(* This file was automatically generated: do not edit *********************)
-include "Basic-1/leq/props.ma".
+include "basic_1/leq/props.ma".
-theorem asucc_repl:
+lemma asucc_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
(asucc g a1) (asucc g a2)))))
\def
(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_:
(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g
a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
-(* COMMENTS
-Initial nodes: 1907
-END *)
-theorem asucc_inj:
+lemma 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
h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2))
-(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _)
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3
-O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g
-(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in
-(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a:
-A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S
-x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
+(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with
+[(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) (ASort O (next g
+n2)) (ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match
+e with [(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with
+[(mk_G next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0)
+H4) in (\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda
+(n3: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0)
+x2))) H3 O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A
+(aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2)
+H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda
+(a: A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0)
+(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g
(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2))
a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3:
(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1:
nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0))
x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
-\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _)
+\Rightarrow n4 | (AHead _ _) \Rightarrow n3])) (ASort n3 n2) (ASort x1 x0)
+H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _
n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1
x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1
(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort
(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1:
nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus
g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1
-x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return
-(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _)
-\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G
-next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in
-(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4:
-nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7)
-in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3
-n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def
-(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
-(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
-(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g
-(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S
-x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in
-(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4:
-nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4
-n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq
-g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4
-n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S
-n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S
-n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
+x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort n4 _)
+\Rightarrow n4 | (AHead _ _) \Rightarrow O])) (ASort O (next g n2)) (ASort x1
+x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort
+_ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G next _)
+\Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in (\lambda
+(H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: nat).(eq A
+(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) in (let H9
+\def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 n0) x2)
+(aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def (eq_ind_r A
+(aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g (ASort O (next g
+n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) (aplus_sort_S_S_simpl g n0
+n3 x2)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) x2)
+(\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) a)) H10 (aplus g
+(ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in (leq_sort g (S n3) O
+n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: nat).(\lambda (_:
+(((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g
+(asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0)
+(ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))))).(\lambda
+(H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4)
+n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S n4)
+n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def
(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind
nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A
(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5:
n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0:
nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g
(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4
-n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A
-return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _)
-\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A
-nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _
-n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1
-x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1
-(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0)
-x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
+n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e with
+[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) (ASort n4 n2)
+(ASort x1 x0) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e
+with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4
+n2) (ASort x1 x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r
+nat x1 (\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5
+x0) x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A
(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10
\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g
(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2))
(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g
(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1
x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match
-ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
-(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g
-(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_:
-(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a)))
-\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0)
-a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g
-(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
-n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g
-(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0)
-a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a
-a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in
-(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2)
-k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a
-(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0))
-(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A
-(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A
-(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc
-g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with
-[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1
-x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8)))))))
-H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2:
-A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0:
-A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to
-(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g
-(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n:
-nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g
-(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0))
-(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda
-(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def
-(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x
-in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda
-(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda
-(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0)
-(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a
-x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next
-g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda
-(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _)
+ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
+(ASort x1 x0) H7) in (False_ind (leq g (ASort O n0) (AHead a a0)) H8)))))))
+H5)))))) (\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0))
+(asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1
+n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort
+n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a
+a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to
+(leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1)
+n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g
+(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H_x \def
+(leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in (let H5 \def H_x in
+(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0))
+(ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) (\lambda (x0:
+nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort
+n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g
+a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda
+(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _)
+\Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g (ASort (S n1)
+n0) (AHead a a0)) H8))))))) H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a:
+A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq
+g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g
+a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3:
+A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a
+a0)) (asucc g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g
+(AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1
+n0)))) (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O
+n0)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (ASort O (next g n0))
+H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_:
+A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq g (asucc g a0) a4)))
+(\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O (next g n0)) (AHead a3
+a4)))) (leq g (AHead a a0) (ASort O n0)) (\lambda (x0: A).(\lambda (x1:
+A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda
+(H6: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A
+(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _)
\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1:
nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0)))
(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0:
A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g
a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1)
-n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g
-(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda
-(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g
-(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g
-(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3
-(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5:
-A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g
-(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g
-a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g
-a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8
-\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
-with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3
-(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e:
-A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow
-((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n)
-\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h)
-\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0
-a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g
-a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def
-(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9)
-in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in
-(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2))))))
-a1)).
