X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Fasucc.ma;h=47a13f362abeb6e6e205d321454e181015c7f6c2;hb=89519c7b52e06304a94019dd528925300380cdc0;hp=25f9dfd074c944c466a5f7639c86be2eb7a4556f;hpb=fdda444a05fe4c68c925cd94e4e3a38c93d0c35f;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
index 25f9dfd07..47a13f362 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
@@ -14,11 +14,7 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/asucc".
-
-include "leq/props.ma".
-
-include "aplus/props.ma".
+include "LambdaDelta-1/leq/props.ma".
 
 theorem asucc_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
@@ -114,196 +110,118 @@ n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))
 (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 
 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g 
 (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g 
-(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 in leq 
-return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a 
-(ASort O (next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort 
-O n0) (ASort O n2))))))) with [(leq_sort h1 h2 n3 n4 k H2) \Rightarrow 
-(\lambda (H3: (eq A (ASort h1 n3) (ASort O (next g n0)))).(\lambda (H4: (eq A 
-(ASort h2 n4) (ASort O (next g n2)))).((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 n3])) (ASort h1 n3) (ASort O (next g 
-n0)) H3) 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 h1])) (ASort h1 n3) (ASort O (next g n0)) H3) in (eq_ind nat O 
-(\lambda (n5: nat).((eq nat n3 (next g n0)) \to ((eq A (ASort h2 n4) (ASort O 
-(next g n2))) \to ((eq A (aplus g (ASort n5 n3) k) (aplus g (ASort h2 n4) k)) 
-\to (leq g (ASort O n0) (ASort O n2)))))) (\lambda (H7: (eq nat n3 (next g 
-n0))).(eq_ind nat (next g n0) (\lambda (n5: nat).((eq A (ASort h2 n4) (ASort 
-O (next g n2))) \to ((eq A (aplus g (ASort O n5) k) (aplus g (ASort h2 n4) 
-k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H8: (eq A (ASort h2 
-n4) (ASort O (next g n2)))).(let H9 \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 h2 n4) (ASort O (next g n2)) H8) in 
-((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda 
-(_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow h2])) 
-(ASort h2 n4) (ASort O (next g n2)) H8) in (eq_ind nat O (\lambda (n5: 
-nat).((eq nat n4 (next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) 
-(aplus g (ASort n5 n4) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda 
-(H11: (eq nat n4 (next g n2))).(eq_ind nat (next g n2) (\lambda (n5: 
-nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n5) k)) \to 
-(leq g (ASort O n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort O 
-(next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A 
-(aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O 
-(next g n2)) k))) H12 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 
-k)) in (let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda 
-(a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g (ASort O n2) (S 
-k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H14)))) n4 
-(sym_eq nat n4 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9))) n3 (sym_eq 
-nat n3 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head 
-a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort O 
-(next g n0)))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g 
-n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ 
-_) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A 
-(AHead a3 a5) (ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5) 
-\to (leq g (ASort O n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2 
-(refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2)))))) 
-(\lambda (n3: nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g 
-(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq 
-g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def (match H1 
-in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a 
-a0)).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g 
-(ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n4 n5 k H2) 
-\Rightarrow (\lambda (H3: (eq A (ASort h1 n4) (ASort O (next g 
-n0)))).(\lambda (H4: (eq A (ASort h2 n5) (ASort n3 n2))).((let H5 \def 
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with 
-[(ASort _ n6) \Rightarrow n6 | (AHead _ _) \Rightarrow n4])) (ASort h1 n4) 
-(ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e: 
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow 
-n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4) (ASort O (next g n0)) H3) in 
-(eq_ind nat O (\lambda (n6: nat).((eq nat n4 (next g n0)) \to ((eq A (ASort 
-h2 n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort n6 n4) k) (aplus g (ASort h2 
-n5) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H7: (eq nat 
-n4 (next g n0))).(eq_ind nat (next g n0) (\lambda (n6: nat).((eq A (ASort h2 
-n5) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n6) k) (aplus g (ASort h2 n5) 
-k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H8: (eq A (ASort 
-h2 n5) (ASort n3 n2))).(let H9 \def (f_equal A nat (\lambda (e: A).(match e 
-in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _ 
-_) \Rightarrow n5])) (ASort h2 n5) (ASort n3 n2) H8) in ((let H10 \def 
+(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 
+g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind 
+nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A 
