(Flat Appl) x (THead (Bind b) t1 x0))) (sn3 c (THead (Flat Appl) x (THead
+(Bind b) t1 x0)))) x1 H33)))) (\lambda (H33: (((eq T x x1) \to (\forall (P:
+Prop).P)))).(H8 (THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H34: (eq T
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x0))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x0) H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H33 x (lift_inj x x1 (S O) O
+H35)) in (let H37 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
+(lift_inj x x1 (S O) O H35)) in (H36 (refl_equal T x) P)))))) (pr3_flat
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl) x1 x0
+(refl_equal T (THead (Flat Appl) (lift (S O) O x1) x0)))) H32))) x4 H29))))
+(\lambda (H29: (((eq T x0 x4) \to (\forall (P: Prop).P)))).(H8 (THead (Flat
+Appl) (lift (S O) O x1) x4) (\lambda (H30: (eq T (THead (Flat Appl) (lift (S
+O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4))).(\lambda (P:
+Prop).(let H31 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x4) H30) in ((let H32 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H30) in
+(\lambda (H33: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H34 \def
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
+H29 x0 H32) in (let H35 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T (THead
+(Flat Appl) x (THead (Bind b) t1 x0)) (THead (Flat Appl) x1 (THead (Bind b)
+t1 t0))) \to (\forall (P0: Prop).P0))) H26 x0 H32) in (let H36 \def (eq_ind_r
+T x4 (\lambda (t0: T).(pr3 (CHead c (Bind b) t1) x0 t0)) H22 x0 H32) in (let
+H37 \def (eq_ind_r T x1 (\lambda (t0: T).((eq T (THead (Flat Appl) x (THead
+(Bind b) t1 x0)) (THead (Flat Appl) t0 (THead (Bind b) t1 x0))) \to (\forall
+(P0: Prop).P0))) H35 x (lift_inj x x1 (S O) O H33)) in (let H38 \def
+(eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O
+H33)) in (H34 (refl_equal T x0) P)))))))) H31)))) (pr3_flat (CHead c (Bind b)
+t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S
+O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15))
+x0 x4 H22 Appl) x1 x4 (refl_equal T (THead (Flat Appl) (lift (S O) O x1)
+x4)))) H28))) x3 H25)))) (\lambda (H25: (((eq T t1 x3) \to (\forall (P:
+Prop).P)))).(H2 x3 H25 H21 x4 x1 (sn3_cpr3_trans c t1 x3 H21 (Bind b) (THead
+(Flat Appl) (lift (S O) O x1) x4) (let H_x1 \def (term_dec x0 x4) in (let H26
+\def H_x1 in (or_ind (eq T x0 x4) ((eq T x0 x4) \to (\forall (P: Prop).P))
+(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x4)) (\lambda
+(H27: (eq T x0 x4)).(let H28 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead
+c (Bind b) t1) x0 t0)) H22 x0 H27) in (eq_ind T x0 (\lambda (t0: T).(sn3
+(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) t0))) (let H_x2
+\def (term_dec x x1) in (let H29 \def H_x2 in (or_ind (eq T x x1) ((eq T x
+x1) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl)
+(lift (S O) O x1) x0)) (\lambda (H30: (eq T x x1)).(let H31 \def (eq_ind_r T
+x1 (\lambda (t0: T).(pr2 c x t0)) H15 x H30) in (eq_ind T x (\lambda (t0:
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
+(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
+x1 H30))) (\lambda (H30: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
+(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H31: (eq T (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x0))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x0) H31) in (let H33 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H30 x (lift_inj x x1 (S O) O
+H32)) in (let H34 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
+(lift_inj x x1 (S O) O H32)) in (H33 (refl_equal T x) P)))))) (pr3_flat
