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/sty0/defs.ma".
19 include "LambdaDelta-1/getl/drop.ma".
22 \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((sty0 g e
23 t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c
24 e) \to (sty0 g c (lift h d t1) (lift h d t2))))))))))
26 \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
27 (H: (sty0 g e t1 t2)).(sty0_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
28 (t0: T).(\forall (c0: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 c)
29 \to (sty0 g c0 (lift h d t) (lift h d t0))))))))) (\lambda (c: C).(\lambda
30 (n: nat).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_:
31 (drop h d c0 c)).(eq_ind_r T (TSort n) (\lambda (t: T).(sty0 g c0 t (lift h d
32 (TSort (next g n))))) (eq_ind_r T (TSort (next g n)) (\lambda (t: T).(sty0 g
33 c0 (TSort n) t)) (sty0_sort g c0 n) (lift h d (TSort (next g n))) (lift_sort
34 (next g n) h d)) (lift h d (TSort n)) (lift_sort n h d)))))))) (\lambda (c:
35 C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c
36 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (H1: (sty0 g d v
37 w)).(\lambda (H2: ((\forall (c0: C).(\forall (h: nat).(\forall (d0:
38 nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift h d0
39 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda (H3:
40 (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0
41 (lift (S i) O w))) (\lambda (H4: (lt i d0)).(let H5 \def (drop_getl_trans_le
42 i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d (Bind Abbr) v) H0)
43 in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c0 e0)))
44 (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1))) (\lambda (_:
45 C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abbr) v)))) (sty0 g c0 (lift h
46 d0 (TLRef i)) (lift h d0 (lift (S i) O w))) (\lambda (x0: C).(\lambda (x1:
47 C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h (minus d0 i) x0
48 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abbr) v))).(let H9 \def (eq_ind
49 nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S (minus d0 (S i)))
50 (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0 h (minus d0 (S i))
51 H9 Abbr d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind
52 Abbr) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h (minus d0 (S
53 i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O w)))
54 (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abbr) (lift h (minus
55 d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x d)).(eq_ind_r T
56 (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind
57 nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g c0 (TLRef i)
58 (lift h n (lift (S i) O w)))) (eq_ind_r T (lift (S i) O (lift h (minus d0 (S
59 i)) w)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0 (\lambda (_:
60 nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i)) w))))
61 (sty0_abbr g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead x
62 (Bind Abbr) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S i))
63 w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
64 (le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
65 i) O w)) (lift_d w h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
66 (le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
67 H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
68 h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O w)))) (eq_ind nat
69 (S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
70 O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(sty0 g c0
71 (TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
72 c0 (TLRef (plus i h)) (lift n O w))) (sty0_abbr g c0 d v (plus i h)
73 (drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
74 h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
75 h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
76 i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
77 i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
78 H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
79 (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
80 T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
81 nat).(\forall (d0: nat).((drop h d0 c0 d) \to (sty0 g c0 (lift h d0 v) (lift
82 h d0 w)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d0: nat).(\lambda
83 (H3: (drop h d0 c0 c)).(lt_le_e i d0 (sty0 g c0 (lift h d0 (TLRef i)) (lift h
84 d0 (lift (S i) O v))) (\lambda (H4: (lt i d0)).(let H5 \def
85 (drop_getl_trans_le i d0 (le_S_n i d0 (le_S (S i) d0 H4)) c0 c h H3 (CHead d
86 (Bind Abst) v) H0) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
87 O c0 e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d0 i) e0 e1)))
88 (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d (Bind Abst) v)))) (sty0 g
89 c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S i) O v))) (\lambda (x0:
90 C).(\lambda (x1: C).(\lambda (H6: (drop i O c0 x0)).(\lambda (H7: (drop h
91 (minus d0 i) x0 x1)).(\lambda (H8: (clear x1 (CHead d (Bind Abst) v))).(let
92 H9 \def (eq_ind nat (minus d0 i) (\lambda (n: nat).(drop h n x0 x1)) H7 (S
