update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / ty3 / fwd.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
index bf6634e451e6e3b7b78acab46041119e3ec42adf..318a97986dce30bbc3c811a81b4f5dca43a121ce 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/ty3/fwd.ma
@@ -14,11 +14,55 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/ty3/defs.ma".
+include "basic_1/ty3/defs.ma".
 
-include "Basic-1/pc3/props.ma".
+include "basic_1/pc3/props.ma".
 
-theorem ty3_gen_sort:
+implied rec lemma ty3_ind (g: G) (P: (C \to (T \to (T \to Prop)))) (f: 
+(\forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t) \to ((P c t2 
+t) \to (\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to ((P c u t1) \to 
+((pc3 c t1 t2) \to (P c u t2)))))))))))) (f0: (\forall (c: C).(\forall (m: 
+nat).(P c (TSort m) (TSort (next g m)))))) (f1: (\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 ((P d u t) \to (P c (TLRef n) (lift (S 
+n) O t))))))))))) (f2: (\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 ((P d u t) \to (P c (TLRef n) (lift (S n) O 
+u))))))))))) (f3: (\forall (c: C).(\forall (u: T).(\forall (t: T).((ty3 g c u 
+t) \to ((P c u t) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 
+g (CHead c (Bind b) u) t1 t2) \to ((P (CHead c (Bind b) u) t1 t2) \to (P c 
+(THead (Bind b) u t1) (THead (Bind b) u t2))))))))))))) (f4: (\forall (c: 
+C).(\forall (w: T).(\forall (u: T).((ty3 g c w u) \to ((P c w u) \to (\forall 
+(v: T).(\forall (t: T).((ty3 g c v (THead (Bind Abst) u t)) \to ((P c v 
+(THead (Bind Abst) u t)) \to (P c (THead (Flat Appl) w v) (THead (Flat Appl) 
+w (THead (Bind Abst) u t))))))))))))) (f5: (\forall (c: C).(\forall (t1: 
+T).(\forall (t2: T).((ty3 g c t1 t2) \to ((P c t1 t2) \to (\forall (t0: 
+T).((ty3 g c t2 t0) \to ((P c t2 t0) \to (P c (THead (Flat Cast) t2 t1) 
+(THead (Flat Cast) t0 t2))))))))))) (c: C) (t: T) (t0: T) (t1: ty3 g c t t0) 
+on t1: P c t t0 \def match t1 with [(ty3_conv c0 t2 t3 t4 u t5 t6 p) 
+\Rightarrow (f c0 t2 t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t2 t3 t4) u 
+t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 u t5 t6) p) | (ty3_sort c0 m) 
+\Rightarrow (f0 c0 m) | (ty3_abbr n c0 d u g0 t2 t3) \Rightarrow (f1 n c0 d u 
+g0 t2 t3 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) d u t2 t3)) | (ty3_abst n c0 d u 
+g0 t2 t3) \Rightarrow (f2 n c0 d u g0 t2 t3 ((ty3_ind g P f f0 f1 f2 f3 f4 
+f5) d u t2 t3)) | (ty3_bind c0 u t2 t3 b t4 t5 t6) \Rightarrow (f3 c0 u t2 t3 
+((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 u t2 t3) b t4 t5 t6 ((ty3_ind g P f f0 
+f1 f2 f3 f4 f5) (CHead c0 (Bind b) u) t4 t5 t6)) | (ty3_appl c0 w u t2 v t3 
+t4) \Rightarrow (f4 c0 w u t2 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 w u t2) v 
+t3 t4 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 v (THead (Bind Abst) u t3) t4)) | 
+(ty3_cast c0 t2 t3 t4 t5 t6) \Rightarrow (f5 c0 t2 t3 t4 ((ty3_ind g P f f0 
+f1 f2 f3 f4 f5) c0 t2 t3 t4) t5 t6 ((ty3_ind g P f f0 f1 f2 f3 f4 f5) c0 t3 
+t5 t6))].
+
+implied rec lemma tys3_ind (g: G) (c: C) (P: (TList \to (T \to Prop))) (f: 
+(\forall (u: T).(\forall (u0: T).((ty3 g c u u0) \to (P TNil u))))) (f0: 
+(\forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts: 
+TList).((tys3 g c ts u) \to ((P ts u) \to (P (TCons t ts) u)))))))) (t: 
+TList) (t0: T) (t1: tys3 g c t t0) on t1: P t t0 \def match t1 with 
+[(tys3_nil u u0 t2) \Rightarrow (f u u0 t2) | (tys3_cons t2 u t3 ts t4) 
+\Rightarrow (f0 t2 u t3 ts t4 ((tys3_ind g c P f f0) ts u t4))].
