author | nipkow |
Thu, 24 Apr 1997 18:06:46 +0200 | |
changeset 3040 | 7d48671753da |
parent 3011 | a3b73ba44a11 |
child 3196 | c522bc46aea7 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/List |
923 | 2 |
ID: $Id$ |
1465 | 3 |
Author: Tobias Nipkow |
923 | 4 |
Copyright 1994 TU Muenchen |
5 |
||
6 |
List lemmas |
|
7 |
*) |
|
8 |
||
3011 | 9 |
goal thy "!x. xs ~= x#xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
10 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
11 |
by (ALLGOALS Asm_simp_tac); |
2608 | 12 |
qed_spec_mp "not_Cons_self"; |
2512 | 13 |
Addsimps [not_Cons_self]; |
923 | 14 |
|
3011 | 15 |
goal thy "(xs ~= []) = (? y ys. xs = y#ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
16 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
17 |
by (Simp_tac 1); |
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
18 |
by (Asm_simp_tac 1); |
923 | 19 |
qed "neq_Nil_conv"; |
20 |
||
21 |
||
2608 | 22 |
(** list_case **) |
23 |
||
3011 | 24 |
goal thy |
2608 | 25 |
"P(list_case a f xs) = ((xs=[] --> P(a)) & \ |
2891 | 26 |
\ (!y ys. xs=y#ys --> P(f y ys)))"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
27 |
by (induct_tac "xs" 1); |
2608 | 28 |
by (ALLGOALS Asm_simp_tac); |
2891 | 29 |
by (Blast_tac 1); |
2608 | 30 |
qed "expand_list_case"; |
31 |
||
3011 | 32 |
val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
33 |
by(induct_tac "xs" 1); |
2608 | 34 |
by(REPEAT(resolve_tac prems 1)); |
35 |
qed "list_cases"; |
|
36 |
||
3011 | 37 |
goal thy "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
38 |
by (induct_tac "xs" 1); |
2891 | 39 |
by (Blast_tac 1); |
40 |
by (Blast_tac 1); |
|
2608 | 41 |
bind_thm("list_eq_cases", |
42 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp)))))); |
|
43 |
||
44 |
||
923 | 45 |
(** @ - append **) |
46 |
||
3011 | 47 |
goal thy "(xs@ys)@zs = xs@(ys@zs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
48 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
49 |
by (ALLGOALS Asm_simp_tac); |
923 | 50 |
qed "append_assoc"; |
2512 | 51 |
Addsimps [append_assoc]; |
923 | 52 |
|
3011 | 53 |
goal thy "xs @ [] = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
54 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
55 |
by (ALLGOALS Asm_simp_tac); |
923 | 56 |
qed "append_Nil2"; |
2512 | 57 |
Addsimps [append_Nil2]; |
923 | 58 |
|
3011 | 59 |
goal thy "(xs@ys = []) = (xs=[] & ys=[])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
60 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
61 |
by (ALLGOALS Asm_simp_tac); |
2608 | 62 |
qed "append_is_Nil_conv"; |
63 |
AddIffs [append_is_Nil_conv]; |
|
64 |
||
3011 | 65 |
goal thy "([] = xs@ys) = (xs=[] & ys=[])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
66 |
by (induct_tac "xs" 1); |
2608 | 67 |
by (ALLGOALS Asm_simp_tac); |
2891 | 68 |
by(Blast_tac 1); |
2608 | 69 |
qed "Nil_is_append_conv"; |
70 |
AddIffs [Nil_is_append_conv]; |
|
923 | 71 |
|
3011 | 72 |
goal thy "(xs @ ys = xs @ zs) = (ys=zs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
73 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
74 |
by (ALLGOALS Asm_simp_tac); |
923 | 75 |
qed "same_append_eq"; |
2608 | 76 |
AddIffs [same_append_eq]; |
77 |
||
3011 | 78 |
goal thy "!ys. (xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
79 |
by(induct_tac "xs" 1); |
2608 | 80 |
br allI 1; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
81 |
by(induct_tac "ys" 1); |
2608 | 82 |
by(ALLGOALS Asm_simp_tac); |
83 |
br allI 1; |
|
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
84 |