+(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: 
+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 n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h2])) (ASort h2 n5) 
-(ASort n3 n2) H8) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n5 n2) \to 
-((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n6 n5) k)) \to (leq 
-g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H11: (eq nat n5 n2)).(eq_ind 
-nat n2 (\lambda (n6: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g 
-(ASort n3 n6) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H12: 
-(eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H13 
-\def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a 
-(aplus g (ASort n3 n2) k))) H12 (aplus g (ASort O n0) (S k)) 
-(aplus_sort_O_S_simpl g n0 k)) in (let H14 \def (eq_ind_r A (aplus g (ASort 
-n3 n2) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H13 (aplus g 
-(ASort (S n3) n2) (S k)) (aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O 
-(S n3) n0 n2 (S k) H14)))) n5 (sym_eq nat n5 n2 H11))) h2 (sym_eq nat h2 n3 
-H10))) H9))) n4 (sym_eq nat n4 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) 
-H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A 
-(AHead a0 a4) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a3 a5) 
-(ASort n3 n2))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match 
-e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | 
-(AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind 
-((eq A (AHead a3 a5) (ASort n3 n2)) \to ((leq g a0 a3) \to ((leq g a4 a5) \to 
-(leq g (ASort O n0) (ASort (S n3) n2))))) H6)) H5 H2 H3)))]) in (H2 
-(refl_equal A (ASort O (next g n0))) (refl_equal A (ASort n3 n2))))))) n1 
-H0)) (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) 
-(asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda 
-(H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind 
-(\lambda (n4: nat).((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 O 
+[(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: 
+nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) 
+\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g 
+(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O 
+(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat 
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda 
+(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g 
+(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 _ 
+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 
+n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: 
+nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 
+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 n3 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 n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 
+(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in 
+(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) 
+(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g 
+(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq 
+g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda 
+(n4: nat).((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 O 
 n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) 
-\to (leq g (ASort n3 n0) (ASort O n2))))).(let H2 \def (match H1 in leq 
-return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a 
-(ASort n3 n0)) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort (S n3) 
-n0) (ASort O n2))))))) with [(leq_sort h1 h2 n4 n5 k H2) \Rightarrow (\lambda 
-(H3: (eq A (ASort h1 n4) (ASort n3 n0))).(\lambda (H4: (eq A (ASort h2 n5) 
-(ASort O (next g n2)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e 
-in A return (\lambda (_: A).nat) with [(ASort _ n6) \Rightarrow n6 | (AHead _ 
-_) \Rightarrow n4])) (ASort h1 n4) (ASort n3 n0) H3) in ((let H6 \def 
+\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 
+n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat 
+(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort 
+n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: 
+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 n6 _) \Rightarrow n6 | (AHead _ _) \Rightarrow h1])) (ASort h1 n4) 
-(ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n6: nat).((eq nat n4 n0) \to 
-((eq A (ASort h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n6 n4) 
-k) (aplus g (ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))))) 
-(\lambda (H7: (eq nat n4 n0)).(eq_ind nat n0 (\lambda (n6: nat).((eq A (ASort 
-h2 n5) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n3 n6) k) (aplus g 
-(ASort h2 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H8: 
-(eq A (ASort h2 n5) (ASort O (next g n2)))).(let H9 \def (f_equal A nat 
-(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n6) 
-\Rightarrow n6 | (AHead _ _) \Rightarrow n5])) (ASort h2 n5) (ASort O (next g 
-n2)) H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A 
-return (\lambda (_: A).nat) with [(ASort n6 _) \Rightarrow n6 | (AHead _ _) 
-\Rightarrow h2])) (ASort h2 n5) (ASort O (next g n2)) H8) in (eq_ind nat O 
-(\lambda (n6: nat).((eq nat n5 (next g n2)) \to ((eq A (aplus g (ASort n3 n0) 
-k) (aplus g (ASort n6 n5) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) 
-(\lambda (H11: (eq nat n5 (next g n2))).(eq_ind nat (next g n2) (\lambda (n6: 
-nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O n6) k)) \to (leq g 
-(ASort (S n3) n0) (ASort O n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0) 
