new definitions and new theorems

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
index 5954560ed6ffd4f743db53dc7b4fc5578f4b9195..377501693001d94b40b947b9d955f9c5194ee0dc 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma
@@ -18,3 +18,1166 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/spare".
 
 include "theory.ma".
 
+definition nfs2:
+ C \to (TList \to Prop)
+\def
+ let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil 
+\Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))]) 
+in nfs2.
+
+theorem nf2_gen_beta:
+ \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c 
+(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: 
+((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) 
+\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: 
+Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | 
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind 
+Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead 
+(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in 
+(False_ind P H0))))))).
+
+theorem nf2_gen__aux:
+ \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T 
+(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: 
+T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to 
+(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort 
+n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) 
+d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
+_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: 
+nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u 
+(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind 
+T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in 
+(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall 
+(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to 
+(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: 
+T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to 
+(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: 
+(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
+t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e 
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef 
+_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u 
+(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T 
+T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) 
+(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let 
+H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) 
+(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort 
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i 
+| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 
+(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: 
+nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) 
+(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | 
+(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | 
+false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map 
+f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus 
+x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map 
+(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort 
+n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) 
+with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) 
+\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in 
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: 
+((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) 
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with 
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) 
+\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in 
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) 
+\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t 
+t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) 
+H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t 
+t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift 
+(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 
+P)))))) H3)) H2))))))))))) x)).
+
+theorem nf2_gen_abbr:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u 
+t)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
+T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) 
+t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x 
+in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t 
+(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift 
+(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O 
+x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O 
+x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda 
+(_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ 
+_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S 
+O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind 
+Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) 
+t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda 
+(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in 
+(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) 
+O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c 
+(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H 
+(lift (S O) O x) H2) in (nf2_gen__aux Abbr x u O (H3 x (pr2_free c (THead 
+(Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl 
+x) u))) P))) H1))) H0))))))).
+
+theorem nf2_gen_void:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u 
+(lift (S O) O t))) \to (\forall (P: Prop).P))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: 
+T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind 
+Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__aux Void t u O 
+(H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t (pr0_zeta Void 
+not_void_abst t t (pr0_refl t) u))) P))))).
+
+theorem arity_nf2_inv_all:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t 
+a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat 
+(\lambda (n: nat).(eq T t (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c 
+ws))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: 
+(arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: 
+A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat 
+(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 
+ws))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (_: (nf2 c0 (TSort 
+n))).(or3_intro1 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (TSort n) 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat 
+(\lambda (n0: nat).(eq T (TSort n) (TSort n0)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(TSort n) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda 
+(i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) 
+v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: 
+T).(nfs2 c0 ws)))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort 
+n0))) n (refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind 
+Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: 
+(((nf2 d u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads 
+(Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0: 
+nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 d (CHead d0 (Bind Abst) 
+v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: 
+T).(nfs2 d ws)))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(nf2_gen_lref c0 d 
+u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (TLRef i) 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_4 TList nat C T (\lambda 
+(ws: TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TLRef 
+i) (THeads (Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0: 
+nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 c0 (CHead d0 (Bind Abst) 
+v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: 
+T).(nfs2 c0 ws))))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) 
+u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: 
+(((nf2 d u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u 
+(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads 
+(Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0: 
+nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 d (CHead d0 (Bind Abst) 
+v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: 
+T).(nfs2 d ws)))))))))).(\lambda (_: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T 
+T (\lambda (w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(TLRef i) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda 
+(i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TLRef i) (THeads (Flat Appl) 
+ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0: nat).(\lambda (d0: 
+C).(\lambda (v: T).(getl i0 c0 (CHead d0 (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))) 
+(ex3_4_intro TList nat C T (\lambda (ws: TList).(\lambda (i0: nat).(\lambda 
+(_: C).(\lambda (_: T).(eq T (TLRef i) (THeads (Flat Appl) ws (TLRef 
+i0))))))) (\lambda (_: TList).(\lambda (i0: nat).(\lambda (d0: C).(\lambda 
+(v: T).(getl i0 c0 (CHead d0 (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) 
+TNil i d u (refl_equal T (TLRef i)) H0 I))))))))))) (\lambda (b: B).(\lambda 
+(H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: 
+A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 
+T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind b) u) t0 
+a2)).(\lambda (_: (((nf2 (CHead c0 (Bind b) u) t0) \to (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind b) u) w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind b) u) (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda 
+(ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 
+(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i (CHead c0 (Bind b) u) (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 (CHead c0 (Bind b) u) ws)))))))))).(\lambda (H5: 
+(nf2 c0 (THead (Bind b) u t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) 
+\to ((arity g (CHead c0 (Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u 
+t0)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind 
+b0) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 
+w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_4 
+TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda 
+(_: T).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws (TLRef i))))))) 
+(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i 
+c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))))) (\lambda (_: (not 
+(eq B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0 
+a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0 
+H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr) 
+u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 
+w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_4 
+TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda 
+(_: T).(eq T (THead (Bind Abbr) u t0) (THeads (Flat Appl) ws (TLRef i))))))) 
+(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i 
+c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))))) (\lambda (H6: 
+(not (eq B Abst Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 
+a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u t0))).(let H9 \def (match (H6 
+(refl_equal B Abst)) in False return (\lambda (_: False).(or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))))) with []) in H9)))) (\lambda (_: (not 
+(eq B Void Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 
+a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let H9 \def 
+(arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void 
+c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T 
+T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t0) (THead 
+(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda 
+(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda 
+(n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x: T).(\lambda (H10: (eq T t0 
+(lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead 
+(Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x) 
+(\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
+(THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda 
+(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind 
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t1) 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind Void) u t1) (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))) 
+(nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq 
+T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 
+(CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind 
+Void) u (lift (S O) O x)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind 
+Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))))) t0 H10)))) H9))))) b H0 H3 
+H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda 
+(_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T 
+T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
+T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0: 
+T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 
+a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda 
+(w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w) 
+u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i (CHead c0 (Bind Abst) u) (CHead 
+d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 (CHead c0 (Bind Abst) u) ws)))))))))).(\lambda (H4: 
+(nf2 c0 (THead (Bind Abst) u t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in 
+(and_ind (nf2 c0 u) (nf2 (CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: 
+(nf2 (CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind 
+Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_2_intro T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind 
+Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: 
+T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) 
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda 
+(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws 
+(TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: 
+C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 
+ws)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g c0 t0 
+(AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) 
+(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: 
+C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda 
+(_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 
+(CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda 
+(_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (H4: (nf2 c0 (THead 
+(Flat Appl) u t0))).(let H5 \def (nf2_gen_flat Appl c0 u t0 H4) in (and_ind 
+(nf2 c0 u) (nf2 c0 t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T 
+(THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda 
+(_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind 
+Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t0) 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) 
+(\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def (H3 H7) in 
+(let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq 
+T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))) 
+(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u 
+t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat 
+(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_4 TList 
+nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: 
+T).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))))) 
+(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i 
+c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H9: (ex3_2 
+T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: 
+T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq 
+T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
+t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t0 (THead (Bind Abst) 
+x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: (nf2 (CHead c0 (Bind Abst) 
+x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat 
+Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in (let H14 \def (eq_ind T t0 
+(\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THead (Bind Abst) x0 x1) 
+H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T 
+T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead 
+(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda 
+(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda 
+(n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead 
+(Bind Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind 
+Abst) x0 x1)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda 
+(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (THead 
+(Bind Abst) x0 x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))))) t0 H10)))))))) H9)) (\lambda (H9: (ex 
+nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind nat (\lambda (n: 
+nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: 
+T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead 
+c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u 
+t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) 
+(\lambda (x: nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T 
+t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in 
+(let H12 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 
+(TSort x) H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))))) (let H13 \def (match (arity_gen_sort g 
+c0 x (AHead a1 a2) H12) in leq return (\lambda (a0: A).(\lambda (a3: 
+A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) \to ((eq A a3 (ASort 
+O x)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat 
+Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) 
+(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 
+c0 ws)))))))))))) with [(leq_sort h1 h2 n1 n2 k H13) \Rightarrow (\lambda 
+(H14: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H15: (eq A (ASort h2 n2) 
+(ASort O x))).((let H16 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e 
+in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead 
+_ _) \Rightarrow False])) I (AHead a1 a2) H14) in (False_ind ((eq A (ASort h2 
+n2) (ASort O x)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat 
+Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: 
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) 
+w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) 
+(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 
+c0 ws))))))))) H16)) H15 H13))) | (leq_head a0 a3 H13 a4 a5 H14) \Rightarrow 
+(\lambda (H15: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H16: (eq A 
+(AHead a3 a5) (ASort O x))).((let H17 \def (f_equal A A (\lambda (e: 
+A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | 
+(AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H15) in ((let H18 
+\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) 
+with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 
+a4) (AHead a1 a2) H15) in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to 
+((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to 
+(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u 
+(TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) 
+(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: 
+C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) 
+ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: 
+C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 
+ws)))))))))))) (\lambda (H19: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6: 
+A).((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a1 a3) \to ((leq g a6 a5) 
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) 
+u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) 
+(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: 
+C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) 
+ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: 
+C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 
+ws))))))))))) (\lambda (H20: (eq A (AHead a3 a5) (ASort O x))).(let H21 \def 
+(eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: 
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
+True])) I (ASort O x) H20) in (False_ind ((leq g a1 a3) \to ((leq g a2 a5) 