by(induct_tac "ys" 1); |
2608 | 85 |
by(ALLGOALS Asm_simp_tac); |
86 |
qed_spec_mp "append1_eq_conv"; |
|
87 |
AddIffs [append1_eq_conv]; |
|
88 |
||
3011 | 89 |
goal thy "xs ~= [] --> hd xs # tl xs = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
90 |
by(induct_tac "xs" 1); |
2608 | 91 |
by(ALLGOALS Asm_simp_tac); |
92 |
qed_spec_mp "hd_Cons_tl"; |
|
93 |
Addsimps [hd_Cons_tl]; |
|
923 | 94 |
|
3011 | 95 |
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
96 |
by (induct_tac "xs" 1); |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
97 |
by (ALLGOALS Asm_simp_tac); |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
98 |
qed "hd_append"; |
923 | 99 |
|
3011 | 100 |
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; |
2608 | 101 |
by(simp_tac (!simpset setloop(split_tac[expand_list_case])) 1); |
102 |
qed "tl_append"; |
|
103 |
||
104 |
(** map **) |
|
105 |
||
3011 | 106 |
goal thy |
2608 | 107 |
"(!x. x : set_of_list xs --> f x = g x) --> map f xs = map g xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
108 |
by(induct_tac "xs" 1); |
2608 | 109 |
by(ALLGOALS Asm_simp_tac); |
110 |
bind_thm("map_ext", impI RS (allI RS (result() RS mp))); |
|
111 |
||
3011 | 112 |
goal thy "map (%x.x) = (%xs.xs)"; |
2608 | 113 |
by (rtac ext 1); |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
114 |
by (induct_tac "xs" 1); |
2608 | 115 |
by (ALLGOALS Asm_simp_tac); |
116 |
qed "map_ident"; |
|
117 |
Addsimps[map_ident]; |
|
118 |
||
3011 | 119 |
goal thy "map f (xs@ys) = map f xs @ map f ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
120 |
by (induct_tac "xs" 1); |
2608 | 121 |
by (ALLGOALS Asm_simp_tac); |
122 |
qed "map_append"; |
|
123 |
Addsimps[map_append]; |
|
124 |
||
3011 | 125 |
goalw thy [o_def] "map (f o g) xs = map f (map g xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
126 |
by (induct_tac "xs" 1); |
2608 | 127 |
by (ALLGOALS Asm_simp_tac); |
128 |
qed "map_compose"; |
|
129 |
Addsimps[map_compose]; |
|
130 |
||
3011 | 131 |
goal thy "rev(map f xs) = map f (rev xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
132 |
by (induct_tac "xs" 1); |
2608 | 133 |
by (ALLGOALS Asm_simp_tac); |
134 |
qed "rev_map"; |
|
135 |
||
1169 | 136 |
(** rev **) |
137 |
||
3011 | 138 |
goal thy "rev(xs@ys) = rev(ys) @ rev(xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
139 |
by (induct_tac "xs" 1); |
2512 | 140 |
by (ALLGOALS Asm_simp_tac); |
1169 | 141 |
qed "rev_append"; |
2512 | 142 |
Addsimps[rev_append]; |
1169 | 143 |
|
3011 | 144 |
goal thy "rev(rev l) = l"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
145 |
by (induct_tac "l" 1); |
2512 | 146 |
by (ALLGOALS Asm_simp_tac); |
1169 | 147 |
qed "rev_rev_ident"; |
2512 | 148 |
Addsimps[rev_rev_ident]; |
1169 | 149 |
|
2608 | 150 |
|
923 | 151 |
(** mem **) |
152 |
||
3011 | 153 |
goal thy "x mem (xs@ys) = (x mem xs | x mem ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
154 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
155 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
923 | 156 |
qed "mem_append"; |
2512 | 157 |
Addsimps[mem_append]; |
923 | 158 |
|
3011 | 159 |
goal thy "x mem [x:xs.P(x)] = (x mem xs & P(x))"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
160 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
161 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
923 | 162 |
qed "mem_filter"; |
2512 | 163 |
Addsimps[mem_filter]; |
923 | 164 |
|
1908 | 165 |
(** set_of_list **) |
1812 | 166 |
|
1908 | 167 |
goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
168 |