-k) (aplus g (ASort O (next g n2)) k))).(let H13 \def (eq_ind_r A (aplus g 
-(ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) 
-H12 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in 
-(let H14 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: 
-A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H13 (aplus g (ASort O n2) (S 
-k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g (S n3) O n0 n2 (S k) 
-H14)))) n5 (sym_eq nat n5 (next g n2) H11))) h2 (sym_eq nat h2 O H10))) H9))) 
-n4 (sym_eq nat n4 n0 H7))) h1 (sym_eq nat h1 n3 H6))) H5)) H4 H2))) | 
-(leq_head a0 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a0 a4) 
-(ASort n3 n0))).(\lambda (H5: (eq A (AHead a3 a5) (ASort O (next g 
-n2)))).((let H6 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ 
-_) \Rightarrow True])) I (ASort n3 n0) H4) in (False_ind ((eq A (AHead a3 a5) 
-(ASort O (next g n2))) \to ((leq g a0 a3) \to ((leq g a4 a5) \to (leq g 
-(ASort (S n3) n0) (ASort O n2))))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A 
-(ASort n3 n0)) (refl_equal A (ASort O (next g n2))))))) (\lambda (n4: 
+[(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 H2 \def (match 
-H1 in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a 
-a0)).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 n2)) \to (leq g (ASort 
-(S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1 h2 n5 n6 k H2) 
-\Rightarrow (\lambda (H3: (eq A (ASort h1 n5) (ASort n3 n0))).(\lambda (H4: 
-(eq A (ASort h2 n6) (ASort n4 n2))).((let H5 \def (f_equal A nat (\lambda (e: 
-A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7) \Rightarrow 
-n7 | (AHead _ _) \Rightarrow n5])) (ASort h1 n5) (ASort n3 n0) H3) in ((let 
-H6 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: 
-A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _) \Rightarrow h1])) 
-(ASort h1 n5) (ASort n3 n0) H3) in (eq_ind nat n3 (\lambda (n7: nat).((eq nat 
-n5 n0) \to ((eq A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n7 
-n5) k) (aplus g (ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) 
-n2)))))) (\lambda (H7: (eq nat n5 n0)).(eq_ind nat n0 (\lambda (n7: nat).((eq 
-A (ASort h2 n6) (ASort n4 n2)) \to ((eq A (aplus g (ASort n3 n7) k) (aplus g 
-(ASort h2 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda 
-(H8: (eq A (ASort h2 n6) (ASort n4 n2))).(let H9 \def (f_equal A nat (\lambda 
-(e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n7) 
-\Rightarrow n7 | (AHead _ _) \Rightarrow n6])) (ASort h2 n6) (ASort n4 n2) 
-H8) in ((let H10 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort n7 _) \Rightarrow n7 | (AHead _ _) 
-\Rightarrow h2])) (ASort h2 n6) (ASort n4 n2) H8) in (eq_ind nat n4 (\lambda 
-(n7: nat).((eq nat n6 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g 
-(ASort n7 n6) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda 
-(H11: (eq nat n6 n2)).(eq_ind nat n2 (\lambda (n7: nat).((eq A (aplus g 
-(ASort n3 n0) k) (aplus g (ASort n4 n7) k)) \to (leq g (ASort (S n3) n0) 
-(ASort (S n4) n2)))) (\lambda (H12: (eq A (aplus g (ASort n3 n0) k) (aplus g 
-(ASort n4 n2) k))).(let H13 \def (eq_ind_r A (aplus g (ASort n3 n0) k) 
-(\lambda (a: A).(eq A a (aplus g (ASort n4 n2) k))) H12 (aplus g (ASort (S 
-n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H14 \def (eq_ind_r A 
-(aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S 
-k)) a)) H13 (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 
-k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k) H14)))) n6 (sym_eq nat n6 n2 
-H11))) h2 (sym_eq nat h2 n4 H10))) H9))) n5 (sym_eq nat n5 n0 H7))) h1 
-(sym_eq nat h1 n3 H6))) H5)) H4 H2))) | (leq_head a0 a3 H2 a4 a5 H3) 
-\Rightarrow (\lambda (H4: (eq A (AHead a0 a4) (ASort n3 n0))).(\lambda (H5: 
-(eq A (AHead a3 a5) (ASort n4 n2))).((let H6 \def (eq_ind A (AHead a0 a4) 
-(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) 
-\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H4) in 
-(False_ind ((eq A (AHead a3 a5) (ASort n4 n2)) \to ((leq g a0 a3) \to ((leq g 
-a4 a5) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) H6)) H5 H2 H3)))]) 
-in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort n4 n2)))))))) n1 H0 
-IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) 
-(asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: 
-(((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) 
+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: 
+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 
+(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)) 
+(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g 
+(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) 
+a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) 
+in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 
+H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n 
+n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda 
+(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) 
 a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a 
 a0)))).(nat_ind (\lambda (n1: nat).((((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)) 
@@ -312,185 +230,91 @@ n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))
 (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O 
 n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq 
 g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g 
-(AHead a a0)))).(let H5 \def (match H4 in leq return (\lambda (a3: 
