X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2FT%2Fdec.ma;h=ae783b82210f1ee9363cb924ba45b746ecf88174;hb=8f5533bd34e93eee2a14cdcfd0595be65651bfa7;hp=a088c40e3bb410629eb0c8e932c9aca9ecce0b8a;hpb=88a68a9c334646bc17314d5327cd3b790202acd6;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
index a088c40e3..ae783b822 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/T/dec.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/T/defs.ma".
+include "basic_1/T/fwd.ma".
 
-theorem terms_props__bind_dec:
+fact terms_props__bind_dec:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -26,41 +26,36 @@ B).(or (eq B Abbr b) ((eq B Abbr b) \to (\forall (P: Prop).P)))) (or_introl
 (eq B Abbr Abbr) ((eq B Abbr Abbr) \to (\forall (P: Prop).P)) (refl_equal B 
 Abbr)) (or_intror (eq B Abbr Abst) ((eq B Abbr Abst) \to (\forall (P: 
 Prop).P)) (\lambda (H: (eq B Abbr Abst)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind B Abbr (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) 
-with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow 
-False])) I Abst H) in (False_ind P H0))))) (or_intror (eq B Abbr Void) ((eq B 
-Abbr Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda 
-(P: Prop).(let H0 \def (eq_ind B Abbr (\lambda (ee: B).(match ee in B return 
-(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | 
-Void \Rightarrow False])) I Void H) in (False_ind P H0))))) b2)) (\lambda 
-(b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B Abst b) \to (\forall 
-(P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst Abbr) \to (\forall 
-(P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind B Abst (\lambda (ee: B).(match ee in B return (\lambda (_: B).Prop) 
-with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow 
-False])) I Abbr H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B 
-Abst Abst) \to (\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B 
-Abst Void) ((eq B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B 
-Abst Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: 
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | 
-Abst \Rightarrow True | Void \Rightarrow False])) I Void H) in (False_ind P 
-H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Void b) ((eq B 
-Void b) \to (\forall (P: Prop).P)))) (or_intror (eq B Void Abbr) ((eq B Void 
-Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Void Abbr)).(\lambda (P: 
-Prop).(let H0 \def (eq_ind B Void (\lambda (ee: B).(match ee in B return 
-(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | 
-Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror (eq B 
-Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: (eq B 
-Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda (ee: 
-B).(match ee in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | 
-Abst \Rightarrow False | Void \Rightarrow True])) I Abst H) in (False_ind P 
-H0))))) (or_introl (eq B Void Void) ((eq B Void Void) \to (\forall (P: 
-Prop).P)) (refl_equal B Void)) b2)) b1).
-(* COMMENTS
-Initial nodes: 559
-END *)
+(eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst 
+\Rightarrow False | Void \Rightarrow False])) I Abst H) in (False_ind P 
+H0))))) (or_intror (eq B Abbr Void) ((eq B Abbr Void) \to (\forall (P: 
+Prop).P)) (\lambda (H: (eq B Abbr Void)).(\lambda (P: Prop).(let H0 \def 
+(eq_ind B Abbr (\lambda (ee: B).(match ee with [Abbr \Rightarrow True | Abst 
+\Rightarrow False | Void \Rightarrow False])) I Void H) in (False_ind P 
+H0))))) b2)) (\lambda (b2: B).(B_ind (\lambda (b: B).(or (eq B Abst b) ((eq B 
+Abst b) \to (\forall (P: Prop).P)))) (or_intror (eq B Abst Abbr) ((eq B Abst 
