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 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tau0/props".
19 include "tau0/defs.ma".
21 include "getl/drop.ma".
24 \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e
25 t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
26 e) \to (tau0 g c (lift h d t1) (lift h d t2))))))))))
28 \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
29 (H: (tau0 g e t1 t2)).(tau0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
30 (t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
31 \to (tau0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
32 (n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
33 (drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(tau0 g c0 t (lift h d
34 (TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(tau0 g
35 c0 (TSort n) t)) (tau0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
36 (next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
37 C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
38 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (tau0 g d v
39 w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0:
40 nat).((drop h d0 c0 d) \to (tau0 g c0 (lift h d0 v) (lift h d0
41 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
42 (drop h d0 c0 c)).(lt_le_e i d0 (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0
43 (lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
44 i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
45 in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
46 (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
47 C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (tau0 g c0 (lift h
48 d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
49 C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
50 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
51 nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
52 (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
53 H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
54 Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
55 i)) c1 d)) (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
56 (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
57 d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
58 (TLRef i) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
59 nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 g c0 (TLRef i)
60 (lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
61 i)) w)) (\lambda (t: T).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
62 nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
63 (tau0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
64 (Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
65 w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
66 (le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
67 i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
68 (le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
69 H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
70 h)) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
71 (S i) (\lambda (_: nat).(tau0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
72 O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(tau0 g c0
73 (TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
74 c0 (TLRef (plus i h)) (lift n O w))) (tau0_abbr g c0 d v (plus i h)
75 (drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
76 h (S i)) (plus_comm h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
77 h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
78 i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
79 i (S O)) (plus_comm i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
80 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
81 (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
82 T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
83 nat).(\forall (d0: nat).((drop h d0 c0 d) \to (tau0 g c0 (lift h d0 v) (lift
84 h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
85 (H3: (drop h d0 c0 c)).(lt_le_e i d0 (tau0 g c0 (lift h d0 (TLRef i)) (lift h
86 d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
87 (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
88 (Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
89 O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
90 (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (tau0 g
91 c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
92 C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
93 (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
94 H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
95 (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
96 h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
97 (CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
98 (minus d0 (S i)) c1 d)) (tau0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
99 i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
100 h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
101 d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i)
102 O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(tau0 g
103 c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
104 (minus d0 (S i)) v)) (\lambda (t: T).(tau0 g c0 (TLRef i) t)) (eq_ind nat d0
105 (\lambda (_: nat).(tau0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
106 v)))) (tau0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
107 x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
108 i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
109 (le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
110 i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
111 (le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
112 H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
113 h)) (\lambda (t: T).(tau0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
114 (S i) (\lambda (_: nat).(tau0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
115 O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(tau0 g c0
116 (TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
117 c0 (TLRef (plus i h)) (lift n O v))) (tau0_abst g c0 d v (plus i h)
118 (drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
119 h (S i)) (plus_comm h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
120 h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
121 i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
122 i (S O)) (plus_comm i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
123 H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
124 (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g (CHead c (Bind b) v) t3
125 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
126 nat).((drop h d c0 (CHead c (Bind b) v)) \to (tau0 g c0 (lift h d t3) (lift h
127 d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
128 (H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
129 (Bind b) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Bind b) v
130 t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
131 (\lambda (t: T).(tau0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
132 d) t3)) t)) (tau0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
133 t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
134 b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
135 h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
136 (c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g
137 c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
138 nat).((drop h d c0 c) \to (tau0 g c0 (lift h d t3) (lift h d
139 t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
140 (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
141 Appl) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Flat Appl) v
142 t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
143 t4)) (\lambda (t: T).(tau0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
144 (Flat Appl) d) t3)) t)) (tau0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
145 d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
146 d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
147 (THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
148 (\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c v1
149 v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
150 nat).((drop h d c0 c) \to (tau0 g c0 (lift h d v1) (lift h d
151 v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g c t3
152 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
153 nat).((drop h d c0 c) \to (tau0 g c0 (lift h d t3) (lift h d
154 t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
155 (drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
156 (Flat Cast) d) t3)) (\lambda (t: T).(tau0 g c0 t (lift h d (THead (Flat Cast)
157 v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
158 d) t4)) (\lambda (t: T).(tau0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
159 (s (Flat Cast) d) t3)) t)) (tau0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
160 h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
161 (s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
162 Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
163 v1 t3 h d))))))))))))))) e t1 t2 H))))).
165 theorem tau0_correct:
166 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c
167 t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2)))))))
169 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
170 (tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
171 T).(ex T (\lambda (t3: T).(tau0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
172 (n: nat).(ex_intro T (\lambda (t2: T).(tau0 g c0 (TSort (next g n)) t2))
173 (TSort (next g (next g n))) (tau0_sort g c0 (next g n))))) (\lambda (c0:
174 C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
175 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (tau0 g d v
176 w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g d w t2)))).(let H3 \def H2
177 in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex T (\lambda (t2: T).(tau0 g
178 c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (tau0 g d w
179 x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O w) t2)) (lift (S i)
180 O x) (tau0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
181 H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
182 nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
183 T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(tau0 g
184 d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(tau0 g d w t2)) (ex
185 T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
186 (_: (tau0 g d w x)).(ex_intro T (\lambda (t2: T).(tau0 g c0 (lift (S i) O v)
187 t2)) (lift (S i) O w) (tau0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
188 i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
189 T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g (CHead c0 (Bind b)
190 v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(tau0 g (CHead c0 (Bind b) v)
191 t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(tau0 g (CHead c0
192 (Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3)
193 t4))) (\lambda (x: T).(\lambda (H3: (tau0 g (CHead c0 (Bind b) v) t3
194 x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Bind b) v t3) t4)) (THead
195 (Bind b) v x) (tau0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
196 C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0
197 t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(tau0 g c0 t3 t4)))).(let H2
198 \def H1 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4:
199 T).(tau0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
200 (tau0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Appl)
201 v t3) t4)) (THead (Flat Appl) v x) (tau0_appl g c0 v t3 x H3)))) H2))))))))
202 (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (tau0 g c0 v1
203 v2)).(\lambda (H1: (ex T (\lambda (t2: T).(tau0 g c0 v2 t2)))).(\lambda (t2:
204 T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H3: (ex T
205 (\lambda (t4: T).(tau0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda
206 (t4: T).(tau0 g c0 v2 t4)) (ex T (\lambda (t4: T).(tau0 g c0 (THead (Flat
207 Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (tau0 g c0 v2 x)).(let H6
208 \def H3 in (ex_ind T (\lambda (t4: T).(tau0 g c0 t3 t4)) (ex T (\lambda (t4:
209 T).(tau0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
210 (tau0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(tau0 g c0 (THead (Flat Cast)
211 v2 t3) t4)) (THead (Flat Cast) x x0) (tau0_cast g c0 v2 x H5 t3 x0 H7))))
212 H6)))) H4))))))))))) c t1 t H))))).