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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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/clen".
19 include "pr2/props.ma".
21 include "clen/getl.ma".
23 theorem pr2_gen_ctail:
24 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
25 (t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_:
26 T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0
27 (clen c) u t t2)))))))))
29 \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
30 (t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c)
31 (\lambda (c0: C).(pr2 c0 t1 t2)) (or (pr2 c t1 t2) (ex3 T (\lambda (_: T).(eq
32 K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen
33 c) u t t2)))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda
34 (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (or
35 (pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t3:
36 T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 t0)))))))) (\lambda
37 (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
38 (_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) (ex3 T (\lambda (_:
39 T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(subst0
40 (clen c) u t t4))) (pr2_free c t3 t4 H1))))))) (\lambda (c0: C).(\lambda (d:
41 C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind
42 Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3
43 t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0
44 (CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead
45 d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let H_x \def (getl_gen_tail k
46 Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq
47 C d (CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))))
48 (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k
49 (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort
50 n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda
51 (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda
52 (H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c
53 (CHead e (Bind Abbr) u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u
54 e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t)
55 (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
56 (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq
57 C d (CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr)
58 u0))).(or_introl (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr)))
59 (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))
60 (pr2_delta c x u0 i H9 t3 t4 H2 t H3))))) H7)) (\lambda (H7: (ex4 nat
61 (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind
62 Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort
63 n))))).(ex4_ind nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_:
64 nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n:
65 nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k
66 (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c)
67 u t0 t)))) (\lambda (x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda
68 (H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d
69 (CSort x0))).(let H12 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u0 t4
70 t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 (\lambda (t0: T).(subst0
71 (clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or
72 (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind Abbr))) (\lambda (t0:
73 T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))))) (or_intror (pr2
74 c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0:
75 T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (ex3_intro T
76 (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0))
77 (\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 (refl_equal K (Bind Abbr)) H2
78 H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
80 theorem pr2_gen_cbind:
81 \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
82 (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1)
83 (THead (Bind b) v t2)))))))
85 \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
86 (t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H
87 in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_:
88 (pr2 c0 t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0
89 t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with
90 [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b)
91 v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead
92 c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0
93 t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4:
94 (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
95 (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3
96 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v
97 t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead
98 (Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6
99 (Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0
100 (CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2)
101 \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T
102 t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda
103 (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr)
104 u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1)
105 (THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1
106 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d
107 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead
108 (Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind
109 T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u))
110 \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1)
111 (THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead
112 d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3
113 t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let
114 H11 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u)
115 (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda
116 (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1)
117 (THead (Bind b) v t2)) (\lambda (H12: (land (eq nat i O) (eq C (CHead d (Bind
118 Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind
119 Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v
120 t2)) (\lambda (H13: (eq nat i O)).(\lambda (H14: (eq C (CHead d (Bind Abbr)
121 u) (CHead c (Bind b) v))).(let H15 \def (f_equal C C (\lambda (e: C).(match e
122 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _)
123 \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in ((let
124 H16 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
125 with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K
126 return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
127 \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in
128 ((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
129 C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead
130 d (Bind Abbr) u) (CHead c (Bind b) v) H14) in (\lambda (H18: (eq B Abbr
131 b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind nat i (\lambda (n:
132 nat).(subst0 n u t3 t2)) H10 O H13) in (let H21 \def (eq_ind T u (\lambda
133 (t4: T).(subst0 O t4 t3 t2)) H20 v H17) in (eq_ind B Abbr (\lambda (b0:
134 B).(pr2 c (THead (Bind b0) v t1) (THead (Bind b0) v t2))) (pr2_free c (THead
135 (Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3
136 H9 t2 H21)) b H18)))))) H16)) H15)))) H12)) (\lambda (H12: (ex2 nat (\lambda
137 (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr)
138 u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j:
139 nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead
140 (Bind b) v t2)) (\lambda (x: nat).(\lambda (H13: (eq nat i (S x))).(\lambda
141 (H14: (getl x c (CHead d (Bind Abbr) u))).(let H15 \def (f_equal nat nat
142 (\lambda (e: nat).e) i (S x) H13) in (let H16 \def (eq_ind nat i (\lambda (n:
143 nat).(subst0 n u t3 t2)) H10 (S x) H15) in (pr2_head_2 c v t1 t2 (Bind b)
144 (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind
145 b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H14) t1 t3 H9 t2
146 H16))))))) H12)) H11)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0
147 (sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal
148 C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))).
150 theorem pr2_gen_cflat:
151 \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
152 (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
154 \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
155 (t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H
156 in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_:
157 (pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0
158 t2) \to (pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
159 (\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0
160 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_:
161 C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2)))))
162 (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to
163 ((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
164 (\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1
165 t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4)))
166 c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i
167 H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f)
168 v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead
169 c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1
170 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c
171 t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
172 t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4
173 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t
174 t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d
175 (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1
176 t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr)
177 u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y
178 \def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i
179 H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0
180 (sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal
181 C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).