author | nipkow |
Thu, 11 Nov 1999 11:43:14 +0100 | |
changeset 8009 | 29a7a79ee7f4 |
parent 7570 | a9391550eea1 |
child 8064 | 357652a08ee0 |
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 |
||
4935 | 9 |
Goal "!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); |
5316 | 11 |
by Auto_tac; |
2608 | 12 |
qed_spec_mp "not_Cons_self"; |
3574 | 13 |
bind_thm("not_Cons_self2",not_Cons_self RS not_sym); |
14 |
Addsimps [not_Cons_self,not_Cons_self2]; |
|
923 | 15 |
|
4935 | 16 |
Goal "(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
|
17 |
by (induct_tac "xs" 1); |
5316 | 18 |
by Auto_tac; |
923 | 19 |
qed "neq_Nil_conv"; |
20 |
||
4830 | 21 |
(* Induction over the length of a list: *) |
4935 | 22 |
val [prem] = Goal |
4911 | 23 |
"(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)"; |
5132 | 24 |
by (rtac measure_induct 1 THEN etac prem 1); |
4911 | 25 |
qed "length_induct"; |
26 |
||
923 | 27 |
|
3468 | 28 |
(** "lists": the list-forming operator over sets **) |
3342
ec3b55fcb165
New operator "lists" for formalizing sets of lists
paulson
parents:
3292
diff
changeset
|
29 |
|
5043 | 30 |
Goalw lists.defs "A<=B ==> lists A <= lists B"; |
3342
ec3b55fcb165
New operator "lists" for formalizing sets of lists
paulson
parents:
3292
diff
changeset
|
31 |
by (rtac lfp_mono 1); |
ec3b55fcb165
New operator "lists" for formalizing sets of lists
paulson
parents:
3292
diff
changeset
|
32 |
by (REPEAT (ares_tac basic_monos 1)); |
ec3b55fcb165
New operator "lists" for formalizing sets of lists
paulson
parents:
3292
diff
changeset
|
33 |
qed "lists_mono"; |
3196 | 34 |
|
6141 | 35 |
val listsE = lists.mk_cases "x#l : lists A"; |
3468 | 36 |
AddSEs [listsE]; |
37 |
AddSIs lists.intrs; |
|
38 |
||
5043 | 39 |
Goal "l: lists A ==> l: lists B --> l: lists (A Int B)"; |
3468 | 40 |
by (etac lists.induct 1); |
41 |
by (ALLGOALS Blast_tac); |
|
42 |
qed_spec_mp "lists_IntI"; |
|
43 |
||
4935 | 44 |
Goal "lists (A Int B) = lists A Int lists B"; |
4423 | 45 |
by (rtac (mono_Int RS equalityI) 1); |
4089 | 46 |
by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1); |
47 |
by (blast_tac (claset() addSIs [lists_IntI]) 1); |
|
3468 | 48 |
qed "lists_Int_eq"; |
49 |
Addsimps [lists_Int_eq]; |
|
50 |
||
3196 | 51 |
|
4643 | 52 |
(** Case analysis **) |
53 |
section "Case analysis"; |
|
2608 | 54 |
|
4935 | 55 |
val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; |
3457 | 56 |
by (induct_tac "xs" 1); |
57 |
by (REPEAT(resolve_tac prems 1)); |
|
2608 | 58 |
qed "list_cases"; |
59 |
||
4935 | 60 |
Goal "(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
|
61 |
by (induct_tac "xs" 1); |
2891 | 62 |
by (Blast_tac 1); |
63 |
by (Blast_tac 1); |
|
2608 | 64 |
bind_thm("list_eq_cases", |
65 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp)))))); |
|
66 |
||
3860 | 67 |
(** length **) |
68 |
(* needs to come before "@" because of thm append_eq_append_conv *) |
|
69 |
||
70 |
section "length"; |
|
71 |
||
4935 | 72 |
Goal "length(xs@ys) = length(xs)+length(ys)"; |
3860 | 73 |
by (induct_tac "xs" 1); |
5316 | 74 |
by Auto_tac; |
3860 | 75 |
qed"length_append"; |
76 |
Addsimps [length_append]; |
|
77 |
||
5129 | 78 |
Goal "length (map f xs) = length xs"; |
79 |
by (induct_tac "xs" 1); |
|
5316 | 80 |
by Auto_tac; |
3860 | 81 |
qed "length_map"; |
82 |
Addsimps [length_map]; |
|
83 |
||
4935 | 84 |
Goal "length(rev xs) = length(xs)"; |