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
+H29))) x4 H27))) (\lambda (H27: (((eq T x0 x4) \to (\forall (P:
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x1) x4) (\lambda (H28: (eq T
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x4))).(\lambda (P: Prop).(let H29 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x5: nat).(plus x5 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x4) H28) in ((let H30 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1) x4) H28) in
+(\lambda (H31: (eq T (lift (S O) O x) (lift (S O) O x1))).(let H32 \def
+(eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0)))
+H27 x0 H30) in (let H33 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 (CHead c
+(Bind b) t1) x0 t0)) H22 x0 H30) in (let H34 \def (eq_ind_r T x1 (\lambda
+(t0: T).(pr2 c x t0)) H15 x (lift_inj x x1 (S O) O H31)) in (H32 (refl_equal
+T x0) P)))))) H29)))) (pr3_flat (CHead c (Bind b) t1) (lift (S O) O x) (lift
+(S O) O x1) (pr3_lift (CHead c (Bind b) t1) c (S O) O (drop_drop (Bind b) O c
+c (drop_refl c) t1) x x1 (pr3_pr2 c x x1 H15)) x0 x4 H22 Appl))) H26))))))
+H24))) x2 H20))))))) H19)) (\lambda (H19: (pr3 (CHead c (Bind b) t1) x0 (lift
+(S O) O x2))).(sn3_gen_lift (CHead c (Bind b) t1) (THead (Flat Appl) x1 x2)
+(S O) O (eq_ind_r T (THead (Flat Appl) (lift (S O) O x1) (lift (S O) (s (Flat
+Appl) O) x2)) (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) t0)) (sn3_pr3_trans
+(CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x1) x0) (let H_x0 \def
+(term_dec x x1) in (let H20 \def H_x0 in (or_ind (eq T x x1) ((eq T x x1) \to
+(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
+O) O x1) x0)) (\lambda (H21: (eq T x x1)).(let H22 \def (eq_ind_r T x1
+(\lambda (t0: T).(pr2 c x t0)) H15 x H21) in (eq_ind T x (\lambda (t0:
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x0)))
+(sn3_sing (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O x) x0) H9)
+x1 H21))) (\lambda (H21: (((eq T x x1) \to (\forall (P: Prop).P)))).(H9
+(THead (Flat Appl) (lift (S O) O x1) x0) (\lambda (H22: (eq T (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x1)
+x0))).(\lambda (P: Prop).(let H23 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x3: nat).(plus x3 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x1) x0) H22) in (let H24 \def (eq_ind_r T x1 (\lambda (t0:
+T).((eq T x t0) \to (\forall (P0: Prop).P0))) H21 x (lift_inj x x1 (S O) O
+H23)) in (let H25 \def (eq_ind_r T x1 (\lambda (t0: T).(pr2 c x t0)) H15 x
+(lift_inj x x1 (S O) O H23)) in (H24 (refl_equal T x) P)))))) (pr3_flat
+(CHead c (Bind b) t1) (lift (S O) O x) (lift (S O) O x1) (pr3_lift (CHead c
+(Bind b) t1) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) t1) x x1
+(pr3_pr2 c x x1 H15)) x0 x0 (pr3_refl (CHead c (Bind b) t1) x0) Appl)))
+H20))) (THead (Flat Appl) (lift (S O) O x1) (lift (S O) O x2)) (pr3_thin_dx
+(CHead c (Bind b) t1) x0 (lift (S O) O x2) H19 (lift (S O) O x1) Appl)) (lift
+(S O) O (THead (Flat Appl) x1 x2)) (lift_head (Flat Appl) x1 x2 (S O) O)) c
+(drop_drop (Bind b) O c c (drop_refl c) t1))) 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 (Bind b) t1 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 x u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b0: B).(\forall (u0:
+T).(pr2 (CHead c (Bind b0) u0) z1 t4))))))))).(ex4_4_ind T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
+b) t1 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 x
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
+T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) z1 t4)))))))
+(sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H14: (eq T (THead (Bind b) t1 x0) (THead (Bind Abst) x1