-A).(\lambda (a4: A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort O (next g 
-n0))) \to ((eq A a4 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a 
-a0))))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A 
-(ASort h1 n1) (ASort O (next g n0)))).(\lambda (H7: (eq A (ASort h2 n2) 
-(AHead a (asucc g a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match 
-e in A return (\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead 
-_ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H6) in ((let H9 
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) 
-with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow h1])) (ASort h1 
-n1) (ASort O (next g n0)) H6) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1 
-(next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A 
-(aplus g (ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) 
-(AHead a a0)))))) (\lambda (H10: (eq nat n1 (next g n0))).(eq_ind nat (next g 
-n0) (\lambda (n3: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq 
-A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) 
-(AHead a a0))))) (\lambda (H11: (eq A (ASort h2 n2) (AHead a (asucc g 
-a0)))).(let H12 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead a (asucc g a0)) H11) in (False_ind ((eq A 
-(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g 
-(ASort O n0) (AHead a a0))) H12))) n1 (sym_eq nat n1 (next g n0) H10))) h1 
-(sym_eq nat h1 O H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 H6) 
-\Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort O (next g 
-n0)))).(\lambda (H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9 
-\def (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda 
-(_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
-True])) I (ASort O (next g n0)) H7) in (False_ind ((eq A (AHead a4 a6) (AHead 
-a (asucc g a0))) \to ((leq g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort O 
-n0) (AHead a a0))))) H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort O (next g 
-n0))) (refl_equal A (AHead a (asucc g a0)))))))) (\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 H5 \def (match H4 in leq return (\lambda (a3: A).(\lambda (a4: 
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (ASort n1 n0)) \to ((eq A a4 (AHead 
-a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0))))))) with 
-[(leq_sort h1 h2 n2 n3 k H5) \Rightarrow (\lambda (H6: (eq A (ASort h1 n2) 
-(ASort n1 n0))).(\lambda (H7: (eq A (ASort h2 n3) (AHead a (asucc g 
-a0)))).((let H8 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort _ n4) \Rightarrow n4 | (AHead _ _) 
-\Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) H6) in ((let H9 \def (f_equal A 
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort n4 
-_) \Rightarrow n4 | (AHead _ _) \Rightarrow h1])) (ASort h1 n2) (ASort n1 n0) 
-H6) in (eq_ind nat n1 (\lambda (n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2 
-n3) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n4 n2) k) (aplus g 
-(ASort h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda 
-(H10: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort h2 n3) 
-(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g (ASort 
-h2 n3) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H11: (eq A 
-(ASort h2 n3) (AHead a (asucc g a0)))).(let H12 \def (eq_ind A (ASort h2 n3) 
-(\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) 
-\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) 
-H11) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n3) 
-k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H12))) n2 (sym_eq nat n2 n0 
-H10))) h1 (sym_eq nat h1 n1 H9))) H8)) H7 H5))) | (leq_head a3 a4 H5 a5 a6 
-H6) \Rightarrow (\lambda (H7: (eq A (AHead a3 a5) (ASort n1 n0))).(\lambda 
-(H8: (eq A (AHead a4 a6) (AHead a (asucc g a0)))).((let H9 \def (eq_ind A 
-(AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 
-n0) H7) in (False_ind ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to ((leq 
-g a3 a4) \to ((leq g a5 a6) \to (leq g (ASort (S n1) n0) (AHead a a0))))) 
-H9)) H8 H5 H6)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a 
-(asucc g a0)))))))))) 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 H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4: 
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A 
-a4 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with 
-[(leq_sort h1 h2 n1 n2 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n1) 
-(AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n2) (ASort O (next g 
-n0)))).((let H6 \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 a (asucc g a0)) H4) in (False_ind ((eq A (ASort 
-h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H6)) H5 H3))) | 
-(leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5) 
-(AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort O (next g 
-n0)))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) 
-\Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5) 
-(AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g 
-a0)) \to ((eq A (AHead a4 a6) (ASort O (next g n0))) \to ((leq g a7 a4) \to 
-((leq g a5 a6) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H9: (eq A 
-a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 
-a6) (ASort O (next g n0))) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g 
-(AHead a a0) (ASort O n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort O 