+Abbr) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst Abbr)).(\lambda (P: 
+Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match ee with [Abbr 
+\Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False])) I Abbr 
+H) in (False_ind P H0))))) (or_introl (eq B Abst Abst) ((eq B Abst Abst) \to 
+(\forall (P: Prop).P)) (refl_equal B Abst)) (or_intror (eq B Abst Void) ((eq 
+B Abst Void) \to (\forall (P: Prop).P)) (\lambda (H: (eq B Abst 
+Void)).(\lambda (P: Prop).(let H0 \def (eq_ind B Abst (\lambda (ee: B).(match 
+ee with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow 
+False])) I Void H) in (False_ind P H0))))) b2)) (\lambda (b2: B).(B_ind 
+(\lambda (b: B).(or (eq B Void b) ((eq B Void b) \to (\forall (P: Prop).P)))) 
+(or_intror (eq B Void Abbr) ((eq B Void Abbr) \to (\forall (P: Prop).P)) 
+(\lambda (H: (eq B Void Abbr)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void 
+(\lambda (ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow 
+False | Void \Rightarrow True])) I Abbr H) in (False_ind P H0))))) (or_intror 
+(eq B Void Abst) ((eq B Void Abst) \to (\forall (P: Prop).P)) (\lambda (H: 
+(eq B Void Abst)).(\lambda (P: Prop).(let H0 \def (eq_ind B Void (\lambda 
+(ee: B).(match ee with [Abbr \Rightarrow False | Abst \Rightarrow False | 
+Void \Rightarrow True])) I Abst H) in (False_ind P H0))))) (or_introl (eq B 
+Void Void) ((eq B Void Void) \to (\forall (P: Prop).P)) (refl_equal B Void)) 
+b2)) b1).
 
-theorem bind_dec_not:
+lemma bind_dec_not:
  \forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) (not (eq B b1 b2))))
 \def
  \lambda (b1: B).(\lambda (b2: B).(let H_x \def (terms_props__bind_dec b1 b2) 
@@ -69,11 +64,8 @@ Prop).P)) (or (eq B b1 b2) ((eq B b1 b2) \to False)) (\lambda (H0: (eq B b1
 b2)).(or_introl (eq B b1 b2) ((eq B b1 b2) \to False) H0)) (\lambda (H0: 
 (((eq B b1 b2) \to (\forall (P: Prop).P)))).(or_intror (eq B b1 b2) ((eq B b1 
 b2) \to False) (\lambda (H1: (eq B b1 b2)).(H0 H1 False)))) H)))).
-(* COMMENTS
-Initial nodes: 131
-END *)
 
-theorem terms_props__flat_dec:
+fact terms_props__flat_dec:
  \forall (f1: F).(\forall (f2: F).(or (eq F f1 f2) ((eq F f1 f2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -83,21 +75,17 @@ F).(or (eq F Appl f) ((eq F Appl f) \to (\forall (P: Prop).P)))) (or_introl
 (eq F Appl Appl) ((eq F Appl Appl) \to (\forall (P: Prop).P)) (refl_equal F 
 Appl)) (or_intror (eq F Appl Cast) ((eq F Appl Cast) \to (\forall (P: 
 Prop).P)) (\lambda (H: (eq F Appl Cast)).(\lambda (P: Prop).(let H0 \def 
-(eq_ind F Appl (\lambda (ee: F).(match ee in F return (\lambda (_: F).Prop) 
-with [Appl \Rightarrow True | Cast \Rightarrow False])) I Cast H) in 
-(False_ind P H0))))) f2)) (\lambda (f2: F).(F_ind (\lambda (f: F).(or (eq F 
-Cast f) ((eq F Cast f) \to (\forall (P: Prop).P)))) (or_intror (eq F Cast 
-Appl) ((eq F Cast Appl) \to (\forall (P: Prop).P)) (\lambda (H: (eq F Cast 
-Appl)).(\lambda (P: Prop).(let H0 \def (eq_ind F Cast (\lambda (ee: F).(match 
-ee in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast 
+(eq_ind F Appl (\lambda (ee: F).(match ee with [Appl \Rightarrow True | Cast 
+\Rightarrow False])) I Cast H) in (False_ind P H0))))) f2)) (\lambda (f2: 
+F).(F_ind (\lambda (f: F).(or (eq F Cast f) ((eq F Cast f) \to (\forall (P: 
+Prop).P)))) (or_intror (eq F Cast Appl) ((eq F Cast Appl) \to (\forall (P: 
+Prop).P)) (\lambda (H: (eq F Cast Appl)).(\lambda (P: Prop).(let H0 \def 
+(eq_ind F Cast (\lambda (ee: F).(match ee with [Appl \Rightarrow False | Cast 
 \Rightarrow True])) I Appl H) in (False_ind P H0))))) (or_introl (eq F Cast 
 Cast) ((eq F Cast Cast) \to (\forall (P: Prop).P)) (refl_equal F Cast)) f2)) 
 f1).