+x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c
+x x3)).(\lambda (H17: ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind
+b0) u0) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t0: T).((eq T (THead
+(Flat Appl) x (THead (Bind b) t1 x0)) t0) \to (\forall (P: Prop).P))) H10
+(THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4)
+(\lambda (t0: T).(sn3 c t0)) (let H19 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in ((let H20
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
+\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14) in
+((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind Abst) x1 x2) H14)
+in (\lambda (_: (eq T t1 x1)).(\lambda (H23: (eq B b Abst)).(let H24 \def
+(eq_ind_r T x2 (\lambda (t0: T).(\forall (b0: B).(\forall (u0: T).(pr2 (CHead
+c (Bind b0) u0) t0 x4)))) H17 x0 H21) in (let H25 \def (eq_ind B b (\lambda
+(b0: B).((eq T (THead (Flat Appl) x (THead (Bind b0) t1 x0)) (THead (Bind
+Abbr) x3 x4)) \to (\forall (P: Prop).P))) H18 Abst H23) in (let H26 \def
+(eq_ind B b (\lambda (b0: B).(\forall (t4: T).((((eq T (THead (Flat Appl)
+(lift (S O) O x) x0) t4) \to (\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind
+b0) t1) (THead (Flat Appl) (lift (S O) O x) x0) t4) \to (sn3 (CHead c (Bind
+b0) t1) t4))))) H9 Abst H23) in (let H27 \def (eq_ind B b (\lambda (b0:
+B).(\forall (t4: T).((((eq T (THead (Flat Appl) (lift (S O) O x) x0) t4) \to
+(\forall (P: Prop).P))) \to ((pr3 (CHead c (Bind b0) t1) (THead (Flat Appl)
+(lift (S O) O x) x0) t4) \to (\forall (x5: T).(\forall (x6: T).((eq T t4
+(THead (Flat Appl) (lift (S O) O x5) x6)) \to (sn3 c (THead (Flat Appl) x5
+(THead (Bind b0) t1 x6)))))))))) H8 Abst H23) in (let H28 \def (eq_ind B b
+(\lambda (b0: B).(\forall (t4: T).((((eq T t1 t4) \to (\forall (P: Prop).P)))
+\to ((pr3 c t1 t4) \to (\forall (t0: T).(\forall (v0: T).((sn3 (CHead c (Bind
+b0) t4) (THead (Flat Appl) (lift (S O) O v0) t0)) \to (sn3 c (THead (Flat
+Appl) v0 (THead (Bind b0) t4 t0)))))))))) H2 Abst H23) in (let H29 \def
+(eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H23) in (let H30
+\def (match (H29 (refl_equal B Abst)) in False return (\lambda (_:
+False).(sn3 c (THead (Bind Abbr) x3 x4))) with []) in H30)))))))))) H20))
+H19)) t3 H15)))))))))) H13)) (\lambda (H13: (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)
+t1 x0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t3 (THead (Bind b0) 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 (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b0) y2) z1
+z2))))))))).(ex6_6_ind 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) t1 x0) (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b0) 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 (b0:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c (Bind b0) 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 (Bind b) t1 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
+(H17: (pr2 c x x5)).(\lambda (H18: (pr2 c x2 x6)).(\lambda (H19: (pr2 (CHead
+c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t0: T).((eq T
+(THead (Flat Appl) x (THead (Bind b) t1 x0)) t0) \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 (t0: T).(sn3 c t0)) (let H21 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in ((let H22 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ t0 _)
+\Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
+((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _
+t0) \Rightarrow t0])) (THead (Bind b) t1 x0) (THead (Bind x1) x2 x3) H15) in
+(\lambda (H24: (eq T t1 x2)).(\lambda (H25: (eq B b x1)).(let H26 \def
