(* 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))
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
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)
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)
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)).
projects / helm.git / blobdiff