-(* COMMENTS
-Initial nodes: 263
-END *)
 
-theorem terms_props__kind_dec:
+fact terms_props__kind_dec:
  \forall (k1: K).(\forall (k2: K).(or (eq K k1 k2) ((eq K k1 k2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -114,39 +102,35 @@ Prop).P)))) (or_introl (eq K (Bind b) (Bind b)) ((eq K (Bind b) (Bind b)) \to
 b b0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Bind b) (Bind b0)) ((eq 
 K (Bind b) (Bind b0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Bind b) 
 (Bind b0))).(\lambda (P: Prop).(let H2 \def (f_equal K B (\lambda (e: 
-K).(match e in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | 
-(Flat _) \Rightarrow b])) (Bind b) (Bind b0) H1) in (let H3 \def (eq_ind_r B 
-b0 (\lambda (b1: B).((eq B b b1) \to (\forall (P0: Prop).P0))) H0 b H2) in 
-(H3 (refl_equal B b) P))))))) H)))) (\lambda (f: F).(or_intror (eq K (Bind b) 
-(Flat f)) ((eq K (Bind b) (Flat f)) \to (\forall (P: Prop).P)) (\lambda (H: 
-(eq K (Bind b) (Flat f))).(\lambda (P: Prop).(let H0 \def (eq_ind K (Bind b) 
-(\lambda (ee: K).(match ee in K return (\lambda (_: K).Prop) with [(Bind _) 
-\Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in (False_ind 
-P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda (k: K).(or 
-(eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) (\lambda 
-(b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) \to 
-(\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda (P: 
-Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return 
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
-True])) I (Bind b) H) in (False_ind P H0)))))) (\lambda (f0: F).(let H_x \def 
-(terms_props__flat_dec f f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F 
-f f0) \to (\forall (P: Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat 
-f) (Flat f0)) \to (\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind 
-F f (\lambda (f1: F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) 
-\to (\forall (P: Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat 
-f) (Flat f)) \to (\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) 
-(\lambda (H0: (((eq F f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K 
-(Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) 
-(\lambda (H1: (eq K (Flat f) (Flat f0))).(\lambda (P: Prop).(let H2 \def 
-(f_equal K F (\lambda (e: K).(match e in K return (\lambda (_: K).F) with 
-[(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat f) (Flat f0) H1) 
-in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) \to (\forall 
-(P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) k2))) k1).