+
+lemma ty3_gen_sort:
  \forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c 
 (TSort n) x) \to (pc3 c (TSort (next g n)) x)))))
 \def
@@ -37,59 +81,53 @@ T).e) u (TSort n) H6) in (let H8 \def (eq_ind T u (\lambda (t0: T).((eq T t0
 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TSort n) H7) in 
 (pc3_t t1 c0 (TSort (next g n)) (H8 (refl_equal T (TSort n))) t2 
 H5))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
-(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e in 
-T return (\lambda (_: T).nat) with [(TSort n0) \Rightarrow n0 | (TLRef _) 
-\Rightarrow m | (THead _ _ _) \Rightarrow m])) (TSort m) (TSort n) H1) in 
-(eq_ind_r nat n (\lambda (n0: nat).(pc3 c0 (TSort (next g n)) (TSort (next g 
-n0)))) (pc3_refl c0 (TSort (next g n))) m H2))))) (\lambda (n0: nat).(\lambda 
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d 
-(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: 
-(((eq T u (TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T 
-(TLRef n0) (TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) (lift (S n0) O t)) 
-H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) u))).(\lambda (t: 
-T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u (TSort n)) \to (pc3 d 
-(TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) (TSort n))).(let H5 
-\def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 
-(TSort (next g n)) (lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda 
-(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u 
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 
-t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort 
-(next g n)) t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let 
-H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+(TSort m) (TSort n))).(let H2 \def (f_equal T nat (\lambda (e: T).(match e 
+with [(TSort n0) \Rightarrow n0 | (TLRef _) \Rightarrow m | (THead _ _ _) 
+\Rightarrow m])) (TSort m) (TSort n) H1) in (eq_ind_r nat n (\lambda (n0: 
+nat).(pc3 c0 (TSort (next g n)) (TSort (next g n0)))) (pc3_refl c0 (TSort 
+(next g n))) m H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abbr) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
+(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) 
+(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) 
+(lift (S n0) O t)) H5))))))))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda 
+(d: C).(\lambda (u: T).(\lambda (_: (getl n0 c0 (CHead d (Bind Abst) 
+u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
+(TSort n)) \to (pc3 d (TSort (next g n)) t)))).(\lambda (H4: (eq T (TLRef n0) 
+(TSort n))).(let H5 \def (eq_ind T (TLRef n0) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (TSort n) H4) in (False_ind (pc3 c0 (TSort (next g n)) 
+(lift (S n0) O u)) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda 
+(t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TSort n)) \to 
+(pc3 c0 (TSort (next g n)) t)))).(\lambda (b: B).(\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq 
+T t1 (TSort n)) \to (pc3 (CHead c0 (Bind b) u) (TSort (next g n)) 
+t2)))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TSort n))).(let H6 \def 
+(eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Bind 
+b) u t2)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 
+(TSort (next g n)) u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c0 v (THead (Bind Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 
+(TSort (next g n)) (THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead 
+(Flat Appl) w v) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
-(False_ind (pc3 c0 (TSort (next g n)) (THead (Bind b) u t2)) H6))))))))))))) 
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w 
-u)).(\lambda (_: (((eq T w (TSort n)) \to (pc3 c0 (TSort (next g n)) 
-u)))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind 
-Abst) u t))).(\lambda (_: (((eq T v (TSort n)) \to (pc3 c0 (TSort (next g n)) 
-(THead (Bind Abst) u t))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) 
-(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w 
-(THead (Bind Abst) u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 
-(TSort n)) \to (pc3 c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda 
-(_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort 
-(next g n)) t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort 
-n))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
-H5) in (False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Cast) t0 t2)) 
-H6))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 1179
-END *)
+(False_ind (pc3 c0 (TSort (next g n)) (THead (Flat Appl) w (THead (Bind Abst) 
+u t))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (pc3 
+c0 (TSort (next g n)) t2)))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 
+t0)).(\lambda (_: (((eq T t2 (TSort n)) \to (pc3 c0 (TSort (next g n)) 
+t0)))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TSort n))).(let H6 \def 
+(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H5) in (False_ind (pc3 c0 (TSort (next g n)) 
+(THead (Flat Cast) t0 t2)) H6))))))))))) c y x H0))) H))))).
 
-theorem ty3_gen_lref:
+lemma 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 
@@ -215,28 +253,27 @@ T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda
 C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) x0 x1 x2 (pc3_t t1 c0 
 (lift (S n) O x1) H12 t2 H5) H13 H14)))))))) H11)) H10)))))))))))))))) 
 (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (TLRef 
-n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H1) in 
-(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: 
-T).(pc3 c0 (lift (S n) O t) (TSort (next g m)))))) (\lambda (e: C).(\lambda 
-(u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
-(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (TSort (next 
-g m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t)))))) H2))))) (\lambda (n0: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
-(u: T).(\lambda (H1: (getl n0 c0 (CHead d (Bind Abbr) u))).(\lambda (t: 
-T).(\lambda (H2: (ty3 g d u t)).(\lambda (_: (((eq T u (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S 
-n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
-(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(pc3 d (lift (S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl n d (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H4: 
-(eq T (TLRef n0) (TLRef n))).(let H5 \def (f_equal T nat (\lambda (e: 
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n0 | 
+n))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+False])) I (TLRef n) H1) in (False_ind (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (TSort (next g 
+m)))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 
+(lift (S n) O u) (TSort (next g m)))))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) H2))))) (\lambda (n0: 
+nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H1: (getl n0 
+c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d u 
+t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 d (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 d (lift (S 
+n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n0 | 
 (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
 n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
 (Bind Abbr) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
@@ -269,50 +306,49 @@ u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))
 n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n d 
 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
 T).(ty3 g e u0 t0))))))))).(\lambda (H4: (eq T (TLRef n0) (TLRef n))).(let H5 
-\def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) 
-with [(TSort _) \Rightarrow n0 | (TLRef n1) \Rightarrow n1 | (THead _ _ _) 
-\Rightarrow n0])) (TLRef n0) (TLRef n) H4) in (let H6 \def (eq_ind nat n0 
-(\lambda (n1: nat).(getl n1 c0 (CHead d (Bind Abst) u))) H1 n H5) in 
-(eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C T T (\lambda (_: C).(\lambda 
-(_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n1) O u))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
-Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 
-(lift (S n) O u0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))) (or_intror (ex3_3 
-C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
-t0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+\def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow n0 | 
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n0])) (TLRef n0) (TLRef 
+n) H4) in (let H6 \def (eq_ind nat n0 (\lambda (n1: nat).(getl n1 c0 (CHead d 
+(Bind Abst) u))) H1 n H5) in (eq_ind_r nat n (\lambda (n1: nat).(or (ex3_3 C 
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O 
+t0) (lift (S n1) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
 T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
