X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fleq%2Fprops.ma;h=e1a1510be95e82d18adca1847b797821df5629cf;hb=20377cd037f6cc5eb9c6a5664354a8a0189d3f4f;hp=0ad16f9e7c681689f09156afbe372878275f8035;hpb=b58315ef220a130a44acbf528cd6885ddadad642;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
index 0ad16f9e7..e1a1510be 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/leq/props.ma
@@ -14,49 +14,29 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/leq/props".
+include "LambdaDelta-1/leq/fwd.ma".
 
-include "leq/defs.ma".
-
-include "aplus/props.ma".
+include "LambdaDelta-1/aplus/props.ma".
 
 theorem ahead_inj_snd:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall 
 (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
 \def
  \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda 
-(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match 
-H in leq return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a 
-a0)).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2 
-a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A 
-(ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3 
-a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A 
-return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) 
-\Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) 
-(AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) 
-k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1) 
-\Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3: 
-(eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a6 | 
-(AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6) 
-(AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A 
-(AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g 
-a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A 
-(AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g 
-a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def 
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with 
-[(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7) 
-(AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e in A 
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _) 
-\Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a: 
-A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4))))) 
-(\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to 
-((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12: 
-(leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8))) 
-a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 
-(refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
+(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def 
+(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A 
+(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda 
+(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) 
+(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda 
+(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead 
+a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in 
+A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) 
+\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((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 a3 a4) (AHead x0 x1) H3) 
+in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: 
+A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: 
+A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))).
 
 theorem leq_refl:
  \forall (g: G).(\forall (a: A).(leq g a a))
@@ -72,7 +52,7 @@ theorem leq_eq:
 a2))))
 \def
  \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 
-a2)).(eq_ind_r A a2 (\lambda (a: A).(leq g a a2)) (leq_refl g a2) a1 H)))).
+a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
 
 theorem leq_sym:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g 
@@ -97,73 +77,40 @@ a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0
 a3) \to (leq g a a3))))) (\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))).(\lambda (a3: A).(\lambda (H1: (leq g 
-(ASort h2 n2) a3)).(let H2 \def (match H1 in leq return (\lambda (a: 
-A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (ASort h2 n2)) \to 
-((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3 
-k0 H2) \Rightarrow (\lambda (H3: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda 
-(H4: (eq A (ASort h3 n3) a3)).((let H5 \def (f_equal A nat (\lambda (e: 
-A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n 
-| (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H3) in ((let H6 
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) 
-with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) 
-(ASort h2 n2) H3) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to 
-((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort 
-h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H7: (eq nat n0 
-n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A 
-(aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 
-n1) a3)))) (\lambda (H8: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) 
-(\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) 
-\to (leq g (ASort h1 n1) a))) (\lambda (H9: (eq A (aplus g (ASort h2 n2) k0) 
-(aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 
-n3)) (\lambda (H10: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) 
-(ASort h2 n2) k k H0 (minus k0 k)) in (let H11 \def (eq_ind_r nat (plus 
-(minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g 
-(ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_trans k (S k) k0 (le_S 
-k k (le_n k)) H10))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g 
-(ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H11 
-H9))))) (\lambda (H10: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) 
-(ASort h3 n3) k0 k0 H9 (minus k k0)) in (let H11 \def (eq_ind_r nat (plus 
-(minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g 
-(ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H10)) in (leq_sort g h1 h3 
-n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) 
-(aplus g (ASort h3 n3) k) H0 H11))))))) a3 H8)) n0 (sym_eq nat n0 n2 H7))) h0 
-(sym_eq nat h0 h2 H6))) H5)) H4 H2))) | (leq_head a0 a4 H2 a5 a6 H3) 
-\Rightarrow (\lambda (H4: (eq A (AHead a0 a5) (ASort h2 n2))).(\lambda (H5: 
-(eq A (AHead a4 a6) a3)).((let H6 \def (eq_ind A (AHead a0 a5) (\lambda (e: 
-A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow 
-False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind 
-((eq A (AHead a4 a6) a3) \to ((leq g a0 a4) \to ((leq g a5 a6) \to (leq g 
-(ASort h1 n1) a3)))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (ASort h2 n2)) 
-(refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: 
-(leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3 
-a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 
-a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) \to (leq g a5 
-a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def 
-(match H4 in leq return (\lambda (a: A).(\lambda (a7: A).(\lambda (_: (leq ? 
-a a7)).((eq A a (AHead a4 a6)) \to ((eq A a7 a0) \to (leq g (AHead a3 a5) 
-a0)))))) with [(leq_sort h1 h2 n1 n2 k H5) \Rightarrow (\lambda (H6: (eq A 
-(ASort h1 n1) (AHead a4 a6))).(\lambda (H7: (eq A (ASort h2 n2) a0)).((let H8 
-\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 a4 a6) H6) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq 
-A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 
-a5) a0))) H8)) H7 H5))) | (leq_head a7 a8 H5 a9 a10 H6) \Rightarrow (\lambda 
-(H7: (eq A (AHead a7 a9) (AHead a4 a6))).(\lambda (H8: (eq A (AHead a8 a10) 
-a0)).((let H9 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
-(_: A).A) with [(ASort _ _) \Rightarrow a9 | (AHead _ a) \Rightarrow a])) 
-(AHead a7 a9) (AHead a4 a6) H7) in ((let H10 \def (f_equal A A (\lambda (e: 
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a7 | 
-(AHead a _) \Rightarrow a])) (AHead a7 a9) (AHead a4 a6) H7) in (eq_ind A a4 
-(\lambda (a: A).((eq A a9 a6) \to ((eq A (AHead a8 a10) a0) \to ((leq g a a8) 
-\to ((leq g a9 a10) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H11: (eq A a9 
-a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a8 a10) a0) \to ((leq g a4 
-a8) \to ((leq g a a10) \to (leq g (AHead a3 a5) a0))))) (\lambda (H12: (eq A 
-(AHead a8 a10) a0)).(eq_ind A (AHead a8 a10) (\lambda (a: A).((leq g a4 a8) 
-\to ((leq g a6 a10) \to (leq g (AHead a3 a5) a)))) (\lambda (H13: (leq g a4 
-a8)).(\lambda (H14: (leq g a6 a10)).(leq_head g a3 a8 (H1 a8 H13) a5 a10 (H3 
-a10 H14)))) a0 H12)) a9 (sym_eq A a9 a6 H11))) a7 (sym_eq A a7 a4 H10))) H9)) 
-H8 H5 H6)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A 
-a0))))))))))))) a1 a2 H)))).
+(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 
+\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: 
+nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 
+n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 
+(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: 
+nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus 
+g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def 
+(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 
+x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 
+n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g 
+(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r 
+nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) 
+(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) 
+x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus 
+g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 
+H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) 
+(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus 
+(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g 
+(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 
+x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g 
+(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: 
+A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: 
+A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: 
+A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) 
+\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) 
+a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in 
+(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: 
+A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A 
+a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: 
+(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 
+(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead 
+a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) 
+H5))))))))))))) a1 a2 H)))).
 