-(* COMMENTS
-Initial nodes: 799
-END *)
+K).(match e with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])) (Bind 
+b) (Bind b0) H1) in (let H3 \def (eq_ind_r B b0 (\lambda (b1: B).((eq B b b1) 
+\to (\forall (P0: Prop).P0))) H0 b H2) in (H3 (refl_equal B b) P))))))) H)))) 
+(\lambda (f: F).(or_intror (eq K (Bind b) (Flat f)) ((eq K (Bind b) (Flat f)) 
+\to (\forall (P: Prop).P)) (\lambda (H: (eq K (Bind b) (Flat f))).(\lambda 
+(P: Prop).(let H0 \def (eq_ind K (Bind b) (\lambda (ee: K).(match ee with 
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])) I (Flat f) H) in 
+(False_ind P H0)))))) k2))) (\lambda (f: F).(\lambda (k2: K).(K_ind (\lambda 
+(k: K).(or (eq K (Flat f) k) ((eq K (Flat f) k) \to (\forall (P: Prop).P)))) 
+(\lambda (b: B).(or_intror (eq K (Flat f) (Bind b)) ((eq K (Flat f) (Bind b)) 
+\to (\forall (P: Prop).P)) (\lambda (H: (eq K (Flat f) (Bind b))).(\lambda 
+(P: Prop).(let H0 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee with 
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind b) H) in 
+(False_ind P H0)))))) (\lambda (f0: F).(let H_x \def (terms_props__flat_dec f 
+f0) in (let H \def H_x in (or_ind (eq F f f0) ((eq F f f0) \to (\forall (P: 
+Prop).P)) (or (eq K (Flat f) (Flat f0)) ((eq K (Flat f) (Flat f0)) \to 
+(\forall (P: Prop).P))) (\lambda (H0: (eq F f f0)).(eq_ind F f (\lambda (f1: 
+F).(or (eq K (Flat f) (Flat f1)) ((eq K (Flat f) (Flat f1)) \to (\forall (P: 
+Prop).P)))) (or_introl (eq K (Flat f) (Flat f)) ((eq K (Flat f) (Flat f)) \to 
+(\forall (P: Prop).P)) (refl_equal K (Flat f))) f0 H0)) (\lambda (H0: (((eq F 
+f f0) \to (\forall (P: Prop).P)))).(or_intror (eq K (Flat f) (Flat f0)) ((eq 
+K (Flat f) (Flat f0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq K (Flat f) 
+(Flat f0))).(\lambda (P: Prop).(let H2 \def (f_equal K F (\lambda (e: 
+K).(match e with [(Bind _) \Rightarrow f | (Flat f1) \Rightarrow f1])) (Flat 
+f) (Flat f0) H1) in (let H3 \def (eq_ind_r F f0 (\lambda (f1: F).((eq F f f1) 
+\to (\forall (P0: Prop).P0))) H0 f H2) in (H3 (refl_equal F f) P))))))) H)))) 
+k2))) k1).
 
-theorem term_dec:
+lemma term_dec:
  \forall (t1: T).(\forall (t2: T).(or (eq T t1 t2) ((eq T t1 t2) \to (\forall 
 (P: Prop).P))))
 \def
@@ -163,15 +147,14 @@ n) (TSort n)) \to (\forall (P: Prop).P)) (refl_equal T (TSort n))) n0 H0))
 (\lambda (H0: (((eq nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T 
 (TSort n) (TSort n0)) ((eq T (TSort n) (TSort n0)) \to (\forall (P: Prop).P)) 
 (\lambda (H1: (eq T (TSort n) (TSort n0))).(\lambda (P: Prop).(let H2 \def 
-(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
-[(TSort n1) \Rightarrow n1 | (TLRef _) \Rightarrow n | (THead _ _ _) 
-\Rightarrow n])) (TSort n) (TSort n0) H1) in (let H3 \def (eq_ind_r nat n0 
-(\lambda (n1: nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in 
-(H3 (refl_equal nat n) P))))))) H)))) (\lambda (n0: nat).(or_intror (eq T 
-(TSort n) (TLRef n0)) ((eq T (TSort n) (TLRef n0)) \to (\forall (P: Prop).P)) 
-(\lambda (H: (eq T (TSort n) (TLRef n0))).(\lambda (P: Prop).(let H0 \def 
-(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 | 
+(TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort n0) 
+H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to 
+(\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) 
+(\lambda (n0: nat).(or_intror (eq T (TSort n) (TLRef n0)) ((eq T (TSort n) 
+(TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TSort n) (TLRef 
+n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
 (THead _ _ _) \Rightarrow False])) I (TLRef n0) H) in (False_ind P H0)))))) 
 (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TSort n) t) ((eq T 