by (induct_tac "xs" 1); |
1812 | 169 |
by (ALLGOALS Asm_simp_tac); |
2891 | 170 |
by (Blast_tac 1); |
1908 | 171 |
qed "set_of_list_append"; |
2512 | 172 |
Addsimps[set_of_list_append]; |
1812 | 173 |
|
1908 | 174 |
goal thy "(x mem xs) = (x: set_of_list xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
175 |
by (induct_tac "xs" 1); |
1812 | 176 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
2891 | 177 |
by (Blast_tac 1); |
1908 | 178 |
qed "set_of_list_mem_eq"; |
1812 | 179 |
|
3011 | 180 |
goal thy "set_of_list l <= set_of_list (x#l)"; |
1936 | 181 |
by (Simp_tac 1); |
2891 | 182 |
by (Blast_tac 1); |
1936 | 183 |
qed "set_of_list_subset_Cons"; |
184 |
||
3011 | 185 |
goal thy "(set_of_list xs = {}) = (xs = [])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
186 |
by(induct_tac "xs" 1); |
2608 | 187 |
by(ALLGOALS Asm_simp_tac); |
188 |
qed "set_of_list_empty"; |
|
189 |
Addsimps [set_of_list_empty]; |
|
190 |
||
3011 | 191 |
goal thy "set_of_list(rev xs) = set_of_list(xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
192 |
by(induct_tac "xs" 1); |
2608 | 193 |
by(ALLGOALS Asm_simp_tac); |
2891 | 194 |
by(Blast_tac 1); |
2608 | 195 |
qed "set_of_list_rev"; |
196 |
Addsimps [set_of_list_rev]; |
|
197 |
||
3011 | 198 |
goal thy "set_of_list(map f xs) = f``(set_of_list xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
199 |
by(induct_tac "xs" 1); |
2608 | 200 |
by(ALLGOALS Asm_simp_tac); |
201 |
qed "set_of_list_map"; |
|
202 |
Addsimps [set_of_list_map]; |
|
203 |
||
1812 | 204 |
|
923 | 205 |
(** list_all **) |
206 |
||
3011 | 207 |
goal thy "list_all (%x.True) xs = True"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
208 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
209 |
by (ALLGOALS Asm_simp_tac); |
923 | 210 |
qed "list_all_True"; |
2512 | 211 |
Addsimps [list_all_True]; |
923 | 212 |
|
3011 | 213 |
goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
214 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
215 |
by (ALLGOALS Asm_simp_tac); |
2512 | 216 |
qed "list_all_append"; |
217 |
Addsimps [list_all_append]; |
|
923 | 218 |
|
3011 | 219 |
goal thy "list_all P xs = (!x. x mem xs --> P(x))"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
220 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
221 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
2891 | 222 |
by (Blast_tac 1); |
923 | 223 |
qed "list_all_mem_conv"; |
224 |
||
225 |
||
2608 | 226 |
(** filter **) |
923 | 227 |
|
3011 | 228 |
goal thy "[x:xs@ys . P] = [x:xs . P] @ [y:ys . P]"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
229 |
by(induct_tac "xs" 1); |
2608 | 230 |
by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
231 |
qed "filter_append"; |
|
232 |
Addsimps [filter_append]; |
|
233 |
||
234 |
||
235 |
(** concat **) |
|
236 |
||
3011 | 237 |
goal thy "concat(xs@ys) = concat(xs)@concat(ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
238 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
239 |
by (ALLGOALS Asm_simp_tac); |
2608 | 240 |
qed"concat_append"; |
241 |
Addsimps [concat_append]; |
|
2512 | 242 |
|
3011 | 243 |
goal thy "rev(concat ls) = concat (map rev (rev ls))"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
244 |
by (induct_tac "ls" 1); |
2512 | 245 |
by (ALLGOALS Asm_simp_tac); |
2608 | 246 |
qed "rev_concat"; |
923 | 247 |
|
962 | 248 |
(** length **) |
249 |
||
3011 | 250 |
goal thy "length(xs@ys) = length(xs)+length(ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
251 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