 T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) 
+(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n1) O u))))) 
 (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
 Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0))))) (ex3_3_intro C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 c0 (lift (S n) O u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda 
-(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))) d u t (pc3_refl c0 
-(lift (S n) O u)) H6 H2)) n0 H5)))))))))))) (\lambda (c0: C).(\lambda (u: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef 
-n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda 
-(_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) t)))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
-(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
-t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: 
-(ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to 
-(or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 (CHead 
-c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: C).(\lambda (u0: 
-T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0))))) 
+t0))))))) (or_intror (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t0: T).(pc3 c0 (lift (S n) O t0) (lift (S n) O u))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) 
 (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 
-C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 (CHead c0 (Bind 
-b) u) (lift (S n) O u0) t2)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
-T).(getl n (CHead c0 (Bind b) u) (CHead e (Bind Abst) u0))))) (\lambda (e: 
-C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))))))).(\lambda (H5: 
-(eq T (THead (Bind b) u t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Bind 
-b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O 
+u0) (lift (S n) O u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: 
+T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3_intro C T T (\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) (lift (S n) O 
+u))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0)))) d u t (pc3_refl c0 (lift (S n) O u)) H6 H2)) n0 H5)))))))))))) 
+(\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u 
+t)).(\lambda (_: (((eq T u (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) t)))) (\lambda 
+(e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) 
+u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) 
+(ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift 
+(S n) O u0) t)))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: 
+T).(ty3 g e u0 t0))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
+T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O t0) t2)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
+(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
+T).(pc3 (CHead c0 (Bind b) u) (lift (S n) O u0) t2)))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (_: T).(getl n (CHead c0 (Bind b) u) (CHead e 
+(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
+e u0 t0))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef n))).(let 
+H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
 \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
 (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead 
@@ -341,51 +377,47 @@ T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
 C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) 
 (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 
 t0))))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (TLRef n))).(let H6 
-\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee in T return 
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in 
-(False_ind (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: 
-T).(pc3 c0 (lift (S n) O t0) (THead (Flat Appl) w (THead (Bind Abst) u 
-t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: 
-T).(pc3 c0 (lift (S n) O u0) (THead (Flat Appl) w (THead (Bind Abst) u 
-t)))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl n c0 (CHead e 
-(Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-e u0 t0)))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: 
-T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S 
-n) O t) t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
-(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u) t2)))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (t0: 
-T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef n)) \to (or 
-(ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S 
-n) O t) t0)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
-(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
-T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(pc3 c0 (lift (S n) O u) t0)))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) (\lambda (e: 
-C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))))))).(\lambda (H5: (eq T 
-(THead (Flat Cast) t2 t1) (TLRef n))).(let H6 \def (eq_ind T (THead (Flat 
-Cast) t2 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ 
-_) \Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T 
-(\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) 
-(THead (Flat Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: 
-T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: 
-T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) 
+\def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c0 (lift (S n) O t0) (THead 
+(Flat Appl) w (THead (Bind Abst) u t)))))) (\lambda (e: C).(\lambda (u0: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex3_3 C T T 
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u0) 
+(THead (Flat Appl) w (THead (Bind Abst) u t)))))) (\lambda (e: C).(\lambda 
+(u0: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u0))))) (\lambda (e: 
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))) H6)))))))))))) 
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 
+t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: 
+C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) t2)))) (\lambda 
+(e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T 
+T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) 
+t2)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e 
+(Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
+t))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq 
+T t2 (TLRef n)) \to (or (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda 
+(t: T).(pc3 c0 (lift (S n) O t) t0)))) (\lambda (e: C).(\lambda (u: 
+T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abbr) u))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c0 (lift (S n) O u) t0)))) 
 (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind 
 Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u 
-t)))))) H6))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 5569
-END *)
+t))))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 t1) (TLRef n))).(let H6 
+\def (eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (or (ex3_3 C T T (\lambda 
+(_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c0 (lift (S n) O t) (THead (Flat 
+Cast) t0 t2))))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c0 
+(CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: 
+T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(pc3 c0 (lift (S n) O u) (THead (Flat Cast) t0 t2))))) (\lambda (e: 
+C).(\lambda (u: T).(\lambda (_: T).(getl n c0 (CHead e (Bind Abst) u))))) 
+(\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))) 
+H6))))))))))) c y x H0))) H))))).
 
-theorem ty3_gen_bind:
+lemma 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 (ex3_2 T T (\lambda 
 (t2: T).(\lambda (_: T).(pc3 c (THead (Bind b) u t2) x))) (\lambda (_: 
@@ -433,37 +465,35 @@ u) t1 x0)).(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead
 (pc3_t t0 c0 (THead (Bind b) u x0) H11 t2 H5) H12 H13)))))) 
 H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T 
 (TSort m) (THead (Bind b) u t1))).(let H2 \def (eq_ind T (TSort m) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind b) u t1) H1) in (False_ind (ex3_2 T T (\lambda (t2: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (TSort (next g m))))) 
-(\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda 
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H2))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (_: (getl n 
-c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 
-t)).(\lambda (_: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u t2) t))) (\lambda (_: 
-T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
-(CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind 
-b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) 
-H4) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead 
-(Bind b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 
-u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
