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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "wcpr0/defs.ma".
19 theorem wcpr0_gen_sort:
20 \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
23 \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0
24 \def (match H in wcpr0 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_:
25 (wcpr0 c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort
26 n))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort
27 n))).(\lambda (H1: (eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0
28 x) \to (eq C x (CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C
29 (CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x
30 H2)) c (sym_eq C c (CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k)
31 \Rightarrow (\lambda (H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq
32 C (CHead c2 k u2) x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e:
33 C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
34 False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in (False_ind ((eq
35 C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort
36 n))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C
39 theorem wcpr0_gen_head:
40 \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
41 (CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
42 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
43 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
45 \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
46 (H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H in wcpr0 return (\lambda
47 (c: C).(\lambda (c0: C).(\lambda (_: (wcpr0 c c0)).((eq C c (CHead c1 k u1))
48 \to ((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
49 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
50 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))) with
51 [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda
52 (H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to
53 (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq
54 C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2)))
55 (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead
56 c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead
57 c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k
58 u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_:
59 C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead
60 c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1)
61 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda
62 (_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2))
63 c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0)
64 \Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda
65 (H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e:
66 C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
67 (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let
68 H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
69 with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0
70 k0 u0) (CHead c1 k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e:
71 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
72 (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in
73 (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead
74 c2 k0 u2) x) \to ((wcpr0 c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k
75 u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3))))
76 (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
77 (u3: T).(pr0 u1 u3))))))))))) (\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda
78 (k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to
79 ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3:
80 C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
81 T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))))
82 (\lambda (H8: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k
83 u2) x) \to ((wcpr0 c1 c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1))
84 (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3))))
85 (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
86 (u3: T).(pr0 u1 u3))))))))) (\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C
87 (CHead c2 k u2) (\lambda (c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C
88 c (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c
89 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda
90 (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1
91 c2)).(\lambda (H11: (pr0 u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k
92 u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2)
93 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda
94 (_: C).(\lambda (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3:
95 C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3:
96 C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
97 u3))) c2 u2 (refl_equal C (CHead c2 k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0
98 u1 H8))) k0 (sym_eq K k0 k H7))) c0 (sym_eq C c0 c1 H6))) H5)) H4)) H3 H0
99 H1)))]) in (H0 (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))).