252 |
by (ALLGOALS Asm_simp_tac); |
962 | 253 |
qed"length_append"; |
1301 | 254 |
Addsimps [length_append]; |
255 |
||
3011 | 256 |
goal thy "length (map f l) = length l"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
257 |
by (induct_tac "l" 1); |
1301 | 258 |
by (ALLGOALS Simp_tac); |
259 |
qed "length_map"; |
|
260 |
Addsimps [length_map]; |
|
962 | 261 |
|
3011 | 262 |
goal thy "length(rev xs) = length(xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
263 |
by (induct_tac "xs" 1); |
1301 | 264 |
by (ALLGOALS Asm_simp_tac); |
1169 | 265 |
qed "length_rev"; |
1301 | 266 |
Addsimps [length_rev]; |
1169 | 267 |
|
3011 | 268 |
goal thy "(length xs = 0) = (xs = [])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
269 |
by(induct_tac "xs" 1); |
2608 | 270 |
by(ALLGOALS Asm_simp_tac); |
271 |
qed "length_0_conv"; |
|
272 |
AddIffs [length_0_conv]; |
|
273 |
||
3011 | 274 |
goal thy "(0 < length xs) = (xs ~= [])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
275 |
by(induct_tac "xs" 1); |
2608 | 276 |
by(ALLGOALS Asm_simp_tac); |
277 |
qed "length_greater_0_conv"; |
|
278 |
AddIffs [length_greater_0_conv]; |
|
279 |
||
280 |
||
923 | 281 |
(** nth **) |
282 |
||
3011 | 283 |
goal thy |
2608 | 284 |
"!xs. nth n (xs@ys) = \ |
285 |
\ (if n < length xs then nth n xs else nth (n - length xs) ys)"; |
|
286 |
by(nat_ind_tac "n" 1); |
|
287 |
by(Asm_simp_tac 1); |
|
288 |
br allI 1; |
|
289 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
290 |
by(ALLGOALS Asm_simp_tac); |
|
291 |
br allI 1; |
|
292 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
293 |
by(ALLGOALS Asm_simp_tac); |
|
294 |
qed_spec_mp "nth_append"; |
|
295 |
||
3011 | 296 |
goal thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
297 |
by (induct_tac "xs" 1); |
1301 | 298 |
(* case [] *) |
299 |
by (Asm_full_simp_tac 1); |
|
300 |
(* case x#xl *) |
|
301 |
by (rtac allI 1); |
|
302 |
by (nat_ind_tac "n" 1); |
|
303 |
by (ALLGOALS Asm_full_simp_tac); |
|
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
304 |
qed_spec_mp "nth_map"; |
1301 | 305 |
Addsimps [nth_map]; |
306 |
||
3011 | 307 |
goal thy "!n. n < length xs --> list_all P xs --> P(nth n xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
308 |
by (induct_tac "xs" 1); |
1301 | 309 |
(* case [] *) |
310 |
by (Simp_tac 1); |
|
311 |
(* case x#xl *) |
|
312 |
by (rtac allI 1); |
|
313 |
by (nat_ind_tac "n" 1); |
|
314 |
by (ALLGOALS Asm_full_simp_tac); |
|
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
315 |
qed_spec_mp "list_all_nth"; |
1301 | 316 |
|
3011 | 317 |
goal thy "!n. n < length xs --> (nth n xs) mem xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
318 |
by (induct_tac "xs" 1); |
1301 | 319 |
(* case [] *) |
320 |
by (Simp_tac 1); |
|
321 |
(* case x#xl *) |
|
322 |
by (rtac allI 1); |
|
323 |
by (nat_ind_tac "n" 1); |
|
324 |
(* case 0 *) |
|
325 |
by (Asm_full_simp_tac 1); |
|
326 |
(* case Suc x *) |
|
327 |
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
328 |
qed_spec_mp "nth_mem"; |
1301 | 329 |
Addsimps [nth_mem]; |
330 |
||
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
331 |
|
2608 | 332 |
(** take & drop **) |
333 |
section "take & drop"; |
|
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
334 |
|
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
335 |
goal thy "take 0 xs = []"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
336 |
by (induct_tac "xs" 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
337 |
by (ALLGOALS Asm_simp_tac); |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