-t2)))) H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
-C).(\lambda (u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) 
-u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d 
-(THead (Bind b) u t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) 
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 
-t2))))))).(\lambda (H4: (eq T (TLRef n) (THead (Bind b) u t1))).(let H5 \def 
-(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in (False_ind 
-(ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) 
-(lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) 
-(\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H1) in 
+(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
+b) u t2) (TSort (next g m))))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u 
+t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
+H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: 
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u 
+t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: 
+T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq 
+T (TLRef n) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in 
+(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
+b) u t2) (lift (S n) O t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
+t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda 
+(u0: T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: 
+T).(\lambda (_: (ty3 g d u0 t)).(\lambda (_: (((eq T u0 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 d (THead (Bind b) u 
+t2) t))) (\lambda (_: T).(\lambda (t0: T).(ty3 g d u t0))) (\lambda (t2: 
+T).(\lambda (_: T).(ty3 g (CHead d (Bind b) u) t1 t2))))))).(\lambda (H4: (eq 
+T (TLRef n) (THead (Bind b) u t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda 
+(ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow 
+True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t1) H4) in 
+(False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind 
+b) u t2) (lift (S n) O u0)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u 
+t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) 
 H5))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H1: 
 (ty3 g c0 u0 t)).(\lambda (H2: (((eq T u0 (THead (Bind b) u t1)) \to (ex3_2 T 
 T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) t))) 
@@ -475,93 +505,86 @@ t2)).(\lambda (H4: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda
 t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b0) u0) u t4))) 
 (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind 
 b) u) t1 t3))))))).(\lambda (H5: (eq T (THead (Bind b0) u0 t0) (THead (Bind 
-b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e in T return 
-(\lambda (_: T).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 
-| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with 
-[(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 
-t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | 
-(TLRef _) \Rightarrow u0 | (THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) 
-u0 t0) (THead (Bind b) u t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow t3])) (THead (Bind b0) 
-u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq T u0 u)).(\lambda (H10: 
-(eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: T).((eq T t3 (THead 
+b) u t1))).(let H6 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match 
+k with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind 
+b0) u0 t0) (THead (Bind b) u t1) H5) in ((let H7 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | 
+(THead _ t3 _) \Rightarrow t3])) (THead (Bind b0) u0 t0) (THead (Bind b) u 
+t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t3) \Rightarrow 
+t3])) (THead (Bind b0) u0 t0) (THead (Bind b) u t1) H5) in (\lambda (H9: (eq 
+T u0 u)).(\lambda (H10: (eq B b0 b)).(let H11 \def (eq_ind T t0 (\lambda (t3: 
+T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda 
+(_: T).(pc3 (CHead c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: 
+T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: 
+T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 
+t4)))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g 
+(CHead c0 (Bind b0) u0) t3 t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 
+(\lambda (b1: B).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
+(t3: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b1) u0) (THead (Bind b) u t3) 
+t2))) (\lambda (_: T).(\lambda (t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) 
+(\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind 
+b) u) t1 t3)))))) H11 b H10) in (let H14 \def (eq_ind B b0 (\lambda (b1: 
+B).(ty3 g (CHead c0 (Bind b1) u0) t1 t2)) H12 b H10) in (eq_ind_r B b 
+(\lambda (b1: B).(ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead 
+(Bind b) u t3) (THead (Bind b1) u0 t2)))) (\lambda (_: T).(\lambda (t4: 
+T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind 
+b) u) t1 t3))))) (let H15 \def (eq_ind T u0 (\lambda (t3: T).((eq T t1 (THead 
 (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 (CHead 
-c0 (Bind b0) u0) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: 
-T).(ty3 g (CHead c0 (Bind b0) u0) u t5))) (\lambda (t4: T).(\lambda (_: 
-T).(ty3 g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t1 t4)))))) H4 t1 H8) in 
-(let H12 \def (eq_ind T t0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b0) u0) t3 
-t2)) H3 t1 H8) in (let H13 \def (eq_ind B b0 (\lambda (b1: B).((eq T t1 
-(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 
-(CHead c0 (Bind b1) u0) (THead (Bind b) u t3) t2))) (\lambda (_: T).(\lambda 
-(t4: T).(ty3 g (CHead c0 (Bind b1) u0) u t4))) (\lambda (t3: T).(\lambda (_: 
-T).(ty3 g (CHead (CHead c0 (Bind b1) u0) (Bind b) u) t1 t3)))))) H11 b H10) 
-in (let H14 \def (eq_ind B b0 (\lambda (b1: B).(ty3 g (CHead c0 (Bind b1) u0) 
-t1 t2)) H12 b H10) in (eq_ind_r B b (\lambda (b1: B).(ex3_2 T T (\lambda (t3: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b1) u0 t2)))) 
+c0 (Bind b) t3) (THead (Bind b) u t4) t2))) (\lambda (_: T).(\lambda (t5: 
+T).(ty3 g (CHead c0 (Bind b) t3) u t5))) (\lambda (t4: T).(\lambda (_: 
+T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind b) u) t1 t4)))))) H13 u H9) in 
+(let H16 \def (eq_ind T u0 (\lambda (t3: T).(ty3 g (CHead c0 (Bind b) t3) t1 
+t2)) H14 u H9) in (let H17 \def (eq_ind T u0 (\lambda (t3: T).((eq T t3 
+(THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 
+c0 (THead (Bind b) u t4) t))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u 
+t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
+t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 (\lambda (t3: T).(ty3 g c0 t3 
+t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: T).(ex3_2 T T (\lambda (t4: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Bind b) t3 t2)))) 
+(\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda 
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) (ex3_2_intro T T (\lambda (t3: 
+T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) (THead (Bind b) u t2)))) 
 (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u t4))) (\lambda (t3: T).(\lambda 
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))) (let H15 \def (eq_ind T u0 
-(\lambda (t3: T).((eq T t1 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t4: T).(\lambda (_: T).(pc3 (CHead c0 (Bind b) t3) (THead (Bind b) u t4) 
-t2))) (\lambda (_: T).(\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t3) u t5))) 
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead (CHead c0 (Bind b) t3) (Bind 
-b) u) t1 t4)))))) H13 u H9) in (let H16 \def (eq_ind T u0 (\lambda (t3: 
-T).(ty3 g (CHead c0 (Bind b) t3) t1 t2)) H14 u H9) in (let H17 \def (eq_ind T 
-u0 (\lambda (t3: T).((eq T t3 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t))) (\lambda (_: 
-T).(\lambda (t5: T).(ty3 g c0 u t5))) (\lambda (t4: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) u) t1 t4)))))) H2 u H9) in (let H18 \def (eq_ind T u0 
-(\lambda (t3: T).(ty3 g c0 t3 t)) H1 u H9) in (eq_ind_r T u (\lambda (t3: 
-T).(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) 
-(THead (Bind b) t3 t2)))) (\lambda (_: T).(\lambda (t5: T).(ty3 g c0 u t5))) 
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))) 
-(ex3_2_intro T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u 
-t3) (THead (Bind b) u t2)))) (\lambda (_: T).(\lambda (t4: T).(ty3 g c0 u 
-t4))) (\lambda (t3: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) 
-t2 t (pc3_refl c0 (THead (Bind b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) 
-H7)) H6))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: 
-T).(\lambda (_: (ty3 g c0 w u0)).(\lambda (_: (((eq T w (THead (Bind b) u 
-t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
-u t2) u0))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u0 