+\to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) 
+u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) 
+(ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: 
+C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) 
+ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: 
+C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))) 
+H21))) a4 (sym_eq A a4 a2 H19))) a0 (sym_eq A a0 a1 H18))) H17)) H16 H13 
+H14)))]) in (H13 (refl_equal A (AHead a1 a2)) (refl_equal A (ASort O x)))) t0 
+H10))))) H9)) (\lambda (H9: (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 
+ws))))))).(ex3_4_ind TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x0: TList).(\lambda (x1: 
+nat).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq T t0 (THeads (Flat 
+Appl) x0 (TLRef x1)))).(\lambda (H11: (getl x1 c0 (CHead x2 (Bind Abst) 
+x3))).(\lambda (H12: (nfs2 c0 x0)).(let H13 \def (eq_ind T t0 (\lambda (t1: 
+T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) 
+H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 
+a2))) H2 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (eq_ind_r T (THeads (Flat 
+Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+(THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat 
+Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))))) (or3_intro2 (ex3_2 T T (\lambda (w: 
+T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef 
+x1))) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 
+w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) 
+(ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THeads (Flat Appl) x0 
+(TLRef x1))) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda 
+(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (THeads 
+(Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda 
+(_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 
+(CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda 
+(_: C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_4_intro TList nat C T (\lambda 
+(ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead 
+(Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws 
+(TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: 
+C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) 
+(TCons u x0) x1 x2 x3 (refl_equal T (THead (Flat Appl) u (THeads (Flat Appl) 
+x0 (TLRef x1)))) H11 (conj (nf2 c0 u) (nfs2 c0 x0) H6 H12))) t0 H10)))))))))) 
+H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: 
+A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_: (((nf2 c0 u) \to 
+(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0: 
+T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w 
+u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T 
+t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (H4: (nf2 
+c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 u t0 H4 (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Cast) u t0) (THead (Bind 
+Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Flat Cast) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws))))))))))))))))) (\lambda (c0: C).(\lambda 
+(t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 t0 a1)).(\lambda (H1: 
+(((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 
+(THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat 
+(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 
+ws)))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 
+c0 t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda 
+(_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))))) 
+(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i 
+c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda 
+(_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))))) 
+(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i 
+c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H5: (ex3_2 
+T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: 
+T).(nf2 (CHead c0 (Bind Abst) w) u))))).(ex3_2_ind T T (\lambda (w: 
+T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
+c0 (Bind Abst) w) u))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T 
+t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) 
+(\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat 
+(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads 
+(Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda 
+(d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind Abst) 
+x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind Abst) 
+x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T 
+T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w u)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: 
+T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))) (or3_intro0 (ex3_2 T 
+T (\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind Abst) x0 x1) (THead 
+(Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: 
+nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+(THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_2_intro T T (\lambda (w: 
+T).(\lambda (u: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) 
+(\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: 
+T).(nf2 (CHead c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) 
+x0 x1)) H7 H8)) t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq 
+T t0 (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 
+(ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w 
+u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda 
+(u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 
+(TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x: 
+nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T (TSort x) (\lambda (t1: 
+T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind 
+Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: 
+nat).(eq T t1 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda 
+(i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 (THeads (Flat Appl) ws 
+(TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: 
+C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: 
+TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))) 
+(or3_intro1 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead 
+(Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: 
+T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: 
+nat).(eq T (TSort x) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TSort x) 
+(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 
+c0 ws)))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x 
+(refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_4 TList nat C T 
+(\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T 
+t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 
+c0 ws))))))).(ex3_4_ind TList nat C T (\lambda (ws: TList).(\lambda (i: 
+nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef 
+i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: 
+T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) (or3 (ex3_2 T T 
+(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda 
+(w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 
+(CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort 
+n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda 
+(_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))))) 
+(\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i 
+c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: 
+nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x0: 
+TList).(\lambda (x1: nat).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H6: (eq 
+T t0 (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (H7: (getl x1 c0 (CHead x2 
+(Bind Abst) x3))).(\lambda (H8: (nfs2 c0 x0)).(eq_ind_r T (THeads (Flat Appl) 
+x0 (TLRef x1)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: 
+T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 
+c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) 
+(ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_4 TList nat C T (\lambda 
+(ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 
+(THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: 
+nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) 
+(\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 
+c0 ws)))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T 
+(THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w: 
+T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead 
+c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl) 
+x0 (TLRef x1)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THeads (Flat 
+Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_4_intro TList nat C T (\lambda (ws: 
+TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THeads (Flat 
+Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: 
+TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d 
+(Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: 
+C).(\lambda (_: T).(nfs2 c0 ws))))) x0 x1 x2 x3 (refl_equal T (THeads (Flat 
+Appl) x0 (TLRef x1))) H7 H8)) t0 H6)))))))) H5)) H4))))))))))) c t a H))))).
+
+theorem pc3_gen_sort_abst:
+ \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c 
+(TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda 
+(H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0 
+\def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0: 
+T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c 
+(TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def 
+(pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: 
+T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u 
+x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) 
+u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n))) 
+(pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind 
+T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda 
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P 
+H8)))))))) H3))))) H0))))))).