93 (minus d0 (S i))) (minus_x_Sy d0 i H4)) in (let H10 \def (drop_clear_S x1 x0
94 h (minus d0 (S i)) H9 Abst d v H8) in (ex2_ind C (\lambda (c1: C).(clear x0
95 (CHead c1 (Bind Abst) (lift h (minus d0 (S i)) v)))) (\lambda (c1: C).(drop h
96 (minus d0 (S i)) c1 d)) (sty0 g c0 (lift h d0 (TLRef i)) (lift h d0 (lift (S
97 i) O v))) (\lambda (x: C).(\lambda (H11: (clear x0 (CHead x (Bind Abst) (lift
98 h (minus d0 (S i)) v)))).(\lambda (H12: (drop h (minus d0 (S i)) x
99 d)).(eq_ind_r T (TLRef i) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i)
100 O v)))) (eq_ind nat (plus (S i) (minus d0 (S i))) (\lambda (n: nat).(sty0 g
101 c0 (TLRef i) (lift h n (lift (S i) O v)))) (eq_ind_r T (lift (S i) O (lift h
102 (minus d0 (S i)) v)) (\lambda (t: T).(sty0 g c0 (TLRef i) t)) (eq_ind nat d0
103 (\lambda (_: nat).(sty0 g c0 (TLRef i) (lift (S i) O (lift h (minus d0 (S i))
104 v)))) (sty0_abst g c0 x (lift h (minus d0 (S i)) v) i (getl_intro i c0 (CHead
105 x (Bind Abst) (lift h (minus d0 (S i)) v)) x0 H6 H11) (lift h (minus d0 (S
106 i)) w) (H2 x h (minus d0 (S i)) H12)) (plus (S i) (minus d0 (S i)))
107 (le_plus_minus (S i) d0 H4)) (lift h (plus (S i) (minus d0 (S i))) (lift (S
108 i) O v)) (lift_d v h (S i) (minus d0 (S i)) O (le_O_n (minus d0 (S i))))) d0
109 (le_plus_minus_r (S i) d0 H4)) (lift h d0 (TLRef i)) (lift_lref_lt i h d0
110 H4))))) H10)))))))) H5))) (\lambda (H4: (le d0 i)).(eq_ind_r T (TLRef (plus i
111 h)) (\lambda (t: T).(sty0 g c0 t (lift h d0 (lift (S i) O v)))) (eq_ind nat
112 (S i) (\lambda (_: nat).(sty0 g c0 (TLRef (plus i h)) (lift h d0 (lift (S i)
113 O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(sty0 g c0
114 (TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(sty0 g
115 c0 (TLRef (plus i h)) (lift n O v))) (sty0_abst g c0 d v (plus i h)
116 (drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
117 h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
118 h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
119 i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
120 i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
121 H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
122 (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g (CHead c (Bind b) v) t3
123 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
124 nat).((drop h d c0 (CHead c (Bind b) v)) \to (sty0 g c0 (lift h d t3) (lift h
125 d t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
126 (H2: (drop h d c0 c)).(eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s
127 (Bind b) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Bind b) v
128 t4)))) (eq_ind_r T (THead (Bind b) (lift h d v) (lift h (s (Bind b) d) t4))
129 (\lambda (t: T).(sty0 g c0 (THead (Bind b) (lift h d v) (lift h (s (Bind b)
130 d) t3)) t)) (sty0_bind g b c0 (lift h d v) (lift h (S d) t3) (lift h (S d)
131 t4) (H1 (CHead c0 (Bind b) (lift h d v)) h (S d) (drop_skip_bind h d c0 c H2
132 b v))) (lift h d (THead (Bind b) v t4)) (lift_head (Bind b) v t4 h d)) (lift
133 h d (THead (Bind b) v t3)) (lift_head (Bind b) v t3 h d))))))))))))) (\lambda
134 (c: C).(\lambda (v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g
135 c t3 t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
136 nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
137 t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
138 (drop h d c0 c)).(eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat
139 Appl) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Appl) v
140 t4)))) (eq_ind_r T (THead (Flat Appl) (lift h d v) (lift h (s (Flat Appl) d)
141 t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Appl) (lift h d v) (lift h (s
142 (Flat Appl) d) t3)) t)) (sty0_appl g c0 (lift h d v) (lift h (s (Flat Appl)
143 d) t3) (lift h (s (Flat Appl) d) t4) (H1 c0 h (s (Flat Appl) d) H2)) (lift h
144 d (THead (Flat Appl) v t4)) (lift_head (Flat Appl) v t4 h d)) (lift h d
145 (THead (Flat Appl) v t3)) (lift_head (Flat Appl) v t3 h d))))))))))))
146 (\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c v1
147 v2)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
148 nat).((drop h d c0 c) \to (sty0 g c0 (lift h d v1) (lift h d
149 v2)))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (sty0 g c t3
150 t4)).(\lambda (H3: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
151 nat).((drop h d c0 c) \to (sty0 g c0 (lift h d t3) (lift h d
152 t4)))))))).(\lambda (c0: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H4:
153 (drop h d c0 c)).(eq_ind_r T (THead (Flat Cast) (lift h d v1) (lift h (s
154 (Flat Cast) d) t3)) (\lambda (t: T).(sty0 g c0 t (lift h d (THead (Flat Cast)
155 v2 t4)))) (eq_ind_r T (THead (Flat Cast) (lift h d v2) (lift h (s (Flat Cast)
156 d) t4)) (\lambda (t: T).(sty0 g c0 (THead (Flat Cast) (lift h d v1) (lift h
157 (s (Flat Cast) d) t3)) t)) (sty0_cast g c0 (lift h d v1) (lift h d v2) (H1 c0
158 h d H4) (lift h (s (Flat Cast) d) t3) (lift h (s (Flat Cast) d) t4) (H3 c0 h
159 (s (Flat Cast) d) H4)) (lift h d (THead (Flat Cast) v2 t4)) (lift_head (Flat
160 Cast) v2 t4 h d)) (lift h d (THead (Flat Cast) v1 t3)) (lift_head (Flat Cast)
161 v1 t3 h d))))))))))))))) e t1 t2 H))))).