3860 | 85 |
by (induct_tac "xs" 1); |
5316 | 86 |
by Auto_tac; |
3860 | 87 |
qed "length_rev"; |
88 |
Addsimps [length_rev]; |
|
89 |
||
7028 | 90 |
Goal "length(tl xs) = (length xs) - 1"; |
4423 | 91 |
by (exhaust_tac "xs" 1); |
5316 | 92 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
93 |
qed "length_tl"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
94 |
Addsimps [length_tl]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
95 |
|
4935 | 96 |
Goal "(length xs = 0) = (xs = [])"; |
3860 | 97 |
by (induct_tac "xs" 1); |
5316 | 98 |
by Auto_tac; |
3860 | 99 |
qed "length_0_conv"; |
100 |
AddIffs [length_0_conv]; |
|
101 |
||
4935 | 102 |
Goal "(0 = length xs) = (xs = [])"; |
3860 | 103 |
by (induct_tac "xs" 1); |
5316 | 104 |
by Auto_tac; |
3860 | 105 |
qed "zero_length_conv"; |
106 |
AddIffs [zero_length_conv]; |
|
107 |
||
4935 | 108 |
Goal "(0 < length xs) = (xs ~= [])"; |
3860 | 109 |
by (induct_tac "xs" 1); |
5316 | 110 |
by Auto_tac; |
3860 | 111 |
qed "length_greater_0_conv"; |
112 |
AddIffs [length_greater_0_conv]; |
|
113 |
||
5296 | 114 |
Goal "(length xs = Suc n) = (? y ys. xs = y#ys & length ys = n)"; |
115 |
by (induct_tac "xs" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
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|
116 |
by Auto_tac; |
5296 | 117 |
qed "length_Suc_conv"; |
118 |
||
923 | 119 |
(** @ - append **) |
120 |
||
3467 | 121 |
section "@ - append"; |
122 |
||
4935 | 123 |
Goal "(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
|
124 |
by (induct_tac "xs" 1); |
5316 | 125 |
by Auto_tac; |
923 | 126 |
qed "append_assoc"; |
2512 | 127 |
Addsimps [append_assoc]; |
923 | 128 |
|
4935 | 129 |
Goal "xs @ [] = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
130 |
by (induct_tac "xs" 1); |
5316 | 131 |
by Auto_tac; |
923 | 132 |
qed "append_Nil2"; |
2512 | 133 |
Addsimps [append_Nil2]; |
923 | 134 |
|
4935 | 135 |
Goal "(xs@ys = []) = (xs=[] & ys=[])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
136 |
by (induct_tac "xs" 1); |
5316 | 137 |
by Auto_tac; |
2608 | 138 |
qed "append_is_Nil_conv"; |
139 |
AddIffs [append_is_Nil_conv]; |
|
140 |
||
4935 | 141 |
Goal "([] = xs@ys) = (xs=[] & ys=[])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
142 |
by (induct_tac "xs" 1); |
5316 | 143 |
by Auto_tac; |
2608 | 144 |
qed "Nil_is_append_conv"; |
145 |
AddIffs [Nil_is_append_conv]; |
|
923 | 146 |
|
4935 | 147 |
Goal "(xs @ ys = xs) = (ys=[])"; |
3574 | 148 |
by (induct_tac "xs" 1); |
5316 | 149 |
by Auto_tac; |
3574 | 150 |
qed "append_self_conv"; |
151 |
||
4935 | 152 |
Goal "(xs = xs @ ys) = (ys=[])"; |
3574 | 153 |
by (induct_tac "xs" 1); |
5316 | 154 |
by Auto_tac; |
3574 | 155 |
qed "self_append_conv"; |
156 |
AddIffs [append_self_conv,self_append_conv]; |
|
157 |
||
4935 | 158 |
Goal "!ys. length xs = length ys | length us = length vs \ |
3860 | 159 |
\ --> (xs@us = ys@vs) = (xs=ys & us=vs)"; |
4423 | 160 |
by (induct_tac "xs" 1); |
161 |
by (rtac allI 1); |
|
162 |
by (exhaust_tac "ys" 1); |
|
163 |
by (Asm_simp_tac 1); |
|
5641 | 164 |
by (Force_tac 1); |
4423 | 165 |
by (rtac allI 1); |
166 |
by (exhaust_tac "ys" 1); |
|
5641 | 167 |
by (Force_tac 1); |
4423 | 168 |
by (Asm_simp_tac 1); |
3860 | 169 |
qed_spec_mp "append_eq_append_conv"; |
170 |
Addsimps [append_eq_append_conv]; |
|
171 |
||
4935 | 172 |
Goal "(xs @ ys = xs @ zs) = (ys=zs)"; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