 (TSort n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or 
@@ -179,15 +162,14 @@ T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
 Prop).P)))).(or_intror (eq T (TSort n) (THead k t t0)) ((eq T (TSort n) 
 (THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TSort n) 
 (THead k t t0))).(\lambda (P: Prop).(let H2 \def (eq_ind T (TSort n) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (n: 
-nat).(\lambda (t2: T).(T_ind (\lambda (t: T).(or (eq T (TLRef n) t) ((eq T 
-(TLRef n) t) \to (\forall (P: Prop).P)))) (\lambda (n0: nat).(or_intror (eq T 
-(TLRef n) (TSort n0)) ((eq T (TLRef n) (TSort n0)) \to (\forall (P: Prop).P)) 
-(\lambda (H: (eq T (TLRef n) (TSort n0))).(\lambda (P: Prop).(let H0 \def 
-(eq_ind T (TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead k t t0) H1) in (False_ind 
+P H2)))))))))) t2))) (\lambda (n: nat).(\lambda (t2: T).(T_ind (\lambda (t: 
+T).(or (eq T (TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))) 
+(\lambda (n0: nat).(or_intror (eq T (TLRef n) (TSort n0)) ((eq T (TLRef n) 
+(TSort n0)) \to (\forall (P: Prop).P)) (\lambda (H: (eq T (TLRef n) (TSort 
+n0))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
 (THead _ _ _) \Rightarrow False])) I (TSort n0) H) in (False_ind P H0)))))) 
 (\lambda (n0: nat).(let H_x \def (nat_dec n n0) in (let H \def H_x in (or_ind 
 (eq nat n n0) ((eq nat n n0) \to (\forall (P: Prop).P)) (or (eq T (TLRef n) 
@@ -199,34 +181,32 @@ T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
 nat n n0) \to (\forall (P: Prop).P)))).(or_intror (eq T (TLRef n) (TLRef n0)) 
 ((eq T (TLRef n) (TLRef n0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T 
 (TLRef n) (TLRef n0))).(\lambda (P: Prop).(let H2 \def (f_equal T nat 
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) 
-(TLRef n) (TLRef n0) H1) in (let H3 \def (eq_ind_r nat n0 (\lambda (n1: 
-nat).((eq nat n n1) \to (\forall (P0: Prop).P0))) H0 n H2) in (H3 (refl_equal 
-nat n) P))))))) H)))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: (or (eq T 
-(TLRef n) t) ((eq T (TLRef n) t) \to (\forall (P: Prop).P)))).(\lambda (t0: 
-T).(\lambda (_: (or (eq T (TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: 
-Prop).P)))).(or_intror (eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) 
-(THead k t t0)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (TLRef n) 
-(THead k t t0))).(\lambda (P: Prop).(let H2 \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 k t t0) H1) in (False_ind P H2)))))))))) t2))) (\lambda (k: 
-K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t t2) ((eq T t 
-t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
-(t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow n | (TLRef n1) 
+\Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) H1) in 
+(let H3 \def (eq_ind_r nat n0 (\lambda (n1: nat).((eq nat n n1) \to (\forall 
+(P0: Prop).P0))) H0 n H2) in (H3 (refl_equal nat n) P))))))) H)))) (\lambda 
+(k: K).(\lambda (t: T).(\lambda (_: (or (eq T (TLRef n) t) ((eq T (TLRef n) 
+t) \to (\forall (P: Prop).P)))).(\lambda (t0: T).(\lambda (_: (or (eq T 
+(TLRef n) t0) ((eq T (TLRef n) t0) \to (\forall (P: Prop).P)))).(or_intror 
+(eq T (TLRef n) (THead k t t0)) ((eq T (TLRef n) (THead k t t0)) \to (\forall 
+(P: Prop).P)) (\lambda (H1: (eq T (TLRef n) (THead k t t0))).(\lambda (P: 
+Prop).(let H2 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
+\Rightarrow False])) I (THead k t t0) H1) in (False_ind P H2)))))))))) t2))) 
+(\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).(or (eq T t 
+t2) ((eq T t t2) \to (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda 