+
+theorem ty3_gen_abst_abst:
+ \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall 
+(t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2 
+T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) 
+u) t1 t2))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u 
+t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T 
+(\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) 
+t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2) 
+x)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c 
+(THead (Bind Abst) u t3) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: 
+T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g 
+(CHead c (Bind Abst) u) t2 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda 
+(t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) (ex2 T (\lambda (w: T).(ty3 
+g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind 
+Abst) u x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c 
+(Bind Abst) u) t2 x0)).(\lambda (_: (ty3 g (CHead c (Bind Abst) u) x0 
+x2)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c 
+(THead (Bind Abst) u t3) (THead (Bind Abst) u t2))))) (\lambda (_: 
+T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda 
+(_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t3)))) (\lambda (t3: 
+T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) 
+(ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind 
+Abst) u) t1 t2))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda 
+(H5: (pc3 c (THead (Bind Abst) u x3) (THead (Bind Abst) u t2))).(\lambda (H6: 
+(ty3 g c u x4)).(\lambda (H7: (ty3 g (CHead c (Bind Abst) u) t1 x3)).(\lambda 
+(_: (ty3 g (CHead c (Bind Abst) u) x3 x5)).(and_ind (pc3 c u u) (\forall (b: 
+B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x3 t2))) (ex2 T (\lambda (w: 
+T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) 
+(\lambda (_: (pc3 c u u)).(\lambda (H10: ((\forall (b: B).(\forall (u0: 
+T).(pc3 (CHead c (Bind b) u0) x3 t2))))).(ex_intro2 T (\lambda (w: T).(ty3 g 
+c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x4 H6 
+(ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x3 H7 (H10 Abst u))))) 
+(pc3_gen_abst c u u x3 t2 H5))))))))) (ty3_gen_bind g Abst c u t1 (THead 
+(Bind Abst) u t2) H))))))))) (ty3_gen_bind g Abst c u t2 x H0)))) 
+(ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2) H))))))).
+
+theorem ty3_typecheck:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t 
+v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H: 
+(ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u: 
+T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g 
+c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) v 
+(ty3_cast g c t v H x H0)))) (ty3_correct g c t v H)))))).
+
+inductive sort: T \to Prop \def
+| sort_sort: \forall (n: nat).(sort (TSort n))
+| sort_abst: \forall (u: T).((sort u) \to (\forall (t: T).((sort t) \to (sort 
+(THead (Bind Abst) u t))))).
+
+theorem sort_nf2:
+ \forall (t: T).((sort t) \to (\forall (c: C).(nf2 c t)))
+\def
+ \lambda (t: T).(\lambda (H: (sort t)).(sort_ind (\lambda (t0: T).(\forall 
+(c: C).(nf2 c t0))) (\lambda (n: nat).(\lambda (c: C).(nf2_sort c n))) 
+(\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: ((\forall (c: C).(nf2 c 
+u)))).(\lambda (t0: T).(\lambda (_: (sort t0)).(\lambda (H3: ((\forall (c: 
+C).(nf2 c t0)))).(\lambda (c: C).(let H_y \def (H3 (CHead c (Bind Abst) u)) 
+in (nf2_abst c u (H1 c) Abst u t0 H_y))))))))) t H)).
+
+theorem sort_pc3:
+ \forall (t1: T).((sort t1) \to (\forall (t2: T).((sort t2) \to (\forall (c: 
+C).((pc3 c t1 t2) \to (eq T t1 t2))))))
+\def
+ \lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda (t: T).(\forall 
+(t2: T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t t2)))))) 
+(\lambda (n: nat).(\lambda (t2: T).(\lambda (H0: (sort t2)).(sort_ind 
+(\lambda (t: T).(\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) 
+t)))) (\lambda (n0: nat).(\lambda (c: C).(\lambda (H1: (pc3 c (TSort n) 
+(TSort n0))).(eq_ind nat n (\lambda (n1: nat).(eq T (TSort n) (TSort n1))) 
+(refl_equal T (TSort n)) n0 (pc3_gen_sort c n n0 H1))))) (\lambda (u: 
+T).(\lambda (_: (sort u)).(\lambda (_: ((\forall (c: C).((pc3 c (TSort n) u) 
+\to (eq T (TSort n) u))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda 
+(_: ((\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) t))))).(\lambda 
+(c: C).(\lambda (H5: (pc3 c (TSort n) (THead (Bind Abst) u 
+t))).(pc3_gen_sort_abst c u t n H5 (eq T (TSort n) (THead (Bind Abst) u 
+t))))))))))) t2 H0)))) (\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: 