-t))).(\lambda (_: (((eq T v (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Bind Abst) u0 
-t)))) (\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2))))))).(\lambda (H5: 
-(eq T (THead (Flat Appl) w v) (THead (Bind b) u t1))).(let H6 \def (eq_ind T 
-(THead (Flat Appl) w v) (\lambda (ee: T).(match ee 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 
-b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 
-c0 (THead (Bind b) u t2) (THead (Flat Appl) w (THead (Bind Abst) u0 t))))) 
-(\lambda (_: T).(\lambda (t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda 
-(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)))) H6)))))))))))) (\lambda (c0: 
-C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda 
-(_: (((eq T t0 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda (t3: 
-T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t3) t2))) (\lambda (_: 
-T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: T).(\lambda (_: (ty3 g c0 t2 
-t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
-(t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) t3))) (\lambda (_: 
-T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 g 
-(CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: (eq T (THead (Flat Cast) t2 
-t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Cast) t2 t0) 
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+(_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))) t2 t (pc3_refl c0 (THead (Bind 
+b) u t2)) H18 H16) u0 H9))))) b0 H10)))))))) H7)) H6))))))))))))) (\lambda 
+(c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (_: (ty3 g c0 w 
+u0)).(\lambda (_: (((eq T w (THead (Bind b) u t1)) \to (ex3_2 T T (\lambda 
+(t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) u0))) (\lambda (_: 
+T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t2: T).(\lambda (_: T).(ty3 g 
+(CHead c0 (Bind b) u) t1 t2))))))).(\lambda (v: T).(\lambda (t: T).(\lambda 
+(_: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (_: (((eq T v (THead 
+(Bind b) u t1)) \to (ex3_2 T T (\lambda (t2: T).(\lambda (_: T).(pc3 c0 
+(THead (Bind b) u t2) (THead (Bind Abst) u0 t)))) (\lambda (_: T).(\lambda 
+(t0: T).(ty3 g c0 u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 
+(Bind b) u) t1 t2))))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead 
+(Bind b) u t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat 
+_) \Rightarrow True])])) I (THead (Bind b) u t1) H5) in (False_ind (ex3_2 T T 
+(\lambda (t2: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t2) (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t))))) (\lambda (_: T).(\lambda (t0: T).(ty3 g 
+c0 u t0))) (\lambda (t2: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 
+t2)))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (ty3 g c0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t3: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
+u t3) t2))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t3: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t3))))))).(\lambda (t3: 
+T).(\lambda (_: (ty3 g c0 t2 t3)).(\lambda (_: (((eq T t2 (THead (Bind b) u 
+t1)) \to (ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) 
+u t4) t3))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: 
+T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4))))))).(\lambda (H5: 
+(eq T (THead (Flat Cast) t2 t0) (THead (Bind b) u t1))).(let H6 \def (eq_ind 
+T (THead (Flat Cast) t2 t0) (\lambda (ee: T).(match ee 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 b) u t1) H5) in (False_ind 
-(ex3_2 T T (\lambda (t4: T).(\lambda (_: T).(pc3 c0 (THead (Bind b) u t4) 
-(THead (Flat Cast) t3 t2)))) (\lambda (_: T).(\lambda (t: T).(ty3 g c0 u t))) 
-(\lambda (t4: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t4)))) 
-H6))))))))))) c y x H0))) H))))))).
-(* COMMENTS
-Initial nodes: 3389
-END *)
+(match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I 
+(THead (Bind b) u t1) H5) in (False_ind (ex3_2 T T (\lambda (t4: T).(\lambda 
+(_: T).(pc3 c0 (THead (Bind b) u t4) (THead (Flat Cast) t3 t2)))) (\lambda 
+(_: T).(\lambda (t: T).(ty3 g c0 u t))) (\lambda (t4: T).(\lambda (_: T).(ty3 
+g (CHead c0 (Bind b) u) t1 t4)))) H6))))))))))) c y x H0))) H))))))).
 
-theorem ty3_gen_appl:
+lemma 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))) 
@@ -612,124 +635,116 @@ T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0:
 T).(\lambda (_: T).(ty3 g c0 w u0))) x0 x1 (pc3_t t1 c0 (THead (Flat Appl) w 
 (THead (Bind Abst) x0 x1)) H11 t2 H5) H12 H13)))))) H10)))))))))))))))) 
 (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq T (TSort m) (THead (Flat 
-Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
-\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w 
-v) H1) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 
-(THead (Flat Appl) w (THead (Bind Abst) u t)) (TSort (next g m))))) (\lambda 
-(u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: 
-T).(\lambda (_: T).(ty3 g c0 w u)))) H2))))) (\lambda (n: nat).(\lambda (c0: 
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d (Bind 
-Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: (((eq T u 
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
-T).(pc3 d (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g d v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g d w u0))))))).(\lambda (H4: (eq T (TLRef n) (THead 
-(Flat Appl) w v))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
-(Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O 
-t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 
-t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) 
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda 
-(_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g 
-d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T T 
-(\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead (Bind 
-Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead (Bind 
-Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
+Appl) w v))).(let H2 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Appl) w v) H1) in (False_ind (ex3_2 T T 
+(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) (TSort (next g m))))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v 
+(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) 
+H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 
+T T (\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead 
+(Bind Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
 u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
-\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H4) in 
-(False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat 
-Appl) w (THead (Bind Abst) u0 t0)) (lift (S n) O u)))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0)))) H5))))))))))) (\lambda (c0: C).(\lambda 
-(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u 
-(THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t))) (\lambda (u0: 
-T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (b: B).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t1 
-t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda 
-(u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Appl) w 
-(THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g 
-(CHead c0 (Bind b) u) v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w u0))))))).(\lambda (H5: (eq 
-T (THead (Bind b) u t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T 
-(THead (Bind b) u t1) (\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 True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u0: T).(\lambda (t0: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Bind b) u 
-t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 
-t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) H6))))))))))))) 
-(\lambda (c0: C).(\lambda (w0: T).(\lambda (u: T).(\lambda (H1: (ty3 g c0 w0 
-u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda 
-(u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 
-t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u0 
-t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (v0: 
-T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 (THead (Bind Abst) u 
-t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) \to (ex3_2 T T 
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
+(lift (S n) O t)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) 
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda 
+(_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 
+T T (\lambda (u0: T).(\lambda (t0: T).(pc3 d (THead (Flat Appl) w (THead 
+(Bind Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g d v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g d w 
+u0))))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Appl) w v))).(let H5 
+\def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