-(* COMMENTS
-Initial nodes: 4697
-END *)
+(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in
+(False_ind (leq g (AHead a a0) (ASort (S n1) n0)) H7))))))) H3)))))) n H1))))
+(\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3))
+\to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc
+g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3:
+(leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H_x \def
+(leq_gen_head1 g a (asucc g a0) (AHead a3 (asucc g a4)) H3) in (let H4 \def
+H_x in (ex3_2_ind A A (\lambda (a5: A).(\lambda (_: A).(leq g a a5)))
+(\lambda (_: A).(\lambda (a6: A).(leq g (asucc g a0) a6))) (\lambda (a5:
+A).(\lambda (a6: A).(eq A (AHead a3 (asucc g a4)) (AHead a5 a6)))) (leq g
+(AHead a a0) (AHead a3 a4)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5:
+(leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A
+(AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 \def (f_equal A A (\lambda
+(e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow
+a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A
+(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g a4) | (AHead
+_ a5) \Rightarrow a5])) (AHead a3 (asucc g a4)) (AHead x0 x1) H7) in (\lambda
+(H10: (eq A a3 x0)).(let H11 \def (eq_ind_r A x1 (\lambda (a5: A).(leq g
+(asucc g a0) a5)) H6 (asucc g a4) H9) in (let H12 \def (eq_ind_r A x0
+(\lambda (a5: A).(leq g a a5)) H5 a3 H10) in (leq_head g a a3 H12 a0 a4 (H0
+a4 H11)))))) H8))))))) H4)))))))) a2)))))) a1)).
-theorem leq_asucc:
+lemma leq_asucc:
\forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g
a0)))))
\def
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)).
-(* COMMENTS
-Initial nodes: 221
-END *)
-theorem leq_ahead_asucc_false:
+lemma 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
(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda
(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A
-(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
-False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1:
-nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O
-\Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to
-P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let
-H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1
-\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S
-n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
-A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0:
-A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_:
-(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def
-(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_:
-A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
+(ASort O (next g n0)) (\lambda (ee: A).(match ee with [(ASort _ _)
+\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in
+(False_ind P H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead
+(ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h)
+\Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S
+n1) n0) a2) (ASort n1 n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0)
+a2 (ASort n1 n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3:
+A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda
+(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0)
+(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g
+(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort
+n1 n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort n1 n0) (\lambda (ee:
+A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H))))))
(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g
a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall
(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1:
A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2
x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def
-(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
-[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g
-a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e
-in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc
-(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0
-with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) |
-(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead
-_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda
-(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3))
-H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g
-(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6)))))))
-H2)))))))))) a1)).
-(* COMMENTS
-Initial nodes: 927
-END *)
+(f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a |
+(AHead a3 _) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in
+((let H7 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _)
+\Rightarrow (asucc g a0) | (AHead _ a3) \Rightarrow a3])) (AHead a (asucc g
+a0)) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def (eq_ind_r A
+x1 (\lambda (a3: A).(leq g a2 a3)) H4 (asucc g a0) H7) in (let H10 \def
+(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in
+(leq_ahead_false_1 g a a0 H10 P))))) H6))))))) H2)))))))))) a1)).
-theorem leq_asucc_false:
+lemma leq_asucc_false:
\forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P:
Prop).P)))
\def
(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A
(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e:
-A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow
-n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5
-\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
-with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r
-nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g
-(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1:
-nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7
-n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2)
-(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O
-n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S
-x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1:
-nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda
-(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next
-g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda
-(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g
-n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat
-(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort
-n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2:
-nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda
-(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g
-(ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S
-n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e
-in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _
-_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def
-(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with
-[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1)
-n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def
-(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g
-(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda
-(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2)))
-H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda
-(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1)
-n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g
-(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2:
-nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H)))))
-(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
+A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) \Rightarrow O]))
+(ASort O n0) (ASort x1 x0) H3) in ((let H5 \def (f_equal A nat (\lambda (e:
+A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0]))
+(ASort O n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def
+(eq_ind_r nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2)
+(aplus g (ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0
+(\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O
+n1) x2))) H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g
+n0)) x2) (\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g
+(ASort O n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def
+(aplus_inj g (S x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2
+(\lambda (n1: nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4)))))))
+H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow
+(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to
+P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def
+(leq_gen_sort1 g n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in
+(ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k:
+nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda
+(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort
+h2 n2))))) P (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2:
+nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0)
+x2))).(\lambda (H3: (eq A (ASort (S n1) n0) (ASort x1 x0))).(let H4 \def
+(f_equal A nat (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 |
+(AHead _ _) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in
+((let H5 \def (f_equal A nat (\lambda (e: A).(match e with [(ASort _ n2)
+\Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) n0) (ASort x1
+x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def (eq_ind_r nat x1
+(\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort n2 x0)
+x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n2: nat).(eq A
+(aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) H7 n0 H5) in (let
+H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda (a0: A).(eq A a0
+(aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) n0) (S x2))
+(aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g (S x2) x2
+(ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: nat).(le
+n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) (\lambda
+(a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P:
Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to
(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead
a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1)
(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1)
(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0
x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1)
-(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
-return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _)
-\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A
-A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
-\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1)
-H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2:
-A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0
-(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6)))))))
-H2))))))))) a)).
-(* COMMENTS
-Initial nodes: 1327
-END *)
+(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with
+[(ASort _ _) \Rightarrow a0 | (AHead a2 _) \Rightarrow a2])) (AHead a0 a1)
+(AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with
+[(ASort _ _) \Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1)
+(AHead x0 x1) H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1
+(\lambda (a2: A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def
+(eq_ind_r A x0 (\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P)))))
+H6))))))) H2))))))))) a)).
projects / helm.git / blobdiff