+((\forall (t2: T).((sort t2) \to (\forall (c: C).((pc3 c u t2) \to (eq T u 
+t2))))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: ((\forall (t2: 
+T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t 
+t2))))))).(\lambda (t2: T).(\lambda (H4: (sort t2)).(sort_ind (\lambda (t0: 
+T).(\forall (c: C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind 
+Abst) u t) t0)))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H5: (pc3 c 
+(THead (Bind Abst) u t) (TSort n))).(pc3_gen_sort_abst c u t n (pc3_s c 
+(TSort n) (THead (Bind Abst) u t) H5) (eq T (THead (Bind Abst) u t) (TSort 
+n)))))) (\lambda (u0: T).(\lambda (H5: (sort u0)).(\lambda (_: ((\forall (c: 
+C).((pc3 c (THead (Bind Abst) u t) u0) \to (eq T (THead (Bind Abst) u t) 
+u0))))).(\lambda (t0: T).(\lambda (H7: (sort t0)).(\lambda (_: ((\forall (c: 
+C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind Abst) u t) 
+t0))))).(\lambda (c: C).(\lambda (H9: (pc3 c (THead (Bind Abst) u t) (THead 
+(Bind Abst) u0 t0))).(and_ind (pc3 c u u0) (\forall (b: B).(\forall (u1: 
+T).(pc3 (CHead c (Bind b) u1) t t0))) (eq T (THead (Bind Abst) u t) (THead 
+(Bind Abst) u0 t0)) (\lambda (H10: (pc3 c u u0)).(\lambda (H11: ((\forall (b: 
+B).(\forall (u1: T).(pc3 (CHead c (Bind b) u1) t t0))))).(let H_y \def (H11 
+Abbr u) in (let H_y0 \def (H1 u0 H5 c H10) in (let H_y1 \def (H3 t0 H7 (CHead 
+c (Bind Abbr) u) H_y) in (let H12 \def (eq_ind_r T t0 (\lambda (t3: T).(pc3 
+(CHead c (Bind Abbr) u) t t3)) H_y t H_y1) in (let H13 \def (eq_ind_r T t0 
+(\lambda (t3: T).(sort t3)) H7 t H_y1) in (eq_ind T t (\lambda (t3: T).(eq T 
+(THead (Bind Abst) u t) (THead (Bind Abst) u0 t3))) (let H14 \def (eq_ind_r T 
+u0 (\lambda (t3: T).(pc3 c u t3)) H10 u H_y0) in (let H15 \def (eq_ind_r T u0 
+(\lambda (t3: T).(sort t3)) H5 u H_y0) in (eq_ind T u (\lambda (t3: T).(eq T 
+(THead (Bind Abst) u t) (THead (Bind Abst) t3 t))) (refl_equal T (THead (Bind 
+Abst) u t)) u0 H_y0))) t0 H_y1)))))))) (pc3_gen_abst c u u0 t t0 H9)))))))))) 
+t2 H4))))))))) t1 H)).
+
+theorem sort_correct:
+ \forall (g: G).(\forall (t1: T).((sort t1) \to (\forall (c: C).(ex3 T 
+(\lambda (t2: T).(tau0 g c t1 t2)) (\lambda (t2: T).(ty3 g c t1 t2)) (\lambda 
+(t2: T).(sort t2))))))
+\def
+ \lambda (g: G).(\lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda 
+(t: T).(\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2: 
+T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2))))) (\lambda (n: nat).(\lambda 
+(c: C).(ex3_intro T (\lambda (t2: T).(tau0 g c (TSort n) t2)) (\lambda (t2: 
+T).(ty3 g c (TSort n) t2)) (\lambda (t2: T).(sort t2)) (TSort (next g n)) 
+(tau0_sort g c n) (ty3_sort g c n) (sort_sort (next g n))))) (\lambda (u: 
+T).(\lambda (H0: (sort u)).(\lambda (H1: ((\forall (c: C).(ex3 T (\lambda 
+(t2: T).(tau0 g c u t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: 
+T).(sort t2)))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: 
+((\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2: 
+T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2)))))).(\lambda (c: C).(let H_x 
+\def (H1 c) in (let H4 \def H_x in (ex3_ind T (\lambda (t2: T).(tau0 g c u 
+t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: T).(sort t2)) (ex3 T 
+(\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(ty3 
+g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) (\lambda (x0: 
+T).(\lambda (_: (tau0 g c u x0)).(\lambda (H6: (ty3 g c u x0)).(\lambda (_: 
+(sort x0)).(let H_x0 \def (H3 (CHead c (Bind Abst) u)) in (let H8 \def H_x0 
+in (ex3_ind T (\lambda (t2: T).(tau0 g (CHead c (Bind Abst) u) t t2)) 
+(\lambda (t2: T).(ty3 g (CHead c (Bind Abst) u) t t2)) (\lambda (t2: T).(sort 
+t2)) (ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda 
+(t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) 
+(\lambda (x1: T).(\lambda (H9: (tau0 g (CHead c (Bind Abst) u) t 
+x1)).(\lambda (H10: (ty3 g (CHead c (Bind Abst) u) t x1)).(\lambda (H11: 
+(sort x1)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) x1 t0)) 
+(ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: 
+T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) 
+(\lambda (x: T).(\lambda (H12: (ty3 g (CHead c (Bind Abst) u) x1 
+x)).(ex3_intro T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) 
+(\lambda (t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort 
+t2)) (THead (Bind Abst) u x1) (tau0_bind g Abst c u t x1 H9) (ty3_bind g c u 
+x0 H6 Abst t x1 H10 x H12) (sort_abst u H0 x1 H11)))) (ty3_correct g (CHead c 
+(Bind Abst) u) t x1 H10)))))) H8))))))) H4)))))))))) t1 H))).