+False])) I (THead (Flat Appl) w v) H4) in (False_ind (ex3_2 T T (\lambda (u0: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
+(lift (S n) O u)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead 
+(Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))) 
+H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: 
+(ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Appl) w v)) \to (ex3_2 T 
+T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t0)) t))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind 
+Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w 
+u0))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 
+g (CHead c0 (Bind b) u) t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat Appl) w 
+v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 (CHead c0 (Bind b) 
+u) (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) t2))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g (CHead c0 (Bind b) u) v (THead (Bind Abst) u0 
+t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) w 
+u0))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (THead (Flat Appl) w 
+v))).(let H6 \def (eq_ind T (THead (Bind b) u t1) (\lambda (ee: T).(match ee 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) 
+\Rightarrow False])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T 
 (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
-Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: T).(\lambda (t0: 
-T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: 
-T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat Appl) w0 v0) (THead 
-(Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: T).(match e in T 
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow w0 | (TLRef _) 
-\Rightarrow w0 | (THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) 
-(THead (Flat Appl) w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | 
-(TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow t0])) (THead (Flat 
-Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: (eq T w0 w)).(let 
-H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to 
-(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w 
-(THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) (\lambda (u0: 
+Abst) u0 t0)) (THead (Bind b) u t2)))) (\lambda (u0: T).(\lambda (t0: T).(ty3 
+g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g 
+c0 w u0)))) H6))))))))))))) (\lambda (c0: C).(\lambda (w0: T).(\lambda (u: 
+T).(\lambda (H1: (ty3 g c0 w0 u)).(\lambda (H2: (((eq T w0 (THead (Flat Appl) 
+w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t: T).(pc3 c0 (THead (Flat 
+Appl) w (THead (Bind Abst) u0 t)) u))) (\lambda (u0: T).(\lambda (t: T).(ty3 
+g c0 v (THead (Bind Abst) u0 t)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 
+w u0))))))).(\lambda (v0: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v0 
+(THead (Bind Abst) u t))).(\lambda (H4: (((eq T v0 (THead (Flat Appl) w v)) 
+\to (ex3_2 T T (\lambda (u0: T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w 
+(THead (Bind Abst) u0 t0)) (THead (Bind Abst) u t)))) (\lambda (u0: 
+T).(\lambda (t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: 
+T).(\lambda (_: T).(ty3 g c0 w u0))))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) w0 v0) (THead (Flat Appl) w v))).(let H6 \def (f_equal T T (\lambda (e: 
+T).(match e with [(TSort _) \Rightarrow w0 | (TLRef _) \Rightarrow w0 | 
+(THead _ t0 _) \Rightarrow t0])) (THead (Flat Appl) w0 v0) (THead (Flat Appl) 
+w v) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ _ t0) \Rightarrow 
+t0])) (THead (Flat Appl) w0 v0) (THead (Flat Appl) w v) H5) in (\lambda (H8: 
+(eq T w0 w)).(let H9 \def (eq_ind T v0 (\lambda (t0: T).((eq T t0 (THead 
+(Flat Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 
+(THead (Flat Appl) w (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t)))) 
+(\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) 
+(\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 
+\def (eq_ind T v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 
+v H7) in (let H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat 
+Appl) w v)) \to (ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead 
+(Flat Appl) w (THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: 
+T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: 
+T).(ty3 g c0 w u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: 
+T).(ty3 g c0 t0 u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T 
+(\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u0 t1)) (THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: 
 T).(\lambda (t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: 
-T).(\lambda (_: T).(ty3 g c0 w u0)))))) H4 v H7) in (let H10 \def (eq_ind T 
-v0 (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H3 v H7) in (let 
-H11 \def (eq_ind T w0 (\lambda (t0: T).((eq T t0 (THead (Flat Appl) w v)) \to 
-(ex3_2 T T (\lambda (u0: T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w 
-(THead (Bind Abst) u0 t1)) u))) (\lambda (u0: T).(\lambda (t1: T).(ty3 g c0 v 
-(THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w 
-u0)))))) H2 w H8) in (let H12 \def (eq_ind T w0 (\lambda (t0: T).(ty3 g c0 t0 
-u)) H1 w H8) in (eq_ind_r T w (\lambda (t0: T).(ex3_2 T T (\lambda (u0: 
-T).(\lambda (t1: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t1)) 
-(THead (Flat Appl) t0 (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda 
-(t1: T).(ty3 g c0 v (THead (Bind Abst) u0 t1)))) (\lambda (u0: T).(\lambda 
-(_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: T).(\lambda (t0: 
-T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) (THead (Flat Appl) 
-w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda (t0: T).(ty3 g c0 v 
-(THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda (_: T).(ty3 g c0 w 
-u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind Abst) u t))) H10 
-H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda 
-(t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T t1 (THead (Flat 
-Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead 
-(Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: T).(\lambda (t: 
-T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: 
-T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t2 
-t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda 
-(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) 
-t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) 
-(\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (H5: (eq T 
-(THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def (eq_ind T 
-(THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee 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 f) \Rightarrow (match f in F return 
-(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow 
-True])])])) I (THead (Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda 
-(u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) 
-(THead (Flat Cast) t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v 
-(THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) 
-H6))))))))))) c y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 3171
-END *)
+T).(\lambda (_: T).(ty3 g c0 w u0))))) (ex3_2_intro T T (\lambda (u0: 
+T).(\lambda (t0: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t0)) 
+(THead (Flat Appl) w (THead (Bind Abst) u t))))) (\lambda (u0: T).(\lambda 
+(t0: T).(ty3 g c0 v (THead (Bind Abst) u0 t0)))) (\lambda (u0: T).(\lambda 
+(_: T).(ty3 g c0 w u0))) u t (pc3_refl c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t))) H10 H12) w0 H8))))))) H6)))))))))))) (\lambda (c0: C).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T 
+t1 (THead (Flat Appl) w v)) \to (ex3_2 T T (\lambda (u: T).(\lambda (t: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) t2))) (\lambda (u: 
+T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u t)))) (\lambda (u: 
+T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda (t0: T).(\lambda (_: (ty3 g 
+c0 t2 t0)).(\lambda (_: (((eq T t2 (THead (Flat Appl) w v)) \to (ex3_2 T T 
+(\lambda (u: T).(\lambda (t: T).(pc3 c0 (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) t0))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind 
+Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u))))))).(\lambda 
+(H5: (eq T (THead (Flat Cast) t2 t1) (THead (Flat Appl) w v))).(let H6 \def 
+(eq_ind T (THead (Flat Cast) t2 t1) (\lambda (ee: T).(match ee with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow 
+(match f with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead 
+(Flat Appl) w v) H5) in (False_ind (ex3_2 T T (\lambda (u: T).(\lambda (t: 
+T).(pc3 c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (THead (Flat Cast) 
+t0 t2)))) (\lambda (u: T).(\lambda (t: T).(ty3 g c0 v (THead (Bind Abst) u 
+t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c0 w u)))) H6))))))))))) c y x 
+H0))) H)))))).