-(next g n0)))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e 
-in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | 
-(AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H10) in (False_ind 
-((leq g a a4) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort O 
-n0)))) H11))) a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) 
-H6 H3 H4)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A 
-(ASort O (next g n0)))))) (\lambda (n1: nat).(\lambda (_: (((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 (S n1) 
-n0)))).(let H3 \def (match H2 in leq return (\lambda (a3: A).(\lambda (a4: 
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead a (asucc g a0))) \to ((eq A 
-a4 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) with 
-[(leq_sort h1 h2 n2 n3 k H3) \Rightarrow (\lambda (H4: (eq A (ASort h1 n2) 
-(AHead a (asucc g a0)))).(\lambda (H5: (eq A (ASort h2 n3) (ASort n1 
-n0))).((let H6 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead a (asucc g a0)) H4) in (False_ind ((eq A (ASort 
-h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g (ASort h2 
-n3) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H6)) H5 H3))) | 
-(leq_head a3 a4 H3 a5 a6 H4) \Rightarrow (\lambda (H5: (eq A (AHead a3 a5) 
-(AHead a (asucc g a0)))).(\lambda (H6: (eq A (AHead a4 a6) (ASort n1 
-n0))).((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a7) 
-\Rightarrow a7])) (AHead a3 a5) (AHead a (asucc g a0)) H5) in ((let H8 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) (AHead a3 a5) 
-(AHead a (asucc g a0)) H5) in (eq_ind A a (\lambda (a7: A).((eq A a5 (asucc g 
-a0)) \to ((eq A (AHead a4 a6) (ASort n1 n0)) \to ((leq g a7 a4) \to ((leq g 
-a5 a6) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H9: (eq A a5 
-(asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a7: A).((eq A (AHead a4 a6) 
-(ASort n1 n0)) \to ((leq g a a4) \to ((leq g a7 a6) \to (leq g (AHead a a0) 
-(ASort (S n1) n0)))))) (\lambda (H10: (eq A (AHead a4 a6) (ASort n1 
-n0))).(let H11 \def (eq_ind A (AHead a4 a6) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ 
-_) \Rightarrow True])) I (ASort n1 n0) H10) in (False_ind ((leq g a a4) \to 
-((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H11))) 
-a5 (sym_eq A a5 (asucc g a0) H9))) a3 (sym_eq A a3 a H8))) H7)) H6 H3 H4)))]) 
-in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1 
-n0))))))) 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 H4 \def (match H3 in leq return (\lambda (a5: A).(\lambda (a6: 
-A).(\lambda (_: (leq ? a5 a6)).((eq A a5 (AHead a (asucc g a0))) \to ((eq A 
-a6 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with 
-[(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1) 
-(AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g 
-a4)))).((let H7 \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 a (asucc g a0)) H5) in (False_ind ((eq A (ASort 
-h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | 
-(leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7) 
-(AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a6 a8) (AHead a3 (asucc g 
-a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e in A return 
-(\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a9) 
-\Rightarrow a9])) (AHead a5 a7) (AHead a (asucc g a0)) H6) in ((let H9 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a5 | (AHead a9 _) \Rightarrow a9])) (AHead a5 a7) 
-(AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a9: A).((eq A a7 (asucc g 
-a0)) \to ((eq A (AHead a6 a8) (AHead a3 (asucc g a4))) \to ((leq g a9 a6) \to 
-((leq g a7 a8) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq 
-A a7 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a9: A).((eq A (AHead a6 
-a8) (AHead a3 (asucc g a4))) \to ((leq g a a6) \to ((leq g a9 a8) \to (leq g 
-(AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a6 a8) (AHead a3 
-(asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a8 | (AHead _ a9) 
-\Rightarrow a9])) (AHead a6 a8) (AHead a3 (asucc g a4)) H11) in ((let H13 
+(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g 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 O (next g 
+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 O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda 
+(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 _ _) 
+\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))) 
+\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g 
+(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a 
+(asucc g a0) (ASort n1 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 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 a6 | (AHead a9 _) \Rightarrow a9])) (AHead a6 
-a8) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a9: A).((eq A a8 
-(asucc g a4)) \to ((leq g a a9) \to ((leq g (asucc g a0) a8) \to (leq g 
-(AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a8 (asucc g 
-a4))).(eq_ind A (asucc g a4) (\lambda (a9: A).((leq g a a3) \to ((leq g 
-(asucc g a0) a9) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15: 
-(leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a 
-a3 H15 a0 a4 (H0 a4 H16)))) a8 (sym_eq A a8 (asucc g a4) H14))) a6 (sym_eq A 
-a6 a3 H13))) H12))) a7 (sym_eq A a7 (asucc g a0) H10))) a5 (sym_eq A a5 a 
-H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0))) 
-(refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
+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)).
