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15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/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)
24 x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C
25 x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c:
26 C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c:
27 C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
28 C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0))
29 (refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda
30 (_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
31 c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
32 (k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C
33 (CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
34 with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
35 (CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1))
36 H5))))))))))) y x H0))) H))).
38 theorem wcpr0_gen_head:
39 \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
40 (CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
41 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
42 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
44 \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
45 (H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c:
46 C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2:
47 C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
48 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda
49 (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0:
50 C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2:
51 C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
52 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))
53 (\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal
54 C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1)
55 (\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
56 T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1
57 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C
58 (CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
59 T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
60 T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))
61 (refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2:
62 C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to
63 (or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead
64 c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_:
65 C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2:
66 T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0
67 k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
68 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
69 \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def
70 (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
71 [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
72 (CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
73 C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
74 \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K
75 k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C
76 (CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3:
77 T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
78 T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10
79 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1
80 (\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda
81 (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
82 (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
83 u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k
84 u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C
85 c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3)))
86 (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def
87 (eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1
88 (\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda
89 (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
90 (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
91 u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T
92 (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3))))
93 (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
94 (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq
95 C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0
96 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C
97 (CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x