173 |
by (Simp_tac 1); |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
174 |
qed "same_append_eq"; |
3860 | 175 |
|
4935 | 176 |
Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
177 |
by (Simp_tac 1); |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
178 |
qed "append1_eq_conv"; |
2608 | 179 |
|
4935 | 180 |
Goal "(ys @ xs = zs @ xs) = (ys=zs)"; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
181 |
by (Simp_tac 1); |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
182 |
qed "append_same_eq"; |
2608 | 183 |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
184 |
AddSIs |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
185 |
[same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
186 |
AddSDs |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
187 |
[same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1]; |
3571 | 188 |
|
4935 | 189 |
Goal "(xs @ ys = ys) = (xs=[])"; |
5132 | 190 |
by (cut_inst_tac [("zs","[]")] append_same_eq 1); |
5316 | 191 |
by Auto_tac; |
4647 | 192 |
qed "append_self_conv2"; |
193 |
||
4935 | 194 |
Goal "(ys = xs @ ys) = (xs=[])"; |
5132 | 195 |
by (simp_tac (simpset() addsimps |
4647 | 196 |
[simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1); |
5132 | 197 |
by (Blast_tac 1); |
4647 | 198 |
qed "self_append_conv2"; |
199 |
AddIffs [append_self_conv2,self_append_conv2]; |
|
200 |
||
4935 | 201 |
Goal "xs ~= [] --> hd xs # tl xs = xs"; |
3457 | 202 |
by (induct_tac "xs" 1); |
5316 | 203 |
by Auto_tac; |
2608 | 204 |
qed_spec_mp "hd_Cons_tl"; |
205 |
Addsimps [hd_Cons_tl]; |
|
923 | 206 |
|
4935 | 207 |
Goal "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
|
208 |
by (induct_tac "xs" 1); |
5316 | 209 |
by Auto_tac; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
210 |
qed "hd_append"; |
923 | 211 |
|
5043 | 212 |
Goal "xs ~= [] ==> hd(xs @ ys) = hd xs"; |
4089 | 213 |
by (asm_simp_tac (simpset() addsimps [hd_append] |
5183 | 214 |
addsplits [list.split]) 1); |
3571 | 215 |
qed "hd_append2"; |
216 |
Addsimps [hd_append2]; |
|
217 |
||
4935 | 218 |
Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; |
5183 | 219 |
by (simp_tac (simpset() addsplits [list.split]) 1); |
2608 | 220 |
qed "tl_append"; |
221 |
||
5043 | 222 |
Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"; |
4089 | 223 |
by (asm_simp_tac (simpset() addsimps [tl_append] |
5183 | 224 |
addsplits [list.split]) 1); |
3571 | 225 |
qed "tl_append2"; |
226 |
Addsimps [tl_append2]; |
|
227 |
||
5272 | 228 |
(* trivial rules for solving @-equations automatically *) |
229 |
||
230 |
Goal "xs = ys ==> xs = [] @ ys"; |
|
5318 | 231 |
by (Asm_simp_tac 1); |
5272 | 232 |
qed "eq_Nil_appendI"; |
233 |
||
234 |
Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs"; |
|
5318 | 235 |
by (dtac sym 1); |
236 |
by (Asm_simp_tac 1); |
|
5272 | 237 |
qed "Cons_eq_appendI"; |
238 |
||
239 |
Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us"; |
|
5318 | 240 |
by (dtac sym 1); |
241 |
by (Asm_simp_tac 1); |
|
5272 | 242 |
qed "append_eq_appendI"; |
243 |
||
4830 | 244 |
|
5427 | 245 |
(*** |
246 |
Simplification procedure for all list equalities. |
|
247 |
Currently only tries to rearranges @ to see if |
|
248 |
- both lists end in a singleton list, |
|
249 |
- or both lists end in the same list. |
|
250 |
***) |
|
251 |
local |
|
252 |
||
253 |