+
+definition pchurch_context:
+ T \to (T \to T)
+\def
+ \lambda (t: T).(\lambda (u: T).(THead (Bind Abst) t (THead (Bind Abst) 
+(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) u))).
+
+definition pnat:
+ T \to T
+\def
+ \lambda (t: T).(pchurch_context t (lift (S (S O)) O t)).
+
+definition church_body:
+ nat \to T
+\def
+ let rec church_body (n: nat) on n: T \def (match n with [O \Rightarrow 
+(TLRef (S O)) | (S n0) \Rightarrow (THead (Flat Appl) (church_body n0) (TLRef 
+O))]) in church_body.
+
+definition pchurch:
+ T \to (nat \to T)
+\def
+ \lambda (t: T).(\lambda (n: nat).(pchurch_context t (church_body n))).
+
+theorem pnat_props__lift_SSO_O:
+ \forall (t: T).(eq T (lift (S (S O)) O t) (lift (S O) O (lift (S O) O t)))
+\def
+ \lambda (t: T).(eq_ind_r T (lift (plus (S O) (S O)) O t) (\lambda (t0: 
+T).(eq T (lift (S (S O)) O t) t0)) (refl_equal T (lift (plus (S O) (S O)) O 
+t)) (lift (S O) O (lift (S O) O t)) (lift_free t (S O) (S O) O O (le_O_n 
+(plus O (S O))) (le_n O))).
+
+theorem pnat_props__lift_SO_SO:
+ \forall (t: T).(eq T (lift (S O) (S O) (lift (S O) O t)) (lift (S O) O (lift 
+(S O) O t)))
+\def
+ \lambda (t: T).(eq_ind nat (plus (S O) O) (\lambda (n: nat).(eq T (lift (S 
+O) n (lift (S O) O t)) (lift (S O) O (lift (S O) O t)))) (eq_ind_r T (lift (S 
+O) O (lift (S O) O t)) (\lambda (t0: T).(eq T t0 (lift (S O) O (lift (S O) O 
+t)))) (refl_equal T (lift (S O) O (lift (S O) O t))) (lift (S O) (plus (S O) 
+O) (lift (S O) O t)) (lift_d t (S O) (S O) O O (le_n O))) (S O) (refl_equal 
+nat (S O))).
+
+theorem pnat_ty3:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t 
+u) \to (\forall (n: nat).(ty3 g c (pchurch t n) (pnat t)))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H: 
+(ty3 g c t u)).(ex_ind T (\lambda (t0: T).(ty3 g c u t0)) (\forall (n: 
+nat).(ty3 g c (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) 
+(lift (S O) O t) (lift (S (S O)) O t)) (church_body n))) (THead (Bind Abst) t 
+(THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) 
+(lift (S (S O)) O t))))) (\lambda (x: T).(\lambda (H0: (ty3 g c u 
+x)).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(ty3 g c (THead (Bind Abst) 
+t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O 
+t)) (church_body n0))) (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind 
+Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t))))) 
+(ty3_bind g c t u H Abst (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O 
+t) (lift (S (S O)) O t)) (TLRef (S O))) (THead (Bind Abst) (THead (Bind Abst) 
+(lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)) (ty3_bind g 
+(CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O 
+t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g 
+(CHead c (Bind Abst) t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H 
+(CHead c (Bind Abst) t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) 
+t)) Abst (lift (S (S O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead 
+(CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S 
+Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) 
+(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) 
+(drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) 
+(ty3_lift g c u x H0 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O 
+t)) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift 
+(S O) O t)) c t (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead 
+c (Bind Abst) t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) 
+Abst (TLRef (S O)) (lift (S (S O)) O t) (ty3_abst g (S O) (CHead (CHead c 
+(Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S 
+O)) O t))) c t (getl_head (Bind Abst) O (CHead c (Bind Abst) t) (CHead c 
+(Bind Abst) t) (getl_refl Abst c t) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t))) u H) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead 
+c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S 
+O)) O t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind 
+Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) 
+(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) 
+(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t)))))) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) 
+(lift (S (S O)) O t)) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst) 
+t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Bind 
+Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst) 
+t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead c (Bind Abst) 
+t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst (lift (S (S 
+O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst) 
+t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead (CHead c 
+(Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop (Bind Abst) 
+O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl (CHead c (Bind 
+Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g c u x H0 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) 
+(drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t 
+(S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) 
+t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (lift (S (S 
+O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst) 
+t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) O 
+(S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) (drop_drop 
+(Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl 
+(CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) 