338 |
qed "take_0"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
339 |
|
2608 | 340 |
goal thy "drop 0 xs = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
341 |
by (induct_tac "xs" 1); |
2608 | 342 |
by (ALLGOALS Asm_simp_tac); |
343 |
qed "drop_0"; |
|
344 |
||
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
345 |
goal thy "take (Suc n) (x#xs) = x # take n xs"; |
1552 | 346 |
by (Simp_tac 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
347 |
qed "take_Suc_Cons"; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
348 |
|
2608 | 349 |
goal thy "drop (Suc n) (x#xs) = drop n xs"; |
350 |
by (Simp_tac 1); |
|
351 |
qed "drop_Suc_Cons"; |
|
352 |
||
353 |
Delsimps [take_Cons,drop_Cons]; |
|
354 |
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons]; |
|
355 |
||
3011 | 356 |
goal thy "!xs. length(take n xs) = min (length xs) n"; |
2608 | 357 |
by(nat_ind_tac "n" 1); |
358 |
by(ALLGOALS Asm_simp_tac); |
|
359 |
br allI 1; |
|
360 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
361 |
by(ALLGOALS Asm_simp_tac); |
|
362 |
qed_spec_mp "length_take"; |
|
363 |
Addsimps [length_take]; |
|
923 | 364 |
|
3011 | 365 |
goal thy "!xs. length(drop n xs) = (length xs - n)"; |
2608 | 366 |
by(nat_ind_tac "n" 1); |
367 |
by(ALLGOALS Asm_simp_tac); |
|
368 |
br allI 1; |
|
369 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
370 |
by(ALLGOALS Asm_simp_tac); |
|
371 |
qed_spec_mp "length_drop"; |
|
372 |
Addsimps [length_drop]; |
|
373 |
||
3011 | 374 |
goal thy "!xs. length xs <= n --> take n xs = xs"; |
2608 | 375 |
by(nat_ind_tac "n" 1); |
376 |
by(ALLGOALS Asm_simp_tac); |
|
377 |
br allI 1; |
|
378 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
379 |
by(ALLGOALS Asm_simp_tac); |
|
380 |
qed_spec_mp "take_all"; |
|
923 | 381 |
|
3011 | 382 |
goal thy "!xs. length xs <= n --> drop n xs = []"; |
2608 | 383 |
by(nat_ind_tac "n" 1); |
384 |
by(ALLGOALS Asm_simp_tac); |
|
385 |
br allI 1; |
|
386 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
387 |
by(ALLGOALS Asm_simp_tac); |
|
388 |
qed_spec_mp "drop_all"; |
|
389 |
||
3011 | 390 |
goal thy |
2608 | 391 |
"!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; |
392 |
by(nat_ind_tac "n" 1); |
|
393 |
by(ALLGOALS Asm_simp_tac); |
|
394 |
br allI 1; |
|
395 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
396 |
by(ALLGOALS Asm_simp_tac); |
|
397 |
qed_spec_mp "take_append"; |
|
398 |
Addsimps [take_append]; |
|
399 |
||
3011 | 400 |
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; |
2608 | 401 |
by(nat_ind_tac "n" 1); |
402 |
by(ALLGOALS Asm_simp_tac); |
|
403 |
br allI 1; |
|
404 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
405 |
by(ALLGOALS Asm_simp_tac); |
|
406 |
qed_spec_mp "drop_append"; |
|
407 |
Addsimps [drop_append]; |
|
408 |
||
3011 | 409 |
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; |
2608 | 410 |
by(nat_ind_tac "m" 1); |
411 |
by(ALLGOALS Asm_simp_tac); |
|
412 |
br allI 1; |
|
413 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
414 |
by(ALLGOALS Asm_simp_tac); |
|
415 |
br allI 1; |
|
416 |
by(res_inst_tac [("n","n")]natE 1); |
|
417 |
by(ALLGOALS Asm_simp_tac); |
|
418 |
qed_spec_mp "take_take"; |
|
419 |
||
3011 | 420 |
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; |
2608 | 421 |
by(nat_ind_tac "m" 1); |
422 |
by(ALLGOALS Asm_simp_tac); |
|
423 |
br allI 1; |
|
424 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
425 |
by(ALLGOALS Asm_simp_tac); |
|
426 |
qed_spec_mp "drop_drop"; |
|
923 | 427 |
|
3011 | 428 |
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; |