val list_eq_pattern = |
|
6394 | 254 |
Thm.read_cterm (Theory.sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT); |
5427 | 255 |
|
7224 | 256 |
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = |
257 |
(case xs of Const("List.list.Nil",_) => cons | _ => last xs) |
|
5427 | 258 |
| last (Const("List.op @",_) $ _ $ ys) = last ys |
259 |
| last t = t; |
|
260 |
||
7224 | 261 |
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true |
5427 | 262 |
| list1 _ = false; |
263 |
||
7224 | 264 |
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = |
265 |
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) |
|
5427 | 266 |
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys |
7224 | 267 |
| butlast xs = Const("List.list.Nil",fastype_of xs); |
5427 | 268 |
|
269 |
val rearr_tac = |
|
270 |
simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]); |
|
271 |
||
272 |
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = |
|
273 |
let |
|
274 |
val lastl = last lhs and lastr = last rhs |
|
275 |
fun rearr conv = |
|
276 |
let val lhs1 = butlast lhs and rhs1 = butlast rhs |
|
277 |
val Type(_,listT::_) = eqT |
|
278 |
val appT = [listT,listT] ---> listT |
|
279 |
val app = Const("List.op @",appT) |
|
280 |
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) |
|
281 |
val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) |
|
282 |
val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) |
|
283 |
handle ERROR => |
|
284 |
error("The error(s) above occurred while trying to prove " ^ |
|
285 |
string_of_cterm ct) |
|
286 |
in Some((conv RS (thm RS trans)) RS eq_reflection) end |
|
287 |
||
288 |
in if list1 lastl andalso list1 lastr |
|
289 |
then rearr append1_eq_conv |
|
290 |
else |
|
291 |
if lastl aconv lastr |
|
292 |
then rearr append_same_eq |
|
293 |
else None |
|
294 |
end; |
|
295 |
in |
|
296 |
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq; |
|
297 |
end; |
|
298 |
||
299 |
Addsimprocs [list_eq_simproc]; |
|
300 |
||
301 |
||
2608 | 302 |
(** map **) |
303 |
||
3467 | 304 |
section "map"; |
305 |
||
5278 | 306 |
Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs"; |
3457 | 307 |
by (induct_tac "xs" 1); |
5316 | 308 |
by Auto_tac; |
2608 | 309 |
bind_thm("map_ext", impI RS (allI RS (result() RS mp))); |
310 |
||
4935 | 311 |
Goal "map (%x. x) = (%xs. xs)"; |
2608 | 312 |
by (rtac ext 1); |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
313 |
by (induct_tac "xs" 1); |
5316 | 314 |
by Auto_tac; |
2608 | 315 |
qed "map_ident"; |
316 |
Addsimps[map_ident]; |
|
317 |
||
4935 | 318 |
Goal "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
|
319 |
by (induct_tac "xs" 1); |
5316 | 320 |
by Auto_tac; |
2608 | 321 |
qed "map_append"; |
322 |
Addsimps[map_append]; |
|
323 |
||
4935 | 324 |
Goalw [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
|
325 |
by (induct_tac "xs" 1); |
5316 | 326 |
by Auto_tac; |
2608 | 327 |
qed "map_compose"; |
328 |
Addsimps[map_compose]; |
|
329 |
||
4935 | 330 |
Goal "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
|
331 |
by (induct_tac "xs" 1); |
5316 | 332 |
by Auto_tac; |
2608 | 333 |
qed "rev_map"; |
334 |
||
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
335 |
(* a congruence rule for map: *) |
6451 | 336 |
Goal "xs=ys ==> (!x. x : set ys --> f x = g x) --> map f xs = map g ys"; |
4423 | 337 |
by (hyp_subst_tac 1); |
338 |
by (induct_tac "ys" 1); |
