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15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1/aprem/defs.ma".
19 implied rec lemma aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall
20 (a1: A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2:
21 A).(\forall (a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to
22 (\forall (a1: A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A)
23 (a1: aprem n a a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3)
24 \Rightarrow (f a2 a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (f0 a2 a3 i a4
25 ((aprem_ind P f f0) i a2 a3 a4) a5)].
28 \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem
29 i (ASort h n) x) \to False))))
31 \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n:
32 nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda
33 (a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0:
34 (aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_:
35 A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2:
36 A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A
37 (AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False
38 | (AHead _ _) \Rightarrow True])) I (ASort h n) H1) in (False_ind False
39 H2))))) (\lambda (a2: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (_:
40 (aprem i0 a2 a)).(\lambda (_: (((eq A a2 (ASort h n)) \to False))).(\lambda
41 (a1: A).(\lambda (H3: (eq A (AHead a1 a2) (ASort h n))).(let H4 \def (eq_ind
42 A (AHead a1 a2) (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow
43 False | (AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False
44 H4))))))))) i y x H0))) H))))).
46 lemma aprem_gen_head_O:
47 \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x)
50 \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O
51 (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x))
52 (\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y
53 x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq
54 A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem
55 y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq
56 nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0:
57 A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0
58 a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e with
59 [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3)
60 (AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e with
61 [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3)
62 (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0:
63 A).(\lambda (a: A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda
64 (H3: (((eq nat i O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda
65 (a3: A).(\lambda (H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0)
66 (AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e with
67 [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 a0)
68 (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e with
69 [(ASort _ _) \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0)
70 (AHead a1 a2) H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0
71 (\lambda (a4: A).((eq nat i O) \to ((eq A a4 (AHead a1 a2)) \to (eq A a
72 a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i a4
73 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee
74 with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind
75 (eq A a a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
77 lemma aprem_gen_head_S:
78 \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem
79 (S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
81 \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda
82 (H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a:
83 A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y:
84 A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n:
85 nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2
86 x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n:
87 nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead
88 a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda
89 (H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let
90 H4 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow
91 a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5
92 \def (f_equal A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 |
93 (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6:
94 (eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def
95 (eq_ind nat O (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _)
96 \Rightarrow False])) I (S i) H2) in (False_ind (aprem i a2 a1) H7)) a0 H6)))
97 H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (H2:
98 (aprem i0 a0 a)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq A a0 (AHead a1
99 a2)) \to (aprem i a2 a))))).(\lambda (a3: A).(\lambda (H4: (eq nat (S i0) (S
100 i))).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \def (f_equal
101 A A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a4 _)
102 \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A
103 A (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead _ a4)
104 \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3
105 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i0 (S i)) \to ((eq A
106 a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 a2 H7) in (let H10 \def (eq_ind A
107 a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 H7) in (let H11 \def (f_equal nat
108 nat (\lambda (e: nat).(match e with [O \Rightarrow i0 | (S n) \Rightarrow
109 n])) (S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq
110 nat n (S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in
111 (let H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in
112 H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).