163 theorem sty0_correct:
164 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((sty0 g c
165 t1 t) \to (ex T (\lambda (t2: T).(sty0 g c t t2)))))))
167 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H:
168 (sty0 g c t1 t)).(sty0_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (t2:
169 T).(ex T (\lambda (t3: T).(sty0 g c0 t2 t3)))))) (\lambda (c0: C).(\lambda
170 (n: nat).(ex_intro T (\lambda (t2: T).(sty0 g c0 (TSort (next g n)) t2))
171 (TSort (next g (next g n))) (sty0_sort g c0 (next g n))))) (\lambda (c0:
172 C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0
173 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (sty0 g d v
174 w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g d w t2)))).(let H3 \def H2
175 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex T (\lambda (t2: T).(sty0 g
176 c0 (lift (S i) O w) t2))) (\lambda (x: T).(\lambda (H4: (sty0 g d w
177 x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O w) t2)) (lift (S i)
178 O x) (sty0_lift g d w x H4 c0 (S i) O (getl_drop Abbr c0 d v i H0)))))
179 H3)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i:
180 nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) v))).(\lambda (w:
181 T).(\lambda (H1: (sty0 g d v w)).(\lambda (H2: (ex T (\lambda (t2: T).(sty0 g
182 d w t2)))).(let H3 \def H2 in (ex_ind T (\lambda (t2: T).(sty0 g d w t2)) (ex
183 T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v) t2))) (\lambda (x: T).(\lambda
184 (_: (sty0 g d w x)).(ex_intro T (\lambda (t2: T).(sty0 g c0 (lift (S i) O v)
185 t2)) (lift (S i) O w) (sty0_lift g d v w H1 c0 (S i) O (getl_drop Abst c0 d v
186 i H0))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (v:
187 T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g (CHead c0 (Bind b)
188 v) t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g (CHead c0 (Bind b) v)
189 t3 t4)))).(let H2 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g (CHead c0
190 (Bind b) v) t3 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3)
191 t4))) (\lambda (x: T).(\lambda (H3: (sty0 g (CHead c0 (Bind b) v) t3
192 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Bind b) v t3) t4)) (THead
193 (Bind b) v x) (sty0_bind g b c0 v t3 x H3)))) H2))))))))) (\lambda (c0:
194 C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (sty0 g c0
195 t2 t3)).(\lambda (H1: (ex T (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H2
196 \def H1 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
197 T).(sty0 g c0 (THead (Flat Appl) v t3) t4))) (\lambda (x: T).(\lambda (H3:
198 (sty0 g c0 t3 x)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Appl)
199 v t3) t4)) (THead (Flat Appl) v x) (sty0_appl g c0 v t3 x H3)))) H2))))))))
200 (\lambda (c0: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (sty0 g c0 v1
201 v2)).(\lambda (H1: (ex T (\lambda (t2: T).(sty0 g c0 v2 t2)))).(\lambda (t2:
202 T).(\lambda (t3: T).(\lambda (_: (sty0 g c0 t2 t3)).(\lambda (H3: (ex T
203 (\lambda (t4: T).(sty0 g c0 t3 t4)))).(let H4 \def H1 in (ex_ind T (\lambda
204 (t4: T).(sty0 g c0 v2 t4)) (ex T (\lambda (t4: T).(sty0 g c0 (THead (Flat
205 Cast) v2 t3) t4))) (\lambda (x: T).(\lambda (H5: (sty0 g c0 v2 x)).(let H6
206 \def H3 in (ex_ind T (\lambda (t4: T).(sty0 g c0 t3 t4)) (ex T (\lambda (t4:
207 T).(sty0 g c0 (THead (Flat Cast) v2 t3) t4))) (\lambda (x0: T).(\lambda (H7:
208 (sty0 g c0 t3 x0)).(ex_intro T (\lambda (t4: T).(sty0 g c0 (THead (Flat Cast)
209 v2 t3) t4)) (THead (Flat Cast) x x0) (sty0_cast g c0 v2 x H5 t3 x0 H7))))
210 H6)))) H4))))))))))) c t1 t H))))).