 
-theorem ty3_gen_cast:
+lemma 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 (ex3 T (\lambda (t0: 
 T).(pc3 c (THead (Flat Cast) t0 t2) x)) (\lambda (_: T).(ty3 g c t1 t2)) 
@@ -768,103 +783,97 @@ T).(pc3 c0 (THead (Flat Cast) t4 t2) t0)) (\lambda (_: T).(ty3 g c0 t1 t2))
 (\lambda (t4: T).(ty3 g c0 t2 t4)) x0 (pc3_t t3 c0 (THead (Flat Cast) x0 t2) 
 H11 t0 H5) H12 H13))))) H10)))))))))))))))) (\lambda (c0: C).(\lambda (m: 
 nat).(\lambda (H1: (eq T (TSort m) (THead (Flat Cast) t2 t1))).(let H2 \def 
-(eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H1) in 
-(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (TSort 
-(next g m)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
-t0))) H2))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
-T).(\lambda (_: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda 
+(eq_ind T (TSort m) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow 
+True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
+(THead (Flat Cast) t2 t1) H1) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 
+(THead (Flat Cast) t0 t2) (TSort (next g m)))) (\lambda (_: T).(ty3 g c0 t1 
+t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H2))))) (\lambda (n: nat).(\lambda 
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 (CHead d 
+(Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (_: 
+(((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 d 
+(THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) (\lambda (t0: 
+T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead (Flat Cast) t2 
+t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T 
+(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t))) 
+(\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) 
+H5))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: 
+T).(\lambda (_: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda 
 (_: (ty3 g d u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 
 T (\lambda (t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 
 g d t1 t2)) (\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef 
 n) (THead (Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
-(THead (Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 
-(THead (Flat Cast) t0 t2) (lift (S n) O t))) (\lambda (_: T).(ty3 g c0 t1 
-t2)) (\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (n: 
-nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (_: (getl n c0 
-(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u 
-t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda 
-(t0: T).(pc3 d (THead (Flat Cast) t0 t2) t)) (\lambda (_: T).(ty3 g d t1 t2)) 
-(\lambda (t0: T).(ty3 g d t2 t0)))))).(\lambda (H4: (eq T (TLRef n) (THead 
-(Flat Cast) t2 t1))).(let H5 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match 
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
-(TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead 
-(Flat Cast) t2 t1) H4) in (False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead 
-(Flat Cast) t0 t2) (lift (S n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) 
-(\lambda (t0: T).(ty3 g c0 t2 t0))) H5))))))))))) (\lambda (c0: C).(\lambda 
-(u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u 
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat 
-Cast) t0 t2) t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 
-t2 t0)))))).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: 
-(ty3 g (CHead c0 (Bind b) u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat 
-Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead 
-(Flat Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) 
-(\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T 
-(THead (Bind b) u t0) (THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T 
-(THead (Bind b) u t0) (\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 True | (Flat _) \Rightarrow False])])) I (THead (Flat 
-Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat 
-Cast) t4 t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) 
-(\lambda (t4: T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda 
-(w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w 
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat 
-Cast) t0 t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 
-t2 t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t2 t1) H4) in 
+(False_ind (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (lift (S 
+n) O u))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0))) H5))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda 
+(_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (THead (Flat Cast) t2 t1)) \to 
+(ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) t)) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 t0)))))).(\lambda (b: 
+B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) 
+u) t0 t3)).(\lambda (_: (((eq T t0 (THead (Flat Cast) t2 t1)) \to (ex3 T 
+(\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (THead (Flat Cast) t4 t2) t3)) 
+(\lambda (_: T).(ty3 g (CHead c0 (Bind b) u) t1 t2)) (\lambda (t4: T).(ty3 g 
+(CHead c0 (Bind b) u) t2 t4)))))).(\lambda (H5: (eq T (THead (Bind b) u t0) 
+(THead (Flat Cast) t2 t1))).(let H6 \def (eq_ind T (THead (Bind b) u t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t2 
+t1) H5) in (False_ind (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 
+t2) (THead (Bind b) u t3))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: 
+T).(ty3 g c0 t2 t4))) H6))))))))))))) (\lambda (c0: C).(\lambda (w: 
+T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (_: (((eq T w (THead 
+(Flat Cast) t2 t1)) \to (ex3 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 
+t2) u)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0)))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead 
 (Bind Abst) u t))).(\lambda (_: (((eq T v (THead (Flat Cast) t2 t1)) \to (ex3 
 T (\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u 
 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
 t0)))))).(\lambda (H5: (eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 
 t1))).(let H6 \def (eq_ind T (THead (Flat Appl) w v) (\lambda (ee: T).(match 
-ee 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 f) \Rightarrow 
-(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast 
-\Rightarrow False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T 
-(\lambda (t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w 
-(THead (Bind Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: 
-T).(ty3 g c0 t2 t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: 
-T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 
-(THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat 
-Cast) t4 t2) t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g 
-c0 t2 t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: 
-(((eq T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 
-(THead (Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda 
-(t5: T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) 
-(THead (Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match 
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
-\Rightarrow t3 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) t3 t0) 
-(THead (Flat Cast) t2 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) 
-t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq T t3 t2)).(let H9 