+O t))))) (lift (S (S O)) O x) (ty3_lift g c u x H0 (CHead (CHead c (Bind 
+Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O 
+t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) 
+(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) 
+(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) 
+(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t))))))) (\lambda (n0: nat).(\lambda (H1: (ty3 g c (THead (Bind 
+Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S 
+O)) O t)) (church_body n0))) (THead (Bind Abst) t (THead (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O 
+t))))).(let H_x \def (ty3_gen_abst_abst g c t (THead (Bind Abst) (THead (Bind 
+Abst) (lift (S O) O t) (lift (S (S O)) O t)) (church_body n0)) (THead (Bind 
+Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S 
+O)) O t)) H1) in (let H2 \def H_x in (ex2_ind T (\lambda (w: T).(ty3 g c t 
+w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) t) (THead (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (church_body n0)) (THead 
+(Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift 
+(S (S O)) O t)))) (ty3 g c (THead (Bind Abst) t (THead (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Flat Appl) 
+(church_body n0) (TLRef O)))) (THead (Bind Abst) t (THead (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)))) 
+(\lambda (x0: T).(\lambda (_: (ty3 g c t x0)).(\lambda (H4: (ty3 g (CHead c 
+(Bind Abst) t) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t)) (church_body n0)) (THead (Bind Abst) (THead (Bind Abst) (lift 
+(S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)))).(let H_x0 \def 
+(ty3_gen_abst_abst g (CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O 
+t) (lift (S (S O)) O t)) (church_body n0) (lift (S (S O)) O t) H4) in (let H5 
+\def H_x0 in (ex2_ind T (\lambda (w: T).(ty3 g (CHead c (Bind Abst) t) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) w)) (\lambda (_: T).(ty3 g 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O 
+t) (lift (S (S O)) O t))) (church_body n0) (lift (S (S O)) O t))) (ty3 g c 
+(THead (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) 
+(lift (S (S O)) O t)) (THead (Flat Appl) (church_body n0) (TLRef O)))) (THead 
+(Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S 
+(S O)) O t)) (lift (S (S O)) O t)))) (\lambda (x1: T).(\lambda (H6: (ty3 g 
+(CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O 
+t)) x1)).(\lambda (H7: (ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) 
+(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (church_body n0) 
+(lift (S (S O)) O t))).(ty3_bind g c t u H Abst (THead (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Flat Appl) 
+(church_body n0) (TLRef O))) (THead (Bind Abst) (THead (Bind Abst) (lift (S 
+O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)) (ty3_bind g (CHead c 
+(Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) 
+(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead 
+c (Bind Abst) t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead 
+c (Bind Abst) t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst 
+(lift (S (S O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c 
+(Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead 
+(CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop 
+(Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl 
+(CHead c (Bind Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g 
+c u x H0 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S 
+O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) 
+c t (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind 
+Abst) t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (THead 
+(Flat Appl) (church_body n0) (TLRef O)) (lift (S (S O)) O t) (ex_ind T 
+(\lambda (t0: T).(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (lift (S (S O)) O t) t0)) 
+(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S 
+O) O t) (lift (S (S O)) O t))) (THead (Flat Appl) (church_body n0) (TLRef O)) 
+(lift (S (S O)) O t)) (\lambda (x2: T).(\lambda (H8: (ty3 g (CHead (CHead c 
+(Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S 
+O)) O t))) (lift (S (S O)) O t) x2)).(ty3_conv g (CHead (CHead c (Bind Abst) 
+t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) 
+(lift (S (S O)) O t) x2 H8 (THead (Flat Appl) (church_body n0) (TLRef O)) 
+(THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S (S O)) O t) 
+(lift (S O) (S O) (lift (S O) O (lift (S O) O t))))) (ty3_appl g (CHead 
+(CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t))) (church_body n0) (lift (S (S O)) O t) H7 (TLRef O) (lift (S 
+O) (S O) (lift (S O) O (lift (S O) O t))) (eq_ind_r T (lift (S O) O (lift (S 
+O) O t)) (\lambda (t0: T).(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) 
+(THead (Bind Abst) (lift (S O) O t) t0)) (TLRef O) (THead (Bind Abst) t0 
+(lift (S O) (S O) (lift (S O) O (lift (S O) O t)))))) (let H9 \def (eq_ind T 
+(lift (S (S O)) O t) (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t) (THead 
+(Bind Abst) (lift (S O) O t) t0) x1)) H6 (lift (S O) O (lift (S O) O t)) 
+(pnat_props__lift_SSO_O t)) in (eq_ind T (lift (S O) O (THead (Bind Abst) 
+(lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (\lambda (t0: T).(ty3 g 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O 
+t) (lift (S O) O (lift (S O) O t)))) (TLRef O) t0)) (ty3_abst g O (CHead 
+(CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S O) O (lift (S O) O t)))) (CHead c (Bind Abst) t) (THead (Bind Abst) (lift 