|
5316 | 339 |
by Auto_tac; |
6451 | 340 |
bind_thm("map_cong", impI RSN (2,allI RSN (2, result() RS mp))); |
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
341 |
|
4935 | 342 |
Goal "(map f xs = []) = (xs = [])"; |
8009 | 343 |
by (exhaust_tac "xs" 1); |
5316 | 344 |
by Auto_tac; |
3860 | 345 |
qed "map_is_Nil_conv"; |
346 |
AddIffs [map_is_Nil_conv]; |
|
347 |
||
4935 | 348 |
Goal "([] = map f xs) = (xs = [])"; |
8009 | 349 |
by (exhaust_tac "xs" 1); |
5316 | 350 |
by Auto_tac; |
3860 | 351 |
qed "Nil_is_map_conv"; |
352 |
AddIffs [Nil_is_map_conv]; |
|
353 |
||
8009 | 354 |
Goal "(map f xs = y#ys) = (? x xs'. xs = x#xs' & f x = y & map f xs' = ys)"; |
355 |
by (exhaust_tac "xs" 1); |
|
356 |
by (ALLGOALS Asm_simp_tac); |
|
357 |
qed "map_eq_Cons"; |
|
358 |
||
359 |
Goal "!xs. map f xs = map f ys --> (!x y. f x = f y --> x=y) --> xs=ys"; |
|
360 |
by (induct_tac "ys" 1); |
|
361 |
by (Asm_simp_tac 1); |
|
362 |
by (fast_tac (claset() addss (simpset() addsimps [map_eq_Cons])) 1); |
|
363 |
qed_spec_mp "map_injective"; |
|
364 |
||
365 |
Goal "inj f ==> inj (map f)"; |
|
366 |
by(blast_tac (claset() addDs [map_injective,injD] addIs [injI]) 1); |
|
367 |
qed "inj_mapI"; |
|
368 |
||
369 |
Goalw [inj_on_def] "inj (map f) ==> inj f"; |
|
370 |
by(Clarify_tac 1); |
|
371 |
by(eres_inst_tac [("x","[x]")] ballE 1); |
|
372 |
by(eres_inst_tac [("x","[y]")] ballE 1); |
|
373 |
by(Asm_full_simp_tac 1); |
|
374 |
by(Blast_tac 1); |
|
375 |
by(Blast_tac 1); |
|
376 |
qed "inj_mapD"; |
|
377 |
||
378 |
Goal "inj (map f) = inj f"; |
|
379 |
by(blast_tac (claset() addDs [inj_mapD] addIs [inj_mapI]) 1); |
|
380 |
qed "inj_map"; |
|
3860 | 381 |
|
1169 | 382 |
(** rev **) |
383 |
||
3467 | 384 |
section "rev"; |
385 |
||
4935 | 386 |
Goal "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
|
387 |
by (induct_tac "xs" 1); |
5316 | 388 |
by Auto_tac; |
1169 | 389 |
qed "rev_append"; |
2512 | 390 |
Addsimps[rev_append]; |
1169 | 391 |
|
4935 | 392 |
Goal "rev(rev l) = l"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
393 |
by (induct_tac "l" 1); |
5316 | 394 |
by Auto_tac; |
1169 | 395 |
qed "rev_rev_ident"; |
2512 | 396 |
Addsimps[rev_rev_ident]; |
1169 | 397 |
|
4935 | 398 |
Goal "(rev xs = []) = (xs = [])"; |
4423 | 399 |
by (induct_tac "xs" 1); |
5316 | 400 |
by Auto_tac; |
3860 | 401 |
qed "rev_is_Nil_conv"; |
402 |
AddIffs [rev_is_Nil_conv]; |
|
403 |
||
4935 | 404 |
Goal "([] = rev xs) = (xs = [])"; |
4423 | 405 |
by (induct_tac "xs" 1); |
5316 | 406 |
by Auto_tac; |
3860 | 407 |
qed "Nil_is_rev_conv"; |
408 |
AddIffs [Nil_is_rev_conv]; |
|
409 |
||
6820 | 410 |
Goal "!ys. (rev xs = rev ys) = (xs = ys)"; |
6831 | 411 |
by (induct_tac "xs" 1); |
6820 | 412 |
by (Force_tac 1); |
6831 | 413 |
by (rtac allI 1); |
414 |
by (exhaust_tac "ys" 1); |
|
6820 | 415 |
by (Asm_simp_tac 1); |
416 |
by (Force_tac 1); |
|
417 |
qed_spec_mp "rev_is_rev_conv"; |
|
418 |
AddIffs [rev_is_rev_conv]; |
|
419 |
||
4935 | 420 |
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs"; |
5132 | 421 |
by (stac (rev_rev_ident RS sym) 1); |
6162 | 422 |
by (res_inst_tac [("list", "rev xs")] list.induct 1); |
5132 | 423 |
by (ALLGOALS Simp_tac); |
424 |
by (resolve_tac prems 1); |
|
425 |
by (eresolve_tac prems 1); |
|
4935 | 426 |
qed "rev_induct"; |
427 |
||
5272 | 428 |
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct; |
429 |
||