 
 theorem leq_asucc:
  \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g 
@@ -519,103 +343,51 @@ A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda
 \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: 
 nat).((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 O n0) a2) (ASort O (next g n0)))).(let H1 \def (match H0 in leq 
-return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a 
-(AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O (next g n0))) \to P))))) with 
-[(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) 
-(AHead (ASort O n0) a2))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O (next g 
-n0)))).((let H4 \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 (ASort O n0) a2) H2) in (False_ind ((eq A 
-(ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) 
-(aplus g (ASort h2 n2) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 
-H2) \Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort O n0) 
-a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort O (next g n0)))).((let H5 \def 
+(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 
+g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind 
+A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: 
+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 
+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 
+(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: 
+Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) 
+(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g 
+(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind 
+A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: 
+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 a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) 
-(AHead (ASort O n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | 
-(AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort O n0) a2) H3) in 
-(eq_ind A (ASort O n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) 
-(ASort O (next g n0))) \to ((leq g a a3) \to ((leq g a4 a5) \to P))))) 
-(\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) 
-(ASort O (next g n0))) \to ((leq g (ASort O n0) a3) \to ((leq g a a5) \to 
-P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O (next g n0)))).(let H9 \def 
-(eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: 
-A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow 
-True])) I (ASort O (next g n0)) H8) in (False_ind ((leq g (ASort O n0) a3) 
-\to ((leq g a2 a5) \to P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 
-(ASort O n0) H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort O 
-n0) a2)) (refl_equal A (ASort O (next g n0)))))) (\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 H1 \def (match H0 in leq 
-return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a 
-(AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P))))) with 
-[(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n2) 
-(AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (ASort h2 n3) (ASort n1 
-n0))).((let H4 \def (eq_ind A (ASort h1 n2) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H2) in (False_ind ((eq A 
-(ASort h2 n3) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n2) k) (aplus g 
-(ASort h2 n3) k)) \to P)) H4)) H3 H1))) | (leq_head a0 a3 H1 a4 a5 H2) 
-\Rightarrow (\lambda (H3: (eq A (AHead a0 a4) (AHead (ASort (S n1) n0) 
-a2))).(\lambda (H4: (eq A (AHead a3 a5) (ASort n1 n0))).((let H5 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) 
-(AHead (ASort (S n1) n0) a2) H3) in ((let H6 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | 
-(AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead (ASort (S n1) n0) a2) H3) 
-in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a4 a2) \to ((eq A 
-(AHead a3 a5) (ASort n1 n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to P))))) 
-(\lambda (H7: (eq A a4 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a3 a5) 
-(ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a3) \to ((leq g a a5) \to P)))) 
-(\lambda (H8: (eq A (AHead a3 a5) (ASort n1 n0))).(let H9 \def (eq_ind A 
-(AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_: A).Prop) with 
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 
-n0) H8) in (False_ind ((leq g (ASort (S n1) n0) a3) \to ((leq g a2 a5) \to 
-P)) H9))) a4 (sym_eq A a4 a2 H7))) a0 (sym_eq A a0 (ASort (S n1) n0) H6))) 
-H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2)) 
-(refl_equal A (ASort n1 n0))))))) 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 (a2: A).((leq g (AHead 
-a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda 
-(H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: 
-Prop).(let H2 \def (match H1 in leq return (\lambda (a3: A).(\lambda (a4: 
-A).(\lambda (_: (leq ? a3 a4)).((eq A a3 (AHead (AHead a a0) a2)) \to ((eq A 
-a4 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) 
-\Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) 
-a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 