+(H0: ((\forall (t2: T).(or (eq T t0 t2) ((eq T t0 t2) \to (\forall (P: 
 Prop).P)))))).(\lambda (t2: T).(T_ind (\lambda (t3: T).(or (eq T (THead k t 
 t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (n: 
 nat).(or_intror (eq T (THead k t t0) (TSort n)) ((eq T (THead k t t0) (TSort 
 n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TSort 
 n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: 
-T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TSort n) H1) in (False_ind P H2)))))) (\lambda (n: nat).(or_intror (eq T 
-(THead k t t0) (TLRef n)) ((eq T (THead k t t0) (TLRef n)) \to (\forall (P: 
-Prop).P)) (\lambda (H1: (eq T (THead k t t0) (TLRef n))).(\lambda (P: 
-Prop).(let H2 \def (eq_ind T (THead k t t0) (\lambda (ee: T).(match ee in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H1) in (False_ind P H2)))))) 
+(\lambda (n: nat).(or_intror (eq T (THead k t t0) (TLRef n)) ((eq T (THead k 
+t t0) (TLRef n)) \to (\forall (P: Prop).P)) (\lambda (H1: (eq T (THead k t 
+t0) (TLRef n))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead k t t0) 
+(\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in 
 (False_ind P H2)))))) (\lambda (k0: K).(\lambda (t3: T).(\lambda (H1: (or (eq 
 T (THead k t t0) t3) ((eq T (THead k t t0) t3) \to (\forall (P: 
@@ -257,19 +237,18 @@ k0 H10)) (\lambda (H10: (((eq K k k0) \to (\forall (P: Prop).P)))).(or_intror
 (eq T (THead k t t0) (THead k0 t t0)) ((eq T (THead k t t0) (THead k0 t t0)) 
 \to (\forall (P: Prop).P)) (\lambda (H11: (eq T (THead k t t0) (THead k0 t 
 t0))).(\lambda (P: Prop).(let H12 \def (f_equal T K (\lambda (e: T).(match e 
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t 
-t0) H11) in (let H13 \def (eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to 
-(\forall (P0: Prop).P0))) H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) 
-t4 H7))) (\lambda (H7: (((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror 
-(eq T (THead k t t0) (THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) 
-\to (\forall (P: Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t 
-t4))).(\lambda (P: Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e 
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t 
-t4) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T return 
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 
-| (THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in 
+with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) 
+\Rightarrow k1])) (THead k t t0) (THead k0 t t0) H11) in (let H13 \def 
+(eq_ind_r K k0 (\lambda (k1: K).((eq K k k1) \to (\forall (P0: Prop).P0))) 
+H10 k H12) in (H13 (refl_equal K k) P))))))) H9))) t4 H7))) (\lambda (H7: 
+(((eq T t0 t4) \to (\forall (P: Prop).P)))).(or_intror (eq T (THead k t t0) 
+(THead k0 t t4)) ((eq T (THead k t t0) (THead k0 t t4)) \to (\forall (P: 
+Prop).P)) (\lambda (H8: (eq T (THead k t t0) (THead k0 t t4))).(\lambda (P: 
+Prop).(let H9 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) 
+(THead k t t0) (THead k0 t t4) H8) in ((let H10 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t t4) H8) in 
 (\lambda (_: (eq K k k0)).(let H12 \def (eq_ind_r T t4 (\lambda (t5: T).((eq 
 T t0 t5) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (let H13 \def (eq_ind_r 
 T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) 
@@ -278,26 +257,22 @@ H9)))))) H6))) t3 H4))) (\lambda (H4: (((eq T t t3) \to (\forall (P:
 Prop).P)))).(or_intror (eq T (THead k t t0) (THead k0 t3 t4)) ((eq T (THead k 
 t t0) (THead k0 t3 t4)) \to (\forall (P: Prop).P)) (\lambda (H5: (eq T (THead 
 k t t0) (THead k0 t3 t4))).(\lambda (P: Prop).(let H6 \def (f_equal T K 