2608 | 429 |
by(nat_ind_tac "m" 1); |
430 |
by(ALLGOALS Asm_simp_tac); |
|
431 |
br allI 1; |
|
432 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
433 |
by(ALLGOALS Asm_simp_tac); |
|
434 |
qed_spec_mp "take_drop"; |
|
435 |
||
3011 | 436 |
goal thy "!xs. take n (map f xs) = map f (take n xs)"; |
2608 | 437 |
by(nat_ind_tac "n" 1); |
438 |
by(ALLGOALS Asm_simp_tac); |
|
439 |
br allI 1; |
|
440 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
441 |
by(ALLGOALS Asm_simp_tac); |
|
442 |
qed_spec_mp "take_map"; |
|
443 |
||
3011 | 444 |
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; |
2608 | 445 |
by(nat_ind_tac "n" 1); |
446 |
by(ALLGOALS Asm_simp_tac); |
|
447 |
br allI 1; |
|
448 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
449 |
by(ALLGOALS Asm_simp_tac); |
|
450 |
qed_spec_mp "drop_map"; |
|
451 |
||
3011 | 452 |
goal thy |
2608 | 453 |
"!n i. i < n --> nth i (take n xs) = nth i xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
454 |
by(induct_tac "xs" 1); |
2608 | 455 |
by(ALLGOALS Asm_simp_tac); |
456 |
by(strip_tac 1); |
|
457 |
by(res_inst_tac [("n","n")] natE 1); |
|
2891 | 458 |
by(Blast_tac 1); |
2608 | 459 |
by(res_inst_tac [("n","i")] natE 1); |
460 |
by(ALLGOALS (hyp_subst_tac THEN' Asm_full_simp_tac)); |
|
461 |
qed_spec_mp "nth_take"; |
|
462 |
Addsimps [nth_take]; |
|
923 | 463 |
|
3011 | 464 |
goal thy |
2608 | 465 |
"!xs i. n + i < length xs --> nth i (drop n xs) = nth (n + i) xs"; |
466 |
by(nat_ind_tac "n" 1); |
|
467 |
by(ALLGOALS Asm_simp_tac); |
|
468 |
br allI 1; |
|
469 |
by(res_inst_tac [("xs","xs")]list_cases 1); |
|
470 |
by(ALLGOALS Asm_simp_tac); |
|
471 |
qed_spec_mp "nth_drop"; |
|
472 |
Addsimps [nth_drop]; |
|
473 |
||
474 |
(** takeWhile & dropWhile **) |
|
475 |
||
3011 | 476 |
goal thy |
2608 | 477 |
"x:set_of_list xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
478 |
by(induct_tac "xs" 1); |
2608 | 479 |
by(Simp_tac 1); |
480 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
2891 | 481 |
by(Blast_tac 1); |
2608 | 482 |
bind_thm("takeWhile_append1", conjI RS (result() RS mp)); |
483 |
Addsimps [takeWhile_append1]; |
|
923 | 484 |
|
3011 | 485 |
goal thy |
2608 | 486 |
"(!x:set_of_list xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
487 |
by(induct_tac "xs" 1); |
2608 | 488 |
by(Simp_tac 1); |
489 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
490 |
bind_thm("takeWhile_append2", ballI RS (result() RS mp)); |
|
491 |
Addsimps [takeWhile_append2]; |
|
1169 | 492 |
|
3011 | 493 |
goal thy |
2608 | 494 |
"x:set_of_list xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
495 |
by(induct_tac "xs" 1); |
2608 | 496 |
by(Simp_tac 1); |
497 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
2891 | 498 |
by(Blast_tac 1); |
2608 | 499 |
bind_thm("dropWhile_append1", conjI RS (result() RS mp)); |
500 |
Addsimps [dropWhile_append1]; |
|
501 |
||
3011 | 502 |
goal thy |
2608 | 503 |
"(!x:set_of_list xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
504 |
by(induct_tac "xs" 1); |
2608 | 505 |
by(Simp_tac 1); |
506 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
507 |
bind_thm("dropWhile_append2", ballI RS (result() RS mp)); |
|
508 |
Addsimps [dropWhile_append2]; |
|
509 |
||
3011 | 510 |
goal thy "x:set_of_list(takeWhile P xs) --> x:set_of_list xs & P x"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
511 |
by(induct_tac "xs" 1); |
2608 | 512 |
by(Simp_tac 1); |
513 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
514 |
qed_spec_mp"set_of_list_take_whileD"; |
|
515 |