+(eq_ind_r T x3 (\lambda (t0: T).(pr2 (CHead c (Bind x1) x6) t0 x4)) H19 x0
+H23) in (let H27 \def (eq_ind_r T x2 (\lambda (t0: T).(pr2 c t0 x6)) H18 t1
+H24) in (let H28 \def (eq_ind_r B x1 (\lambda (b0: B).(pr2 (CHead c (Bind b0)
+x6) x0 x4)) H26 b H25) in (eq_ind B b (\lambda (b0: B).(sn3 c (THead (Bind
+b0) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))) (sn3_pr3_trans c (THead
+(Bind b) t1 (THead (Flat Appl) (lift (S O) O x5) x4)) (sn3_bind b c t1
+(sn3_sing c t1 H1) (THead (Flat Appl) (lift (S O) O x5) x4) (let H_x \def
+(term_dec x x5) in (let H29 \def H_x in (or_ind (eq T x x5) ((eq T x x5) \to
+(\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S
+O) O x5) x4)) (\lambda (H30: (eq T x x5)).(let H31 \def (eq_ind_r T x5
+(\lambda (t0: T).(pr2 c x t0)) H17 x H30) in (eq_ind T x (\lambda (t0:
+T).(sn3 (CHead c (Bind b) t1) (THead (Flat Appl) (lift (S O) O t0) x4))) (let
+H_x0 \def (term_dec x0 x4) in (let H32 \def H_x0 in (or_ind (eq T x0 x4) ((eq
+T x0 x4) \to (\forall (P: Prop).P)) (sn3 (CHead c (Bind b) t1) (THead (Flat
+Appl) (lift (S O) O x) x4)) (\lambda (H33: (eq T x0 x4)).(let H34 \def
+(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
+H33) in (eq_ind T x0 (\lambda (t0: T).(sn3 (CHead c (Bind b) t1) (THead (Flat
+Appl) (lift (S O) O x) t0))) (sn3_sing (CHead c (Bind b) t1) (THead (Flat
+Appl) (lift (S O) O x) x0) H9) x4 H33))) (\lambda (H33: (((eq T x0 x4) \to
+(\forall (P: Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x) x4) (\lambda
+(H34: (eq T (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
+(S O) O x) x4))).(\lambda (P: Prop).(let H35 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x) x4) H34) in
+(let H36 \def (eq_ind_r T x4 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0:
+Prop).P0))) H33 x0 H35) in (let H37 \def (eq_ind_r T x4 (\lambda (t0: T).(pr2
+(CHead c (Bind b) x6) x0 t0)) H28 x0 H35) in (H36 (refl_equal T x0) P))))))
+(pr3_pr3_pr3_t c t1 x6 (pr3_pr2 c t1 x6 H27) (THead (Flat Appl) (lift (S O) O
+x) x0) (THead (Flat Appl) (lift (S O) O x) x4) (Bind b) (pr3_pr2 (CHead c
+(Bind b) x6) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
+(S O) O x) x4) (pr2_thin_dx (CHead c (Bind b) x6) x0 x4 H28 (lift (S O) O x)
+Appl))))) H32))) x5 H30))) (\lambda (H30: (((eq T x x5) \to (\forall (P:
+Prop).P)))).(H9 (THead (Flat Appl) (lift (S O) O x5) x4) (\lambda (H31: (eq T
+(THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5)
+x4))).(\lambda (P: Prop).(let H32 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match t0 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t4)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t4))]) in
+lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t0: T) on t0: T \def (match
+t0 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u0 t4) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
+t4))]) in lref_map) (\lambda (x7: nat).(plus x7 (S O))) O x) | (THead _ t0 _)
+\Rightarrow t0])) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl)
+(lift (S O) O x5) x4) H31) in ((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 |
+(TLRef _) \Rightarrow x0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat
+Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift (S O) O x5) x4) H31) in
+(\lambda (H34: (eq T (lift (S O) O x) (lift (S O) O x5))).(let H35 \def
+(eq_ind_r T x5 (\lambda (t0: T).((eq T x t0) \to (\forall (P0: Prop).P0)))
+H30 x (lift_inj x x5 (S O) O H34)) in (let H36 \def (eq_ind_r T x5 (\lambda
+(t0: T).(pr2 c x t0)) H17 x (lift_inj x x5 (S O) O H34)) in (let H37 \def
+(eq_ind_r T x4 (\lambda (t0: T).(pr2 (CHead c (Bind b) x6) x0 t0)) H28 x0