-\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 (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) 
-(AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort 
-h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a3 a4 H2 a5 a6 H3) \Rightarrow 
-(\lambda (H4: (eq A (AHead a3 a5) (AHead (AHead a a0) a2))).(\lambda (H5: (eq 
-A (AHead a4 a6) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda 
-(e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow 
-a5 | (AHead _ a7) \Rightarrow a7])) (AHead a3 a5) (AHead (AHead a a0) a2) H4) 
-in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
-(_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a7 _) \Rightarrow a7])) 
-(AHead a3 a5) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda 
-(a7: A).((eq A a5 a2) \to ((eq A (AHead a4 a6) (AHead a (asucc g a0))) \to 
-((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda (H8: (eq A a5 
-a2)).(eq_ind A a2 (\lambda (a7: A).((eq A (AHead a4 a6) (AHead a (asucc g 
-a0))) \to ((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9: 
-(eq A (AHead a4 a6) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A 
-(\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
-\Rightarrow a6 | (AHead _ a7) \Rightarrow a7])) (AHead a4 a6) (AHead a (asucc 
-g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead a7 _) 
-\Rightarrow a7])) (AHead a4 a6) (AHead a (asucc g a0)) H9) in (eq_ind A a 
-(\lambda (a7: A).((eq A a6 (asucc g a0)) \to ((leq g (AHead a a0) a7) \to 
-((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 (asucc g a0))).(eq_ind A 
-(asucc g a0) (\lambda (a7: A).((leq g (AHead a a0) a) \to ((leq g a2 a7) \to 
-P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g 
-a0))).(leq_ahead_false_1 g a a0 H13 P))) a6 (sym_eq A a6 (asucc g a0) H12))) 
-a4 (sym_eq A a4 a H11))) H10))) a5 (sym_eq A a5 a2 H8))) a3 (sym_eq A a3 
-(AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a 
-a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
+[(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)).
 
 theorem leq_asucc_false:
  \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: 
@@ -627,120 +399,66 @@ a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda
 \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind 
 (\lambda (n1: nat).((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 O (next g n0)) (ASort O n0))).(let H1 \def (match H0 in leq 
-return (\lambda (a0: A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A 
-a0 (ASort O (next g n0))) \to ((eq A a1 (ASort O n0)) \to P))))) with 
-[(leq_sort h1 h2 n1 n2 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) 
-(ASort O (next g n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O 
-n0))).((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return 
-(\lambda (_: A).nat) with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) 
-\Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H2) in ((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 h1])) (ASort h1 n1) 
-(ASort O (next g n0)) H2) in (eq_ind nat O (\lambda (n3: nat).((eq nat n1 
-(next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g 
-(ASort n3 n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H6: (eq nat 
-n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n3: nat).((eq A (ASort h2 
-n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n3) k) (aplus g (ASort h2 n2) 
-k)) \to P))) (\lambda (H7: (eq A (ASort h2 n2) (ASort O n0))).(let H8 \def 
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with 
-[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow n2])) (ASort h2 n2) 
-(ASort O n0) H7) in ((let H9 \def (f_equal A nat (\lambda (e: A).(match e in 
-A return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) 
-\Rightarrow h2])) (ASort h2 n2) (ASort O n0) H7) in (eq_ind nat O (\lambda 
-(n3: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus 
-g (ASort n3 n2) k)) \to P))) (\lambda (H10: (eq nat n2 n0)).(eq_ind nat n0 
-(\lambda (n3: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O 
-n3) k)) \to P)) (\lambda (H11: (eq A (aplus g (ASort O (next g n0)) k) (aplus 
-g (ASort O n0) k))).(let H12 \def (eq_ind_r A (aplus g (ASort O (next g n0)) 
-k) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) k))) H11 (aplus g (ASort O 
-n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k) 
-k (ASort O n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n3: nat).(le n3 
-k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H10))) h2 (sym_eq nat h2 O 
-H9))) H8))) n1 (sym_eq nat n1 (next g n0) H6))) h1 (sym_eq nat h1 O H5))) 
-H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A 
-(AHead a1 a3) (ASort O (next g n0)))).(\lambda (H4: (eq A (AHead a2 a4) 
-(ASort O n0))).((let H5 \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 O (next g n0)) H3) in (False_ind 
-((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to 
-P))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (ASort O (next g n0))) 