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) 
-(THead k t t0) (THead k0 t3 t4) H5) in ((let H7 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t 
-| (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) (THead k t t0) 
-(THead k0 t3 t4) 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 _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 
-t4) H5) in (\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def 
-(eq_ind_r T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k 
-t t0) t5) \to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r 
-T t3 (\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in 
-(let H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) 
-((eq T (THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k t t0) (THead k0 t3 
+t4) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort 
+_) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t5 _) \Rightarrow t5])) 
+(THead k t t0) (THead k0 t3 t4) H5) in ((let H8 \def (f_equal T T (\lambda 
+(e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | 
+(THead _ _ t5) \Rightarrow t5])) (THead k t t0) (THead k0 t3 t4) H5) in 
+(\lambda (H9: (eq T t t3)).(\lambda (_: (eq K k k0)).(let H11 \def (eq_ind_r 
+T t4 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T (THead k t t0) t5) 
+\to (\forall (P0: Prop).P0)))) H2 t0 H8) in (let H12 \def (eq_ind_r T t3 
+(\lambda (t5: T).((eq T t t5) \to (\forall (P0: Prop).P0))) H4 t H9) in (let 
+H13 \def (eq_ind_r T t3 (\lambda (t5: T).(or (eq T (THead k t t0) t5) ((eq T 
+(THead k t t0) t5) \to (\forall (P0: Prop).P0)))) H1 t H9) in (H12 
 (refl_equal T t) P))))))) H7)) H6)))))) H3)))))))) t2))))))) t1).
-(* COMMENTS
-Initial nodes: 2821
-END *)
 
-theorem binder_dec:
+lemma binder_dec:
  \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
 T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall 
 (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
@@ -311,15 +286,14 @@ b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n)
 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda 
 (w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w 
 u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\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) w u) H) in (False_ind P H0))))))))) (\lambda (n: 
-nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
-T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: 
-T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: 
-Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq 
-T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T 
-(TLRef n) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P 
+H0))))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda 
+(w: T).(\lambda (u: T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: 
+B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to 
+(\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: 
+T).(\lambda (H: (eq T (TLRef n) (THead (Bind b) w u))).(\lambda (P: 
+Prop).(let H0 \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) w u) H) in (False_ind P H0))))))))) 
 (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T 
@@ -357,16 +331,12 @@ T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))))))
 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda 
 (w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead 
 (Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 
-t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) 
-\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) 
+t1) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 with [(Bind _) 
 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) 
 in (False_ind P H2))))))))))))) k)) t).