+H33) in (H35 (refl_equal T x) P)))))) H32)))) (pr3_pr3_pr3_t c t1 x6 (pr3_pr2
+c t1 x6 H27) (THead (Flat Appl) (lift (S O) O x) x0) (THead (Flat Appl) (lift
+(S O) O x5) x4) (Bind b) (pr3_flat (CHead c (Bind b) x6) (lift (S O) O x)
+(lift (S O) O x5) (pr3_lift (CHead c (Bind b) x6) c (S O) O (drop_drop (Bind
+b) O c c (drop_refl c) x6) x x5 (pr3_pr2 c x x5 H17)) x0 x4 (pr3_pr2 (CHead c
+(Bind b) x6) x0 x4 H28) Appl)))) H29)))) (THead (Bind b) x6 (THead (Flat
+Appl) (lift (S O) O x5) x4)) (pr3_pr2 c (THead (Bind b) t1 (THead (Flat Appl)
+(lift (S O) O x5) x4)) (THead (Bind b) x6 (THead (Flat Appl) (lift (S O) O
+x5) x4)) (pr2_head_1 c t1 x6 H27 (Bind b) (THead (Flat Appl) (lift (S O) O
+x5) x4)))) x1 H25))))))) H22)) H21)) t3 H16)))))))))))))) H13))
+H12)))))))))))))) y H4))))) H3))))))) u H0))))).
+
+theorem sn3_appl_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((sn3 c
+(THead (Flat Appl) u (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
+\to (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind Abst) w
+t))))))))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
+(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
+T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
+Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind
+Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
+Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
+Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
+H0) u H2 (\lambda (u2: T).(\lambda (H4: (pr3 c (THead (Flat Appl) v (THead
+(Bind Abst) w t)) u2)).(\lambda (H5: (((iso (THead (Flat Appl) v (THead (Bind
+Abst) w t)) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat
+Appl) u (THead (Bind Abbr) v t)) H (THead (Flat Appl) u u2) (pr3_thin_dx c
+(THead (Bind Abbr) v t) u2 (pr3_iso_beta v w t c u2 H4 H5) u Appl))))))))
+H1))))))))).
+
+theorem sn3_appls_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
+T).((sn3 c u) \to (\forall (vs: TList).(\forall (t: T).((sn3 (CHead c (Bind
+b) u) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to (sn3 c (THeads (Flat
+Appl) vs (THead (Bind b) u t))))))))))
+\def
+ \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (c: C).(\lambda
+(u: T).(\lambda (H0: (sn3 c u)).(\lambda (vs: TList).(TList_ind (\lambda (t:
+TList).(\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl) (lifts
+(S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u t0))))))
+(\lambda (t: T).(\lambda (H1: (sn3 (CHead c (Bind b) u) t)).(sn3_bind b c u
+H0 t H1))) (\lambda (v: T).(\lambda (vs0: TList).(TList_ind (\lambda (t:
+TList).(((\forall (t0: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
+(lifts (S O) O t) t0)) \to (sn3 c (THeads (Flat Appl) t (THead (Bind b) u
+t0)))))) \to (\forall (t0: T).((sn3 (CHead c (Bind b) u) (THead (Flat Appl)
+(lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t) t0))) \to (sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) t (THead (Bind b) u t0))))))))
+(\lambda (_: ((\forall (t: T).((sn3 (CHead c (Bind b) u) (THeads (Flat Appl)
+(lifts (S O) O TNil) t)) \to (sn3 c (THeads (Flat Appl) TNil (THead (Bind b)
+u t))))))).(\lambda (t: T).(\lambda (H2: (sn3 (CHead c (Bind b) u) (THead
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O TNil)
+t)))).(sn3_appl_bind b H c u H0 t v H2)))) (\lambda (t: T).(\lambda (t0:
+TList).(\lambda (_: ((((\forall (t1: T).((sn3 (CHead c (Bind b) u) (THeads
+(Flat Appl) (lifts (S O) O t0) t1)) \to (sn3 c (THeads (Flat Appl) t0 (THead
+(Bind b) u t1)))))) \to (\forall (t1: T).((sn3 (CHead c (Bind b) u) (THead
+(Flat Appl) (lift (S O) O v) (THeads (Flat Appl) (lifts (S O) O t0) t1))) \to
