X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fleq%2Fasucc.ma;h=42005d6ba6fb90f8d2f57672b7ed4a32f484f936;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=fd9e7c1d3b261aa920607b83813f23348c7bc9b8;hpb=88a68a9c334646bc17314d5327cd3b790202acd6;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
index fd9e7c1d3..42005d6ba 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma
@@ -14,9 +14,9 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/leq/props.ma".
+include "basic_1/leq/props.ma".
 
-theorem asucc_repl:
+lemma asucc_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
 (asucc g a1) (asucc g a2)))))
 \def
@@ -95,11 +95,8 @@ h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k)
 (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: 
 (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g 
 a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
-(* COMMENTS
-Initial nodes: 1907
-END *)
 
-theorem asucc_inj:
+lemma asucc_inj:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc 
 g a2)) \to (leq g a1 a2))))
 \def
@@ -121,19 +118,18 @@ nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort
 h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: 
 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) 
 x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) 
-(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A 
-return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) 
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def 
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with 
-[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G 
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in 
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3: 
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3 
-O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g 
-(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in 
-(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a: 
-A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S 
-x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g 
+(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with 
+[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) (ASort O (next g 
+n2)) (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match 
+e with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with 
+[(mk_G next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) 
+H4) in (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda 
+(n3: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) 
+x2))) H3 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A 
+(aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) 
+H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda 
+(a: A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) 
+(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g 
 (ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) 
 a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in 
 (leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: 
@@ -147,10 +143,9 @@ O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda
 (ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: 
 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) 
 x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) 
-\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A 
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ 
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) 
+\Rightarrow n4 | (AHead _ _) \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) 
+H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ 
 n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 
 x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 
 (\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort 
@@ -178,28 +173,27 @@ nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g
 (ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: 
 nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus 
 g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) 
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def 
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with 
-[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G 
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in 
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: 
-nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) 
-in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 
-n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def 
-(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g 
-(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) 
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g 
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S 
-x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in 
-(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: 
-nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 
-n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq 
-g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 
-n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S 
-n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S 
-n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def 
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _) 
+\Rightarrow n4 | (AHead _ _) \Rightarrow O])) (ASort O (next g n2)) (ASort x1 
+x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort 
+_ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G next _) 
+\Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in (\lambda 
+(H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: nat).(eq A 
+(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) in (let H9 
+\def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 n0) x2) 
+(aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def (eq_ind_r A 
+(aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g (ASort O (next g 
+n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) (aplus_sort_S_S_simpl g n0 
+n3 x2)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) x2) 
+(\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) a)) H10 (aplus g 
+(ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in (leq_sort g (S n3) O 
+n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: nat).(\lambda (_: 
+(((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g 
+(asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) 
+(ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda 
+(H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4) 
+n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S n4) 
+n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def 
 (leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind 
 nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A 
 (aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: 
@@ -207,14 +201,13 @@ nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2
 n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: 
 nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g 
 (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 
-n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A 
-return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) 
-\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A 
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ 
-n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1 
-x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1 
-(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0) 
-x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A 
+n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with 
+[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) (ASort n4 n2) 
+(ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e 
+with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 
+n2) (ASort x1 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r 
+nat x1 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 
+x0) x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A 
 (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 
 \def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g 
 (ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) 
@@ -242,44 +235,43 @@ nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g
 (x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g 
 (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 
 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match 
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | 
-(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g 
-(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_: 
-(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) 
-\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) 
-a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g 
-(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) 
-n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g 
-(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) 
-a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a 
-a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in 
-(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) 
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a 
-(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) 
-(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A 
-(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A 
-(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc 
-g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1 
-x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8))))))) 
-H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: 
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: 
-A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to 
-(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g 
-(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: 
-nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g 
-(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) 
-(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda 
-(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def 
-(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x 
-in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda 
-(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda 
-(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0) 
-(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a 
-x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next 
-g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda 
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) 
+ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I 
+(ASort x1 x0) H7) in (False_ind (leq g (ASort O n0) (AHead a a0)) H8))))))) 
+H5)))))) (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) 
+(asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 
+n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort 
+n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a 
+a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to 
+(leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) 
+n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g 
+(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H_x \def 
+(leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in (let H5 \def H_x in 
+(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda 
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) 
+(ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) (\lambda (x0: 
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort 
+n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g 
+a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda 
+(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) 
+\Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g (ASort (S n1) 
+n0) (AHead a a0)) H8))))))) H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: 
+A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq 
+g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g 
+a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: 
+A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) 
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a 
+a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g 
+(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 
+n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O 
+n0)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) 
+H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: 
+A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) 
+(\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O (next g n0)) (AHead a3 
+a4)))) (leq g (AHead a a0) (ASort O n0)) (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda 
+(H6: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A 
+(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) 
 \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
 (False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: 
 nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) 
@@ -291,38 +283,30 @@ g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
 (AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: 
 A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g 
 a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def 
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1) 
-n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g 
-(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda 
-(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g 
-(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g 
-(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3 
-(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5: 
-A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g 
-(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g 
-a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g 
-a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 
-\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) 
-with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3 
-(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow 
-((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) 
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h) 
-\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0 
-a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g 
-a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def 
-(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9) 
-in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in 
-(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2)))))) 
-a1)).
-(* COMMENTS
-Initial nodes: 4697
-END *)
+(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in 
+(False_ind (leq g (AHead a a0) (ASort (S n1) n0)) H7))))))) H3)))))) n H1)))) 
+(\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) 
+\to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc 
+g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: 
+(leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H_x \def 
+(leq_gen_head1 g a (asucc g a0) (AHead a3 (asucc g a4)) H3) in (let H4 \def 
+H_x in (ex3_2_ind A A (\lambda (a5: A).(\lambda (_: A).(leq g a a5))) 
+(\lambda (_: A).(\lambda (a6: A).(leq g (asucc g a0) a6))) (\lambda (a5: 
+A).(\lambda (a6: A).(eq A (AHead a3 (asucc g a4)) (AHead a5 a6)))) (leq g 
+(AHead a a0) (AHead a3 a4)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5: 
+(leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A 
+(AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 \def (f_equal A A (\lambda 
+(e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow 
+a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A 
+(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g a4) | (AHead 
+_ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda 
+(H10: (eq A a3 x0)).(let H11 \def (eq_ind_r A x1 (\lambda (a5: A).(leq g 
+(asucc g a0) a5)) H6 (asucc g a4) H9) in (let H12 \def (eq_ind_r A x0 
+(\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0 
+a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)).
 
