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7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/aprem/defs.ma".
19 theorem aprem_gen_sort:
20 \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem
21 i (ASort h n) x) \to False))))
23 \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n:
24 nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda
25 (a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0:
26 (aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_:
27 A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2:
28 A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A
29 (AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop)
30 with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I
31 (ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a:
32 A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2
33 (ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2)
34 (ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee
35 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False |
36 (AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False
37 H4))))))))) i y x H0))) H))))).
39 theorem aprem_gen_head_O:
40 \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x)
43 \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O
44 (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x))
45 (\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y
46 x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq
47 A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem
48 y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq
49 nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0:
50 A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0
51 a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A
52 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _)
53 \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A
54 A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
55 \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3)
56 in (\lambda (H6: (eq A a0 a1)).(eq_ind_r A a1 (\lambda (a: A).(eq A a a1))
57 (refl_equal A a1) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: A).(\lambda
58 (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i O) \to ((eq
59 A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda (H4: (eq
60 nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let H6 \def
61 (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with
62 [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 a0)
63 (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A
64 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a4)
65 \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A a3
66 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A a4
67 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0
68 (\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i)
69 (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
70 \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a
71 a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))).
73 theorem aprem_gen_head_S:
74 \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem
75 (S i) (AHead a1 a2) x) \to (aprem i a2 x)))))
77 \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda
78 (H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a:
79 A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y:
80 A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n:
81 nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2
82 x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n:
83 nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead
84 a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda
85 (H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let
86 H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
87 with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3)
88 (AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A
89 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a)
90 \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0
91 a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \def (eq_ind nat
92 O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
93 \Rightarrow True | (S _) \Rightarrow False])) I (S i) H2) in (False_ind
94 (aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a:
95 A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat
96 i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3:
97 A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0)
98 (AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A
99 return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _)
100 \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A
101 A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
102 \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2)
103 H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4:
104 A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3
105 a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2
106 H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat
107 return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n]))
108 (S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n
109 (S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let
110 H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in
111 H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))).