+(sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (THead (Bind b) u
+t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) u)
+(THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)) \to (sn3 c (THeads
+(Flat Appl) (TCons t t0) (THead (Bind b) u t1))))))).(\lambda (t1:
+T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
+v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
+(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
+(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3
+(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads
+(Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v
+(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3
+(CHead c (Bind b) u) (lift (S O) O v))).(\lambda (H6: (sn3 (CHead c (Bind b)
+u) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1))).(let H_y \def
+(sn3_gen_lift (CHead c (Bind b) u) v (S O) O H5 c) in (sn3_appl_appls t
+(THead (Bind b) u t1) t0 c (H2 t1 H6) v (H_y (drop_drop (Bind b) O c c
+(drop_refl c) u)) (\lambda (u2: T).(\lambda (H7: (pr3 c (THeads (Flat Appl)
+(TCons t t0) (THead (Bind b) u t1)) u2)).(\lambda (H8: (((iso (THeads (Flat
+Appl) (TCons t t0) (THead (Bind b) u t1)) u2) \to (\forall (P:
+Prop).P)))).(let H9 \def (pr3_iso_appls_bind b H (TCons t t0) u t1 c u2 H7
+H8) in (sn3_pr3_trans c (THead (Flat Appl) v (THead (Bind b) u (THeads (Flat
+Appl) (lifts (S O) O (TCons t t0)) t1))) (sn3_appl_bind b H c u H0 (THeads
+(Flat Appl) (lifts (S O) O (TCons t t0)) t1) v H3) (THead (Flat Appl) v u2)
+(pr3_flat c v v (pr3_refl c v) (THead (Bind b) u (THeads (Flat Appl) (lifts
+(S O) O (TCons t t0)) t1)) u2 H9 Appl)))))))))) H4))))))))) vs0))) vs)))))).
+
+theorem sn3_appls_beta:
+ \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (us: TList).((sn3 c
+(THeads (Flat Appl) us (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c
+w) \to (sn3 c (THeads (Flat Appl) us (THead (Flat Appl) v (THead (Bind Abst)
+w t))))))))))
+\def
+ \lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (us:
+TList).(TList_ind (\lambda (t0: TList).((sn3 c (THeads (Flat Appl) t0 (THead
+(Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) (\lambda (H:
+(sn3 c (THead (Bind Abbr) v t))).(\lambda (w: T).(\lambda (H0: (sn3 c
+w)).(sn3_beta c v t H w H0)))) (\lambda (u: T).(\lambda (us0:
+TList).(TList_ind (\lambda (t0: TList).((((sn3 c (THeads (Flat Appl) t0
+(THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads
+(Flat Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3
+c (THead (Flat Appl) u (THeads (Flat Appl) t0 (THead (Bind Abbr) v t)))) \to
+(\forall (w: T).((sn3 c w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat
+Appl) t0 (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))) (\lambda (_:
+(((sn3 c (THeads (Flat Appl) TNil (THead (Bind Abbr) v t))) \to (\forall (w:
+T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) TNil (THead (Flat Appl) v (THead
+(Bind Abst) w t))))))))).(\lambda (H0: (sn3 c (THead (Flat Appl) u (THeads
+(Flat Appl) TNil (THead (Bind Abbr) v t))))).(\lambda (w: T).(\lambda (H1:
+(sn3 c w)).(sn3_appl_beta c u v t H0 w H1))))) (\lambda (t0: T).(\lambda (t1:
+TList).(\lambda (_: (((((sn3 c (THeads (Flat Appl) t1 (THead (Bind Abbr) v
+t))) \to (\forall (w: T).((sn3 c w) \to (sn3 c (THeads (Flat Appl) t1 (THead
+(Flat Appl) v (THead (Bind Abst) w t)))))))) \to ((sn3 c (THead (Flat Appl) u
+(THeads (Flat Appl) t1 (THead (Bind Abbr) v t)))) \to (\forall (w: T).((sn3 c
+w) \to (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) t1 (THead (Flat Appl)
+v (THead (Bind Abst) w t))))))))))).(\lambda (H0: (((sn3 c (THeads (Flat
+Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w)