-theorem leq_asucc:
+lemma leq_asucc:
  \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g 
 a0)))))
 \def
@@ -337,11 +321,8 @@ A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0)
 g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) 
 (AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) 
 a)).
-(* COMMENTS
-Initial nodes: 221
-END *)
 
-theorem leq_ahead_asucc_false:
+lemma leq_ahead_asucc_false:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) 
 (asucc g a1)) \to (\forall (P: Prop).P))))
 \def
@@ -359,20 +340,19 @@ A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
 (ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
 A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda 
 (H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A 
-(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: 
-nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O 
-\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to 
-P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let 
-H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1 
-\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S 
-n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
-A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0: 
-A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_: 
-(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def 
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
+(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in 
+(False_ind P H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead 
+(ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S 
+n1) n0) a2) (ASort n1 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) 
+a2 (ASort n1 n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda 
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
+(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort 
+n1 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort n1 n0) (\lambda (ee: 
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow 
 False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) 
 (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g 
 a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall 
@@ -385,23 +365,16 @@ A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
 (AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
 A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 
 x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g 
-a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e 
-in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc 
-(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0 
-with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) | 
-(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead 
-_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda 
-(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) 
-H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g 
-(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) 
-H2)))))))))) a1)).
-(* COMMENTS
-Initial nodes: 927
-END *)
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a | 
+(AHead a3 _) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in 
+((let H7 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) 
+\Rightarrow (asucc g a0) | (AHead _ a3) \Rightarrow a3])) (AHead a (asucc g 
+a0)) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def (eq_ind_r A 
+x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def 
+(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in 
+(leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)).
 
-theorem leq_asucc_false:
+lemma leq_asucc_false:
  \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: 
 Prop).P)))
 \def
@@ -419,43 +392,42 @@ O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda
 (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g 
 (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A 
 (ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: 
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow 
-n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5 
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) 
-with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O 
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r 
-nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g 
-(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1: 
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7 
-n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) 
-(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O 
-n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S 
-x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1: 
-nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda 
-(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next 
-g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda 
-(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g 
-n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat 
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
-n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda 
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g 
-(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S 
-n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e 
-in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _ 
-_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def 
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with 
-[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) 
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def 
-(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g 
-(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda 
-(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) 
-H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda 
-(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) 
-n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g 
-(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: 
-nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) 
-(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: 
+A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) \Rightarrow O])) 
+(ASort O n0) (ASort x1 x0) H3) in ((let H5 \def (f_equal A nat (\lambda (e: 
+A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) 
+(ASort O n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def 
+(eq_ind_r nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) 
+(aplus g (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 
+(\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O 
+n1) x2))) H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g 
+n0)) x2) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g 
+(ASort O n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def 
+(aplus_inj g (S x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 
+(\lambda (n1: nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) 
+H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow 
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to 
+P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def 
+(leq_gen_sort1 g n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in 
+(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda 
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort 
+h2 n2))))) P (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: 
+nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) 
+x2))).(\lambda (H3: (eq A (ASort (S n1) n0) (ASort x1 x0))).(let H4 \def 
+(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | 
+(AHead _ _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in 
+((let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ n2) 
+\Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) n0) (ASort x1 
+x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def (eq_ind_r nat x1 
+(\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort n2 x0) 
+x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n2: nat).(eq A 
+(aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) H7 n0 H5) in (let 
+H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda (a0: A).(eq A a0 
+(aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) n0) (S x2)) 
+(aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g (S x2) x2 
+(ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: nat).(le 
+n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) (\lambda 
+(a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: 
 Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to 
 (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead 
 a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) 
@@ -464,16 +436,12 @@ A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g
 (asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) 
 (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 
 x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) 
-(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) 
-\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A 
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) 
-H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2: 
-A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0 
-(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6))))))) 
-H2))))))))) a)).
-(* COMMENTS
-Initial nodes: 1327
-END *)
+(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a0 | (AHead a2 _) \Rightarrow a2])) (AHead a0 a1) 
+(AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with 
+[(ASort _ _) \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) 
+(AHead x0 x1) H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 
+(\lambda (a2: A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def 
+(eq_ind_r A x0 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) 
+H6))))))) H2))))))))) a)).