-(* COMMENTS
-Initial nodes: 1063
-END *)
 
-theorem abst_dec:
+lemma abst_dec:
  \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead 
 (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to 
 (\forall (P: Prop).P)))))
@@ -378,69 +348,64 @@ theorem abst_dec:
 t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall 
 (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind 
 Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda 
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
-False])) I (THead (Bind Abst) v t) H) in (False_ind P H0)))))))) (\lambda (n: 
-nat).(\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead 
-(Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) 
-\to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) 
-(THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \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 Abst) v t) H) in (False_ind P H0)))))))) (\lambda (k: 
-K).(\lambda (t: T).(\lambda (_: ((\forall (v: T).(or (ex T (\lambda (t0: 
-T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind 
-Abst) v t0)) \to (\forall (P: Prop).P))))))).(\lambda (t0: T).(\lambda (_: 
-((\forall (v: T).(or (ex T (\lambda (t1: T).(eq T t0 (THead (Bind Abst) v 
-t1)))) (\forall (t1: T).((eq T t0 (THead (Bind Abst) v t1)) \to (\forall (P: 
-Prop).P))))))).(\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind 
-Abst)) in (let H1 \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind 
-Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t 
-t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead 
-(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq K k (Bind 
-Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: 
-T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T 
-(THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let 
-H_x0 \def (term_dec t v) in (let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t 
-v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind 
-Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind 
-Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda 
-(H4: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq 
-T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq 
-T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: 
-Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) 
-(THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) 
-(THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda 
-(t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 
-(refl_equal T (THead (Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) 
-\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead 
-(Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead 
-(Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) 
-(\lambda (t1: T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind 
-Abst) v t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: 
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | 
-(TLRef _) \Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) 
-t t0) (THead (Bind Abst) v 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 _ _ t2) \Rightarrow t2])) (THead (Bind 
-Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T t v)).(H4 H8 
-P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) \to (\forall 
-(P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead 
-(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind 
-Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T 
-(THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def 
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) 
-\Rightarrow k0])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+(ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow 
+False | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in 
+(False_ind P H0)))))))) (\lambda (n: nat).(\lambda (v: T).(or_intror (ex T 
+(\lambda (t: T).(eq T (TLRef n) (THead (Bind Abst) v t)))) (\forall (t: 
+T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) 
+(\lambda (t: T).(\lambda (H: (eq T (TLRef n) (THead (Bind Abst) v 
+t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: 
+T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind 
+P H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall (v: 
+T).(or (ex T (\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall 
+(t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall (P: 
+Prop).P))))))).(\lambda (t0: T).(\lambda (_: ((\forall (v: T).(or (ex T 
+(\lambda (t1: T).(eq T t0 (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T 
+t0 (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))))).(\lambda (v: 
+T).(let H_x \def (terms_props__kind_dec k (Bind Abst)) in (let H1 \def H_x in 
+(or_ind (eq K k (Bind Abst)) ((eq K k (Bind Abst)) \to (\forall (P: Prop).P)) 
+(or (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) 
+(\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall 
+(P: Prop).P)))) (\lambda (H2: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) 
+(\lambda (k0: K).(or (ex T (\lambda (t1: T).(eq T (THead k0 t t0) (THead 
+(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k0 t t0) (THead (Bind 
+Abst) v t1)) \to (\forall (P: Prop).P))))) (let H_x0 \def (term_dec t v) in 
+(let H3 \def H_x0 in (or_ind (eq T t v) ((eq T t v) \to (\forall (P: 
+Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead 
+(Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead 
+(Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H4: (eq T t 
+v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq T (THead 
+(Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: Prop).P))))) 
+(or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind 
+Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind 
+Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t1: T).(eq T 
+(THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 (refl_equal T (THead 
+(Bind Abst) t t0)))) v H4)) (\lambda (H4: (((eq T t v) \to (\forall (P: 
+Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) 
+(THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) 
+(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: 
+T).(\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v 
+t1))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e 
 with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t2 _) 
-\Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H6 
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
+\Rightarrow t2])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) 
+(THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H5) in (\lambda (H8: (eq T 
+t v)).(H4 H8 P))) H6))))))) H3))) k H2)) (\lambda (H2: (((eq K k (Bind Abst)) 
+\to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k 
+t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) 
+(THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: 
+T).(\lambda (H3: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: 
+Prop).(let H4 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k t t0) (THead (Bind Abst) v t1) H3) in ((let H5 \def (f_equal T T 
+(\lambda (e: T).(match e with [(TSort _) \Rightarrow t | (TLRef _) 
+\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead k t t0) (THead (Bind 
+Abst) v t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e with 
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) 
 \Rightarrow t2])) (THead k t t0) (THead (Bind Abst) v t1) H3) in (\lambda (_: 
 (eq T t v)).(\lambda (H8: (eq K k (Bind Abst))).(H2 H8 P)))) H5)) H4))))))) 
 H1))))))))) u).
-(* COMMENTS
-Initial nodes: 1305
-END *)