-(refl_equal A (ASort O n0))))) (\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 H1 \def (match H0 in leq return (\lambda (a0: 
-A).(\lambda (a1: A).(\lambda (_: (leq ? a0 a1)).((eq A a0 (ASort n1 n0)) \to 
-((eq A a1 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n2 n3 k H1) 
-\Rightarrow (\lambda (H2: (eq A (ASort h1 n2) (ASort n1 n0))).(\lambda (H3: 
-(eq A (ASort h2 n3) (ASort (S n1) n0))).((let H4 \def (f_equal A nat (\lambda 
-(e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n4) 
-\Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort h1 n2) (ASort n1 n0) 
-H2) in ((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 h1])) (ASort h1 n2) (ASort n1 n0) H2) in (eq_ind nat n1 (\lambda 
-(n4: nat).((eq nat n2 n0) \to ((eq A (ASort h2 n3) (ASort (S n1) n0)) \to 
-((eq A (aplus g (ASort n4 n2) k) (aplus g (ASort h2 n3) k)) \to P)))) 
-(\lambda (H6: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A (ASort 
-h2 n3) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n4) k) (aplus g 
-(ASort h2 n3) k)) \to P))) (\lambda (H7: (eq A (ASort h2 n3) (ASort (S n1) 
-n0))).(let H8 \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 h2 n3) (ASort (S n1) n0) H7) in ((let H9 \def 
+(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O 
+(next g n0) (ASort O 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 
+O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda 
+(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\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 n4 _) \Rightarrow n4 | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) 
-(ASort (S n1) n0) H7) in (eq_ind nat (S n1) (\lambda (n4: nat).((eq nat n3 
-n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n4 n3) k)) \to P))) 
-(\lambda (H10: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n4: nat).((eq A 
-(aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n4) k)) \to P)) (\lambda 
-(H11: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let 
-H12 \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a0: A).(eq A a0 
-(aplus g (ASort (S n1) n0) k))) H11 (aplus g (ASort (S n1) n0) (S k)) 
-(aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k 
-(ASort (S n1) n0) H12) in (le_Sx_x k (eq_ind_r nat k (\lambda (n4: nat).(le 
-n4 k)) (le_n k) (S k) H_y) P)))) n3 (sym_eq nat n3 n0 H10))) h2 (sym_eq nat 
-h2 (S n1) H9))) H8))) n2 (sym_eq nat n2 n0 H6))) h1 (sym_eq nat h1 n1 H5))) 
-H4)) H3 H1))) | (leq_head a1 a2 H1 a3 a4 H2) \Rightarrow (\lambda (H3: (eq A 
-(AHead a1 a3) (ASort n1 n0))).(\lambda (H4: (eq A (AHead a2 a4) (ASort (S n1) 
-n0))).((let H5 \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 n1 n0) H3) in (False_ind ((eq A (AHead a2 a4) 
-(ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H5)) H4 H1 
-H2)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1) 
-n0))))))) 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 H2 \def (match H1 in leq return 
-(\lambda (a2: A).(\lambda (a3: A).(\lambda (_: (leq ? a2 a3)).((eq A a2 
-(AHead a0 (asucc g a1))) \to ((eq A a3 (AHead a0 a1)) \to P))))) with 
-[(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) 
-(AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0 
-a1))).((let H5 \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 a0 (asucc g a1)) H3) in (False_ind ((eq A 
-(ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g 
-(ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a2 a3 H2 a4 a5 H3) 
-\Rightarrow (\lambda (H4: (eq A (AHead a2 a4) (AHead a0 (asucc g 
-a1)))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a0 a1))).((let H6 \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 a2 a4) 
-(AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a2 | 
-(AHead a6 _) \Rightarrow a6])) (AHead a2 a4) (AHead a0 (asucc g a1)) H4) in 
-(eq_ind A a0 (\lambda (a6: A).((eq A a4 (asucc g a1)) \to ((eq A (AHead a3 
-a5) (AHead a0 a1)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda 
-(H8: (eq A a4 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a6: A).((eq A 
-(AHead a3 a5) (AHead a0 a1)) \to ((leq g a0 a3) \to ((leq g a6 a5) \to P)))) 
-(\lambda (H9: (eq A (AHead a3 a5) (AHead a0 a1))).(let H10 \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 a0 a1) 
-H9) in ((let H11 \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 a0 a1) H9) in (eq_ind A a0 (\lambda 
-(a6: A).((eq A a5 a1) \to ((leq g a0 a6) \to ((leq g (asucc g a1) a5) \to 
-P)))) (\lambda (H12: (eq A a5 a1)).(eq_ind A a1 (\lambda (a6: A).((leq g a0 
-a0) \to ((leq g (asucc g a1) a6) \to P))) (\lambda (_: (leq g a0 
-a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a5 (sym_eq A a5 a1 
-H12))) a3 (sym_eq A a3 a0 H11))) H10))) a4 (sym_eq A a4 (asucc g a1) H8))) a2 
-(sym_eq A a2 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0 
-(asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
+[(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) 
+(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+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)).