 theorem leq_ahead_false_1:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) 
@@ -174,99 +121,45 @@ A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n:
 nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n 
 n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
 (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
-(ASort O n0) a2) (ASort O 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 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 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 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 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 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 n0)) \to ((leq g (ASort O n0) a3) \to ((leq 
-g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O 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 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 n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g 
-(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead 
-(ASort (S n1) n0) a2) (ASort (S 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 (S 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 
-(S 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 (S 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 (S 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 (S 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 (S 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 (S 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 (S 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 (S n1) n0))))))) n H)))))) (\lambda (a: 
-A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: 
+(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) 
+a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: 
+A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
+(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O 
+n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: 
+A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow 
+True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P 
+H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 
+n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) 
+a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 
+(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda 
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) 
+n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
+(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort 
+(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda 
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in 
+(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: 
+((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: 
 Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead 
 a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq 
-g (AHead (AHead a a0) a2) (AHead a 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 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 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 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 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 
-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 a0)) \to 
-((leq g (AHead a a0) a4) \to ((leq g a7 a6) \to P)))) (\lambda (H9: (eq A 
-(AHead a4 a6) (AHead a 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 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 a0) H9) in (eq_ind A a (\lambda (a7: A).((eq A a6 a0) \to ((leq 
-g (AHead a a0) a7) \to ((leq g a2 a6) \to P)))) (\lambda (H12: (eq A a6 
-a0)).(eq_ind A 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 
-a0)).(H a0 H13 P))) a6 (sym_eq A a6 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 
-a0))))))))))) a1)).
+g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def 
+(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in 
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) 
+(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda 
+(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 
+x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A 
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
+\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) 
+in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
+(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) 
+(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def 
+(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def 
+(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 
+H10 P))))) H6))))))) H2)))))))))) a1)).
 