+(S O) O t) (lift (S O) O (lift (S O) O t))) (getl_refl Abst (CHead c (Bind 
+Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O 
+t)))) x1 H9) (THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) 
+(S O) (lift (S O) O (lift (S O) O t)))) (lift_bind Abst (lift (S O) O t) 
+(lift (S O) O (lift (S O) O t)) (S O) O))) (lift (S (S O)) O t) 
+(pnat_props__lift_SSO_O t))) (eq_ind_r T (lift (S O) O (lift (S O) O (lift (S 
+O) O t))) (\lambda (t0: T).(pc3 (CHead (CHead c (Bind Abst) t) (Bind Abst) 
+(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (THead (Flat Appl) 
+(church_body n0) (THead (Bind Abst) (lift (S (S O)) O t) t0)) (lift (S (S O)) 
+O t))) (eq_ind_r T (lift (S O) O (lift (S O) O t)) (\lambda (t0: T).(pc3 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O 
+t) t0)) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) t0 (lift (S O) 
+O (lift (S O) O (lift (S O) O t))))) t0)) (pc3_pr3_r (CHead (CHead c (Bind 
+Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift 
+(S O) O t)))) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S 
+O) O (lift (S O) O t)) (lift (S O) O (lift (S O) O (lift (S O) O t))))) (lift 
+(S O) O (lift (S O) O t)) (pr3_t (THead (Bind Abbr) (church_body n0) (lift (S 
+O) O (lift (S O) O (lift (S O) O t)))) (THead (Flat Appl) (church_body n0) 
+(THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) O (lift (S O) 
+O (lift (S O) O t))))) (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (pr3_pr2 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O 
+t) (lift (S O) O (lift (S O) O t)))) (THead (Flat Appl) (church_body n0) 
+(THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) O (lift (S O) 
+O (lift (S O) O t))))) (THead (Bind Abbr) (church_body n0) (lift (S O) O 
+(lift (S O) O (lift (S O) O t)))) (pr2_free (CHead (CHead c (Bind Abst) t) 
+(Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O 
+t)))) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S O) O 
+(lift (S O) O t)) (lift (S O) O (lift (S O) O (lift (S O) O t))))) (THead 
+(Bind Abbr) (church_body n0) (lift (S O) O (lift (S O) O (lift (S O) O t)))) 
+(pr0_beta (lift (S O) O (lift (S O) O t)) (church_body n0) (church_body n0) 
+(pr0_refl (church_body n0)) (lift (S O) O (lift (S O) O (lift (S O) O t))) 
+(lift (S O) O (lift (S O) O (lift (S O) O t))) (pr0_refl (lift (S O) O (lift 
+(S O) O (lift (S O) O t))))))) (lift (S O) O (lift (S O) O t)) (pr3_pr2 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O 
+t) (lift (S O) O (lift (S O) O t)))) (THead (Bind Abbr) (church_body n0) 
+(lift (S O) O (lift (S O) O (lift (S O) O t)))) (lift (S O) O (lift (S O) O 
+t)) (pr2_free (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) 
+(lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (THead (Bind Abbr) 
+(church_body n0) (lift (S O) O (lift (S O) O (lift (S O) O t)))) (lift (S O) 
+O (lift (S O) O t)) (pr0_zeta Abbr not_abbr_abst (lift (S O) O (lift (S O) O 
+t)) (lift (S O) O (lift (S O) O t)) (pr0_refl (lift (S O) O (lift (S O) O 
+t))) (church_body n0)))))) (lift (S (S O)) O t) (pnat_props__lift_SSO_O t)) 
+(lift (S O) (S O) (lift (S O) O (lift (S O) O t))) (pnat_props__lift_SO_SO 
+(lift (S O) O t)))))) (ty3_correct g (CHead (CHead c (Bind Abst) t) (Bind 
+Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (church_body 
+n0) (lift (S (S O)) O t) H7)) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead 
+(CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) 
+(Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S 
+O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) 
+(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t)))))) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) 
+(lift (S (S O)) O t)) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst) 
+t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Bind 
+Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst) 
+t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead c (Bind Abst) 
+t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst (lift (S (S 
+O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst) 
+t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead (CHead c 
+(Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop (Bind Abst) 
+O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl (CHead c (Bind 
+Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g c u x H0 
+(CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) 
+(drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t 
+(S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) 
+t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (lift (S (S 
+O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst) 
+t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) O 
+(S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead 
+(Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) (drop_drop 
+(Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl 
+(CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) 
+O t))))) (lift (S (S O)) O x) (ty3_lift g c u x H0 (CHead (CHead c (Bind 
+Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O 
+t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) 
+(THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) 
+(drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) 
+(drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift 
+(S (S O)) O t)))))))))) H5)))))) H2))))) n)))) (ty3_correct g c t u H)))))).
+