X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fleq%2Fasucc.ma;h=079e5c2db1bd556ad0e606ad2f88e49a1902c3da;hb=7f82161596bdfccc1179f5edcc0bfd76d34516b5;hp=290c32cc9e6f2ee46d5f921db12ebfae47b8db86;hpb=f9ee4e9041c5ef7dff72da0f6fbe8f2d8204c99e;p=helm.git

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc.ma
index 290c32cc9..079e5c2db 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc.ma
@@ -16,7 +16,7 @@
 
 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/leq/asucc".
 
-include "leq/defs.ma".
+include "leq/props.ma".
 
 include "aplus/props.ma".
 
@@ -81,8 +81,629 @@ 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)))).
 
-axiom asucc_inj:
+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)))).((match n in nat return (\lambda (n3: nat).((leq g 
+(asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) 
+(ASort n1 n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O 
+n0)) (asucc g (ASort n1 n2)))).((match n1 in nat return (\lambda (n3: 
+nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g 
+(ASort O n0) (ASort n3 n2)))) with [O \Rightarrow (\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)))))) 
+| (S n3) \Rightarrow (\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)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0: 
+(leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 in 
+nat return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g 
+(ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O 
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) 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 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)))))) | (S n4) 
+\Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (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)))))]) H0))]) 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)))).((match n in nat return (\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)))))) with [O \Rightarrow (\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)))))))) | (S n1) \Rightarrow 
+(\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))))))))]) 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)))).((match n in nat 
+return (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 
+n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) with [O \Rightarrow (\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)))))) | (S n1) \Rightarrow (\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)))))]) 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).((match n in nat return 
+(\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)) 
+with [O \Rightarrow (\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)))))) | (S n1) \Rightarrow (\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)))))]) 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).((match n in nat 
+return (\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)) with 
+[O \Rightarrow (\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))))) | (S n1) \Rightarrow (\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)))))]) 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)).