X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Fasucc.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Fasucc.ma;h=c996451b46b4f9fed0e2337a5f2b455fe27e35f5;hb=831af787465e1bff886e22ee14b68c8f1bb0177c;hp=0000000000000000000000000000000000000000;hpb=d748e568acf1aaf82b63b6f703d131fcc4a3c994;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
new file mode 100644
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+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/asucc.ma
@@ -0,0 +1,746 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* 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".
+
+theorem asucc_repl:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
+(asucc g a1) (asucc g a2)))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 
+a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g 
+a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: 
+nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g 
+(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) 
+(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O 
+(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow 
+(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq 
+A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: 
+nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g 
+(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S 
+h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) 
+(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind 
+A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O 
+(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq 
+A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) 
+(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) 
+(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) 
+H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g 
+(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: 
+nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) 
+\to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next 
+g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g 
+(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) 
+n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g 
+(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: 
+A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) 
+n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) 
+k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort 
+O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) 
+(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) 
+(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g 
+(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g 
+n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort 
+O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A 
+(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda 
+(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to 
+((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g 
+(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow 
+(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) 
+\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O 
+\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) 
+(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) 
+k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) 
+\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) 
+\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 
+(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A 
+(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) 
+(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g 
+(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O 
+n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort 
+(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) 
+(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda 
+(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort 
+h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) 
+\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) 
+\Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g 
+n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 
+with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h 
+n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort 
+(S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g 
+(ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next 
+g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 
+n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a 
+(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) 
+(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A 
+(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g 
+(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S 
+h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) 
+(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) 
+(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda 
+(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g 
+(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: 
+(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g 
+a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
+
+theorem asucc_inj:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc 
+g a2)) \to (leq g a1 a2))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
+A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: 
+nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g 
+(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda 
+(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) 
+(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((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 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 
+(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 
+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 
+(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: 
+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) 
+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)) 
+(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 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 
+\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)).
+
+theorem leq_asucc:
+ \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g 
+a0)))))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: 
+A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro 
+A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) 
+(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda 
+(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A 
+(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A 
+(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g 
+(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc 
+g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) 
+(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) 
+a)).
+
+theorem leq_ahead_asucc_false:
+ \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) 
+(asucc g a1)) \to (\forall (P: Prop).P))))
+\def
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: 
+A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda 
+(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead 
+(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\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 
+(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 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)).
+
+theorem leq_asucc_false:
+ \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: 
+Prop).P)))
+\def
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) 
+a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda 
+(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) 
+\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 
+(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)).
+