+\to (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
+(Bind Abst) w t))))))))).(\lambda (H1: (sn3 c (THead (Flat Appl) u (THeads
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
+T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
+(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
+(and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
+Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1)
+(THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c
+u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr)
+v t)))).(sn3_appl_appls t0 (THead (Flat Appl) v (THead (Bind Abst) w t)) t1 c
+(H0 H5 w H2) u H4 (\lambda (u2: T).(\lambda (H6: (pr3 c (THeads (Flat Appl)
+(TCons t0 t1) (THead (Flat Appl) v (THead (Bind Abst) w t))) u2)).(\lambda
+(H7: (((iso (THeads (Flat Appl) (TCons t0 t1) (THead (Flat Appl) v (THead
+(Bind Abst) w t))) u2) \to (\forall (P: Prop).P)))).(let H8 \def
+(pr3_iso_appls_beta (TCons t0 t1) v w t c u2 H6 H7) in (sn3_pr3_trans c
+(THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v
+t))) H1 (THead (Flat Appl) u u2) (pr3_thin_dx c (THeads (Flat Appl) (TCons t0
+t1) (THead (Bind Abbr) v t)) u2 H8 u Appl))))))))) H3)))))))))) us0))) us)))).
+
+theorem sn3_appls_abbr:
+ \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) w)) \to (\forall (vs: TList).((sn3 c (THeads (Flat Appl)
+vs (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (w: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) w))).(\lambda (vs: TList).(TList_ind
+(\lambda (t: TList).((sn3 c (THeads (Flat Appl) t (lift (S i) O w))) \to (sn3
+c (THeads (Flat Appl) t (TLRef i))))) (\lambda (H0: (sn3 c (lift (S i) O
+w))).(let H_y \def (sn3_gen_lift c w (S i) O H0 d (getl_drop Abbr c d w i H))
+in (sn3_abbr c d w i H H_y))) (\lambda (v: T).(\lambda (vs0:
+TList).(TList_ind (\lambda (t: TList).((((sn3 c (THeads (Flat Appl) t (lift
+(S i) O w))) \to (sn3 c (THeads (Flat Appl) t (TLRef i))))) \to ((sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) t (lift (S i) O w)))) \to (sn3 c
+(THead (Flat Appl) v (THeads (Flat Appl) t (TLRef i))))))) (\lambda (_:
+(((sn3 c (THeads (Flat Appl) TNil (lift (S i) O w))) \to (sn3 c (THeads (Flat
+Appl) TNil (TLRef i)))))).(\lambda (H1: (sn3 c (THead (Flat Appl) v (THeads
+(Flat Appl) TNil (lift (S i) O w))))).(nf3_appl_abbr c d w i H v H1)))
+(\lambda (t: T).(\lambda (t0: TList).(\lambda (_: (((((sn3 c (THeads (Flat
+Appl) t0 (lift (S i) O w))) \to (sn3 c (THeads (Flat Appl) t0 (TLRef i)))))
+\to ((sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (lift (S i) O w))))
+\to (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) t0 (TLRef
+i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i)
+O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
+(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
+O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
+t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c
+(THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl)
+v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c
+v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O
+w)))).(sn3_appl_appls t (TLRef i) t0 c (H1 H5) v H4 (\lambda (u2: T).(\lambda
+(H6: (pr3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2)).(\lambda (H7:
+(((iso (THeads (Flat Appl) (TCons t t0) (TLRef i)) u2) \to (\forall (P:
+Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) v (THeads (Flat Appl) (TCons
+t t0) (lift (S i) O w))) H2 (THead (Flat Appl) v u2) (pr3_thin_dx c (THeads
+(Flat Appl) (TCons t t0) (lift (S i) O w)) u2 (pr3_iso_appls_abbr c d w i H
+(TCons t t0) u2 H6 H7) v Appl)))))))) H3)))))))) vs0))) vs)))))).