 theorem leq_ahead_false_2:
  \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) 
@@ -277,97 +170,43 @@ A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n:
 nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort 
 n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g 
 (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead 
-a1 (ASort O n0)) (ASort O n0))).(let H1 \def (match H0 in leq return (\lambda 
-(a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a (AHead a1 (ASort 
-O n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k 
-H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 n1) (AHead a1 (ASort O 
-n0)))).(\lambda (H3: (eq A (ASort h2 n2) (ASort O 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 a1 (ASort O n0)) H2) in (False_ind ((eq A (ASort h2 n2) (ASort O 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 a1 (ASort O n0)))).(\lambda (H4: (eq A (AHead a3 a5) 
-(ASort O 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 a1 (ASort O n0)) 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 a1 (ASort O n0)) H3) in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort 
-O n0)) \to ((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a a3) \to ((leq g 
-a4 a5) \to P))))) (\lambda (H7: (eq A a4 (ASort O n0))).(eq_ind A (ASort O 
-n0) (\lambda (a: A).((eq A (AHead a3 a5) (ASort O n0)) \to ((leq g a1 a3) \to 
-((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort O 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 n0) H8) in (False_ind ((leq g a1 a3) \to ((leq g (ASort O 
-n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort O n0) H7))) a0 (sym_eq A a0 a1 
-H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A (AHead a1 (ASort O n0))) 
-(refl_equal A (ASort O n0))))) (\lambda (n1: nat).(\lambda (_: (((leq g 
-(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead 
-a1 (ASort (S n1) n0)) (ASort (S n1) n0))).(let H1 \def (match H0 in leq 
-return (\lambda (a: A).(\lambda (a0: A).(\lambda (_: (leq ? a a0)).((eq A a 
-(AHead a1 (ASort (S n1) n0))) \to ((eq A a0 (ASort (S n1) n0)) \to P))))) 
-with [(leq_sort h1 h2 n2 n3 k H1) \Rightarrow (\lambda (H2: (eq A (ASort h1 
-n2) (AHead a1 (ASort (S n1) n0)))).(\lambda (H3: (eq A (ASort h2 n3) (ASort 
-(S 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 a1 (ASort (S n1) n0)) H2) in (False_ind 
-((eq A (ASort h2 n3) (ASort (S 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 a1 (ASort (S n1) 
-n0)))).(\lambda (H4: (eq A (AHead a3 a5) (ASort (S 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 a1 (ASort (S n1) n0)) 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 a1 (ASort (S n1) n0)) H3) 
-in (eq_ind A a1 (\lambda (a: A).((eq A a4 (ASort (S n1) n0)) \to ((eq A 
-(AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a a3) \to ((leq g a4 a5) \to 
-P))))) (\lambda (H7: (eq A a4 (ASort (S n1) n0))).(eq_ind A (ASort (S n1) n0) 
-(\lambda (a: A).((eq A (AHead a3 a5) (ASort (S n1) n0)) \to ((leq g a1 a3) 
-\to ((leq g a a5) \to P)))) (\lambda (H8: (eq A (AHead a3 a5) (ASort (S 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 (S n1) n0) H8) in (False_ind ((leq g a1 a3) 
-\to ((leq g (ASort (S n1) n0) a5) \to P)) H9))) a4 (sym_eq A a4 (ASort (S n1) 
-n0) H7))) a0 (sym_eq A a0 a1 H6))) H5)) H4 H1 H2)))]) in (H1 (refl_equal A 
-(AHead a1 (ASort (S n1) n0))) (refl_equal A (ASort (S n1) n0))))))) n H)))))) 
-(\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) \to 
-(\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: 
-A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: 
-A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a 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 a1 (AHead a a0))) \to ((eq A 
-a4 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow 
-(\lambda (H3: (eq A (ASort h1 n1) (AHead a1 (AHead a a0)))).(\lambda (H4: (eq 
-A (ASort h2 n2) (AHead a 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 a1 (AHead a a0)) 
-H3) in (False_ind ((eq A (ASort h2 n2) (AHead a 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 a1 
-(AHead a a0)))).(\lambda (H5: (eq A (AHead a4 a6) (AHead a 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 a1 (AHead a a0)) 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 a1 (AHead a a0)) H4) in 
-(eq_ind A a1 (\lambda (a7: A).((eq A a5 (AHead a a0)) \to ((eq A (AHead a4 
-a6) (AHead a a0)) \to ((leq g a7 a4) \to ((leq g a5 a6) \to P))))) (\lambda 
-(H8: (eq A a5 (AHead a a0))).(eq_ind A (AHead a a0) (\lambda (a7: A).((eq A 
-(AHead a4 a6) (AHead a a0)) \to ((leq g a1 a4) \to ((leq g a7 a6) \to P)))) 
-(\lambda (H9: (eq A (AHead a4 a6) (AHead a 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 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 a0) H9) in (eq_ind A a (\lambda (a7: 
-A).((eq A a6 a0) \to ((leq g a1 a7) \to ((leq g (AHead a a0) a6) \to P)))) 
-(\lambda (H12: (eq A a6 a0)).(eq_ind A a0 (\lambda (a7: A).((leq g a1 a) \to 
-((leq g (AHead a a0) a7) \to P))) (\lambda (_: (leq g a1 a)).(\lambda (H14: 
-(leq g (AHead a a0) a0)).(H0 a H14 P))) a6 (sym_eq A a6 a0 H12))) a4 (sym_eq 
-A a4 a H11))) H10))) a5 (sym_eq A a5 (AHead a a0) H8))) a3 (sym_eq A a3 a1 
-H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a1 (AHead a a0))) 
-(refl_equal A (AHead a a0))))))))))) a2)).
+a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O 
+n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g 
+(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) 
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 
+x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) 
+(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match 
+ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | 
+(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P 
+H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 
+n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) 
+n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) 
+(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: 
+A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g 
+(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) 
+n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g 
+a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort 
+(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda 
+(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) 
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in 
+(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: 
+((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: 
+Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead 
+a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq 
+g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def 
+(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in 
+(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: 
+A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: 
+A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: 
+A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) 
+x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A 
+A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) 
+\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) 
+in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda 
+(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) 
+(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def 
+(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let 
+H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 
+P))))) H6))))))) H2)))))))))) a2)).