-\def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to 
-(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t4)) (\lambda (_: 
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H4 t2 H8) in (let 
-H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 t2 H8) in (let H11 
-\def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat Cast) t2 t1)) \to 
-(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) t)) (\lambda (_: 
-T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) H2 t2 H8) in (let 
-H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 t2 H8) in (eq_ind_r 
-T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 
-t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda 
-(t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 (\lambda (t: T).((eq T 
-t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat 
-Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g 
-c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(ty3 g 
-c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: T).(pc3 c0 (THead (Flat 
-Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: T).(ty3 g c0 t1 t2)) 
-(\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 (THead (Flat Cast) t4 t2)) 
-H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 2609
-END *)
+ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k 
+_ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f with [Appl \Rightarrow True | Cast \Rightarrow 
+False])])])) I (THead (Flat Cast) t2 t1) H5) in (False_ind (ex3 T (\lambda 
+(t0: T).(pc3 c0 (THead (Flat Cast) t0 t2) (THead (Flat Appl) w (THead (Bind 
+Abst) u t)))) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t0: T).(ty3 g c0 t2 
+t0))) H6)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H1: (ty3 g c0 t0 t3)).(\lambda (H2: (((eq T t0 (THead (Flat 
+Cast) t2 t1)) \to (ex3 T (\lambda (t4: T).(pc3 c0 (THead (Flat Cast) t4 t2) 
+t3)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t4: T).(ty3 g c0 t2 
+t4)))))).(\lambda (t4: T).(\lambda (H3: (ty3 g c0 t3 t4)).(\lambda (H4: (((eq 
+T t3 (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead 
+(Flat Cast) t5 t2) t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: 
+T).(ty3 g c0 t2 t5)))))).(\lambda (H5: (eq T (THead (Flat Cast) t3 t0) (THead 
+(Flat Cast) t2 t1))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ t _) 
+\Rightarrow t])) (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
+(THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1) H5) in (\lambda (H8: (eq 
+T t3 t2)).(let H9 \def (eq_ind T t3 (\lambda (t: T).((eq T t (THead (Flat 
+Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) 
+t4)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) 
+H4 t2 H8) in (let H10 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t t4)) H3 
+t2 H8) in (let H11 \def (eq_ind T t3 (\lambda (t: T).((eq T t0 (THead (Flat 
+Cast) t2 t1)) \to (ex3 T (\lambda (t5: T).(pc3 c0 (THead (Flat Cast) t5 t2) 
+t)) (\lambda (_: T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5))))) 
+H2 t2 H8) in (let H12 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0 t0 t)) H1 
+t2 H8) in (eq_ind_r T t2 (\lambda (t: T).(ex3 T (\lambda (t5: T).(pc3 c0 
+(THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t))) (\lambda (_: T).(ty3 g 
+c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)))) (let H13 \def (eq_ind T t0 
+(\lambda (t: T).((eq T t (THead (Flat Cast) t2 t1)) \to (ex3 T (\lambda (t5: 
+T).(pc3 c0 (THead (Flat Cast) t5 t2) t2)) (\lambda (_: T).(ty3 g c0 t1 t2)) 
+(\lambda (t5: T).(ty3 g c0 t2 t5))))) H11 t1 H7) in (let H14 \def (eq_ind T 
+t0 (\lambda (t: T).(ty3 g c0 t t2)) H12 t1 H7) in (ex3_intro T (\lambda (t5: 
+T).(pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t2))) (\lambda (_: 
+T).(ty3 g c0 t1 t2)) (\lambda (t5: T).(ty3 g c0 t2 t5)) t4 (pc3_refl c0 
+(THead (Flat Cast) t4 t2)) H14 H10))) t3 H8))))))) H6))))))))))) c y x H0))) 
+H)))))).
 
-theorem tys3_gen_nil:
+lemma tys3_gen_nil:
  \forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T 
 (\lambda (u0: T).(ty3 g c u u0))))))
 \def
@@ -879,14 +888,11 @@ TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda
 TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to 
 (ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t 
 ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee: 
-TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil 
-\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind 
-(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
-(* COMMENTS
-Initial nodes: 255
-END *)
+TList).(match ee with [TNil \Rightarrow False | (TCons _ _) \Rightarrow 
+True])) I TNil H4) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u0 u1))) 
+H5))))))))) y u H0))) H)))).
 
-theorem tys3_gen_cons:
+lemma tys3_gen_cons:
  \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall 
 (u: T).((tys3 g c (TCons t ts) u) \to (land (ty3 g c t u) (tys3 g c ts 
 u)))))))
@@ -898,25 +904,20 @@ u)))))))
 g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to 
 (land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1: 
 T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t 
-ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList 
-return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _) 
-\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0) 
-(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1: 
-(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0 
-u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0) 
-(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t 
-ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList 
-return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _) 
-\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal 
-TList TList (\lambda (e: TList).(match e in TList return (\lambda (_: 
-TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1])) 
-(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def 
-(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land 
-(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList 
-ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind 
-T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3 
-g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
-(* COMMENTS
-Initial nodes: 479
-END *)
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee with 
+[TNil \Rightarrow True | (TCons _ _) \Rightarrow False])) I (TCons t ts) H2) 
+in (False_ind (land (ty3 g c t u0) (tys3 g c ts u0)) H3)))))) (\lambda (t0: 
+T).(\lambda (u0: T).(\lambda (H1: (ty3 g c t0 u0)).(\lambda (ts0: 
+TList).(\lambda (H2: (tys3 g c ts0 u0)).(\lambda (H3: (((eq TList ts0 (TCons 
+t ts)) \to (land (ty3 g c t u0) (tys3 g c ts u0))))).(\lambda (H4: (eq TList 
+(TCons t0 ts0) (TCons t ts))).(let H5 \def (f_equal TList T (\lambda (e: 
+TList).(match e with [TNil \Rightarrow t0 | (TCons t1 _) \Rightarrow t1])) 
+(TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal TList TList 
+(\lambda (e: TList).(match e with [TNil \Rightarrow ts0 | (TCons _ t1) 
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 
+t)).(let H8 \def (eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons 
+t ts)) \to (land (ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def 
+(eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let 
+H10 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj 
+(ty3 g c t u0) (tys3 g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).