+
+theorem sn3_gen_def:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
+\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 c (TLRef
+i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
+(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
+i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
+Abbr c d v i H))))))).
+
+theorem sn3_cdelta:
+ \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
+(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
+C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
+\def
+ \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
+T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
+\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
+C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
+(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
+(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
+(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
+\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
+C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
+v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
+(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
+(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
+c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
+H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
+(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
+(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
+v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
+C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
+v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
+(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
+c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
+(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
+C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
+(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
+(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
+(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
+(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
+(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
+H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
+t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
+(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
+in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
+(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
+H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
+(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
+C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
+v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
+(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
+(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
+H0)))))).
+
+inductive csubn: C \to (C \to Prop) \def
+| csubn_sort: \forall (n: nat).(csubn (CSort n) (CSort n))
+| csubn_head: \forall (c1: C).(\forall (c2: C).((csubn c1 c2) \to (\forall
+(k: K).(\forall (v: T).(csubn (CHead c1 k v) (CHead c2 k v))))))
+| csubn_abst: \forall (c1: C).(\forall (c2: C).((csubn c1 c2) \to (\forall
+(v: T).(\forall (w: T).((sn3 c2 w) \to (csubn (CHead c1 (Bind Abst) v) (CHead
+c2 (Bind Abbr) w))))))).
+
+theorem csubc_csuba:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba
+g c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
+c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csuba g c c0))) (\lambda
+(n: nat).(csuba_refl g (CSort n))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
+(_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (k: K).(\lambda
+(v: T).(csuba_head g c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4:
+C).(\lambda (_: (csubc g c3 c4)).(\lambda (H1: (csuba g c3 c4)).(\lambda (v:
+T).(\lambda (a: A).(\lambda (H2: (sc3 g (asucc g a) c3 v)).(\lambda (w:
+T).(\lambda (H3: (sc3 g a c4 w)).(csuba_abst g c3 c4 H1 v a (sc3_arity_gen g
+c3 v (asucc g a) H2) w (sc3_arity_gen g c4 w a H3))))))))))) c1 c2 H)))).
+
+theorem csubc_csubn:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csubn
+c1 c2))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csubc g c1
+c2)).(csubc_ind g (\lambda (c: C).(\lambda (c0: C).(csubn c c0))) (\lambda
+(n: nat).(csubn_sort n)) (\lambda (c3: C).(\lambda (c4: C).(\lambda (_:
+(csubc g c3 c4)).(\lambda (H1: (csubn c3 c4)).(\lambda (k: K).(\lambda (v:
+T).(csubn_head c3 c4 H1 k v))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda
+(_: (csubc g c3 c4)).(\lambda (H1: (csubn c3 c4)).(\lambda (v: T).(\lambda
+(a: A).(\lambda (_: (sc3 g (asucc g a) c3 v)).(\lambda (w: T).(\lambda (H3:
+(sc3 g a c4 w)).(csubn_abst c3 c4 H1 v w (sc3_sn3 g a c4 w H3))))))))))) c1
+c2 H)))).
+
+theorem ceq_arity_trans:
+ \forall (g: G).(\forall (c1: C).(\forall (c2: C).((ceqc g c2 c1) \to
+(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (arity g c2 t a)))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (ceqc g c2
+c1)).(\lambda (t: T).(\lambda (a: A).(\lambda (H0: (arity g c1 t a)).(let H1
+\def H in (or_ind (csubc g c2 c1) (csubc g c1 c2) (arity g c2 t a) (\lambda
+(H2: (csubc g c2 c1)).(csuba_arity_rev g c1 t a H0 c2 (csubc_csuba g c2 c1
+H2))) (\lambda (H2: (csubc g c1 c2)).(csuba_arity g c1 t a H0 c2 (csubc_csuba
+g c1 c2 H2))) H1)))))))).
projects / helm.git / blobdiff