1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/pr2/props.ma".
19 include "LambdaDelta-1/clen/getl.ma".
21 theorem pr2_gen_ctail:
22 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
23 (t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_:
24 T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0
25 (clen c) u t t2)))))))))
27 \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
28 (t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c)
29 (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(or (pr2 c t1 t2) (ex3 T
30 (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda
31 (t: T).(subst0 (clen c) u t t2))))) (\lambda (y: C).(\lambda (H0: (pr2 y t1
32 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
33 (CTail k u c)) \to (or (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind
34 Abbr))) (\lambda (t3: T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3
35 t0)))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
36 (pr0 t3 t4)).(\lambda (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4)
37 (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t))
38 (\lambda (t: T).(subst0 (clen c) u t t4))) (pr2_free c t3 t4 H1)))))))
39 (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda
40 (H1: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
41 T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4
42 t)).(\lambda (H4: (eq C c0 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda
43 (c1: C).(getl i c1 (CHead d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let
44 H_x \def (getl_gen_tail k Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind
45 (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
46 (CHead e (Bind Abbr) u0)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c)))
47 (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
48 (\lambda (n: nat).(eq C d (CSort n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
49 T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
50 T).(subst0 (clen c) u t0 t)))) (\lambda (H7: (ex2 C (\lambda (e: C).(eq C d
51 (CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr)
52 u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e:
53 C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) (ex3 T (\lambda (_:
54 T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0:
55 T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq C d (CTail k
56 u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) u0))).(or_introl (pr2 c
57 t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3
58 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (pr2_delta c x u0 i H9 t3 t4
59 H2 t H3))))) H7)) (\lambda (H7: (ex4 nat (\lambda (_: nat).(eq nat i (clen
60 c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0))
61 (\lambda (n: nat).(eq C d (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq
62 nat i (clen c))) (\lambda (_: nat).(eq K k (Bind Abbr))) (\lambda (_:
63 nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3
64 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
65 (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x0: nat).(\lambda (H8:
66 (eq nat i (clen c))).(\lambda (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T
67 u u0)).(\lambda (_: (eq C d (CSort x0))).(let H12 \def (eq_ind nat i (\lambda
68 (n: nat).(subst0 n u0 t4 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0
69 (\lambda (t0: T).(subst0 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind
70 Abbr) (\lambda (k0: K).(or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind
71 Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
72 t))))) (or_intror (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
73 Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0
74 t))) (ex3_intro T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda
75 (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4
76 (refl_equal K (Bind Abbr)) H2 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1
79 theorem pr2_gen_cbind:
80 \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
81 (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1)
82 (THead (Bind b) v t2)))))))
84 \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
85 (t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c
86 (Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead
87 (Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y
88 t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
89 (CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v
90 t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
91 (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead
92 (Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1
93 (Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
94 nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
95 T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
96 (subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def
97 (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
98 c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u)
99 v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d
100 (Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S
101 j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead
102 (Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C
103 (CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C
104 (CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3)
105 (THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead
106 d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda
107 (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d
108 | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
109 v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return
110 (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
111 \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
112 \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
113 (CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e:
114 C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
115 (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v)
116 H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def
117 (eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def
118 (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr
119 (\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t)))
120 (pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v
121 (pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7:
122 (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c
123 (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j)))
124 (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b)
125 v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S
126 x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal
127 nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i
128 (\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t
129 (Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead
130 c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t
131 H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
133 theorem pr2_gen_cflat:
134 \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
135 (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
137 \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
138 (t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c
139 (Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2))
140 (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
141 C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2
142 c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
143 (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4
144 H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
145 nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
146 T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
147 (subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def
148 (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
149 c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u)
150 v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0)))