X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Ffwd.ma;h=ddfce504d6ea73d945edd1d2f53877c28eb8a041;hb=f73bd1c1cdd504c2a991071505b2e4f541791a7f;hp=649eb872be605ee2696ce4936472bd390eab1e4c;hpb=5c1b44dfefa085fbb56e23047652d3650be9d855;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma
index 649eb872b..ddfce504d 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/fwd.ma
@@ -14,105 +14,219 @@
 
 (* This file was automatically generated: do not edit *********************)
 
+include "LambdaDelta-1/leq/defs.ma".
 
-
-include "leq/defs.ma".
-
-theorem leq_gen_sort:
+theorem leq_gen_sort1:
  \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
 g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda 
-(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
-h2 n2) k))))))))))
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 
+(ASort h2 n2))))))))))
 \def
  \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
-A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H in leq return 
-(\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort 
-h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda 
-(h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: 
-nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) 
-(aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0) 
-\Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2: 
-(eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match 
-e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ 
-_) \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def 
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with 
-[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) 
-(ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to 
-((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort 
-h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: 
-nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
-h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n: 
-nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g 
-(ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: 
-nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
-h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2 
-n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: 
-nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
-(k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))) 
-(\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
-(_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda 
-(h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort 
-h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq 
-nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0 
-a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1 
-n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead 
-a1 a3) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 
-n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g 
-a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
-(_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: 
-nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal 
-A a2))))))).
+A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda 
+(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort 
+h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A 
+a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g 
+(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat 
+nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a 
+k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: 
+nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: 
+nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) 
+k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal 
+A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort 
+n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 
+n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return 
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) 
+\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 
+h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 
+n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: 
+nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
+nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda 
+(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 
+n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort 
+n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda 
+(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
+(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) 
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) 
+(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 
+n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A 
+(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 
+H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: 
+(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) 
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: 
+A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to 
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 
+n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def 
+(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_: 
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
+True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k) 
+(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0))) 
+H))))).
 
-theorem leq_gen_head:
+theorem leq_gen_head1:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g 
-(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a 
-(AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda 
-(_: A).(\lambda (a4: A).(leq g a2 a4))))))))
+(AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 
+a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
+(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq 
+g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g 
+a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: 
+(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 
+a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda 
+(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq 
+A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) 
+(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match 
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | 
+(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A 
+(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda 
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) 
+(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: 
+(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A 
+(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda 
+(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 
+a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 
+a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: 
+A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 
+a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6 
+a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def 
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
+[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4) 
+(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A 
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) 
+\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 
+a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to 
+(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
+a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: 
+A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6: 
+A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: 
+A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda 
+(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 
+\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A 
+A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda 
+(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) 
+(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5))))))))) 
+H6))))))))))) y a H0))) H))))).
+
+theorem leq_gen_sort2:
+ \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq 
+g a2 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g (ASort h1 n1) 
+k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 
+(ASort h2 n2))))))))))
+\def
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: 
+A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda 
+(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) 
+(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq 
+A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind 
+g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat 
+nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus 
+g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: 
+nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: 
+nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: 
+nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) 
+k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal 
+A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort 
+n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1 
+n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return 
+(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) 
+\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 
+h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 
+n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: 
+nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: 
+nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda 
+(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 
+n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort 
+h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda 
+(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda 
+(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) 
+(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) 
+(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 
+n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A 
+(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 
+H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: 
+(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat 
+(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: 
+A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to 
+(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: 
+nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 
+n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def 
+(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_: 
+A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
+True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: 
+nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) 
+(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda 
+(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0))) 
+H))))).
+
+theorem leq_gen_head2:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a 
+(AHead a1 a2)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g a3 
+a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))))))
 \def
  \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda 
-(H: (leq g (AHead a1 a2) a)).(let H0 \def (match H in leq return (\lambda 
-(a0: A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) 
-\to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a 
-(AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda 
-(_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k 
-H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda 
-(H2: (eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda 
-(e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in 
-(False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k) 
-(aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: 
-A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) 
-(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head 
-a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1 
-a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A 
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) 
-H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) 
-\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda 
-(a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to 
-((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a 
-(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda 
-(_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4 
-a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3) 
-\to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a 
-(AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda 
-(_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3 
-a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a6: A).((leq g a1 a3) \to ((leq g 
-a2 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a6 (AHead a7 
-a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: 
-A).(\lambda (a8: A).(leq g a2 a8))))))) (\lambda (H8: (leq g a1 a3)).(\lambda 
-(H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A 
-(AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 
-a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A 
-(AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1 
-H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A 
-a))))))).
+(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq 
+g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g 
+a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: 
+A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: 
+(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 
+a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda 
+(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq 
+A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: 
+nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort 
+h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) 
+(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match 
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | 
+(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A 
+(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda 
+(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) 
+(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: 
+(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A 
+(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda 
+(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 
+a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 
+a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: 
+A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 
+a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6 
+a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def 
+(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
+[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5) 
+(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A 
+return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) 
+\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 
+a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to 
+(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
+a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: 
+A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6: 
+A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: 
+A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda 
+(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 
+\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A 
+A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda 
+(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) 
+(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4))))))))) 
+H6))))))))))) a y H0))) H))))).