4935 | 430 |
Goal "(xs = [] --> P) --> (!ys y. xs = ys@[y] --> P) --> P"; |
5132 | 431 |
by (res_inst_tac [("xs","xs")] rev_induct 1); |
5316 | 432 |
by Auto_tac; |
4935 | 433 |
bind_thm ("rev_exhaust", |
434 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp))))); |
|
435 |
||
2608 | 436 |
|
3465 | 437 |
(** set **) |
1812 | 438 |
|
3467 | 439 |
section "set"; |
440 |
||
7032 | 441 |
Goal "finite (set xs)"; |
442 |
by (induct_tac "xs" 1); |
|
443 |
by Auto_tac; |
|
444 |
qed "finite_set"; |
|
445 |
AddIffs [finite_set]; |
|
5296 | 446 |
|
4935 | 447 |
Goal "set (xs@ys) = (set xs Un set ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
448 |
by (induct_tac "xs" 1); |
5316 | 449 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
450 |
qed "set_append"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
451 |
Addsimps[set_append]; |
1812 | 452 |
|
4935 | 453 |
Goal "set l <= set (x#l)"; |
5316 | 454 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
455 |
qed "set_subset_Cons"; |
1936 | 456 |
|
4935 | 457 |
Goal "(set xs = {}) = (xs = [])"; |
3457 | 458 |
by (induct_tac "xs" 1); |
5316 | 459 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
460 |
qed "set_empty"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
461 |
Addsimps [set_empty]; |
2608 | 462 |
|
4935 | 463 |
Goal "set(rev xs) = set(xs)"; |
3457 | 464 |
by (induct_tac "xs" 1); |
5316 | 465 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
466 |
qed "set_rev"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
467 |
Addsimps [set_rev]; |
2608 | 468 |
|
4935 | 469 |
Goal "set(map f xs) = f``(set xs)"; |
3457 | 470 |
by (induct_tac "xs" 1); |
5316 | 471 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
472 |
qed "set_map"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
473 |
Addsimps [set_map]; |
2608 | 474 |
|
6433 | 475 |
Goal "set(filter P xs) = {x. x : set xs & P x}"; |
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
476 |
by (induct_tac "xs" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
477 |
by Auto_tac; |
6433 | 478 |
qed "set_filter"; |
479 |
Addsimps [set_filter]; |
|
8009 | 480 |
|
6433 | 481 |
Goal "set[i..j(] = {k. i <= k & k < j}"; |
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
482 |
by (induct_tac "j" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
483 |
by Auto_tac; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
484 |
by (arith_tac 1); |
6433 | 485 |
qed "set_upt"; |
486 |
Addsimps [set_upt]; |
|
487 |
||
488 |
Goal "!i < size xs. set(xs[i := x]) <= insert x (set xs)"; |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
489 |
by (induct_tac "xs" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
490 |
by (Simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
491 |
by (asm_simp_tac (simpset() addsplits [nat.split]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
492 |
by (Blast_tac 1); |
6433 | 493 |
qed_spec_mp "set_list_update_subset"; |
4605 | 494 |
|
5272 | 495 |
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)"; |
5318 | 496 |
by (induct_tac "xs" 1); |
497 |
by (Simp_tac 1); |
|
498 |
by (Asm_simp_tac 1); |
|
499 |
by (rtac iffI 1); |
|
500 |
by (blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1); |
|
501 |
by (REPEAT(etac exE 1)); |
|
502 |
by (exhaust_tac "ys" 1); |
|
5316 | 503 |
by Auto_tac; |
5272 | 504 |
qed "in_set_conv_decomp"; |
505 |
||
8009 | 506 |
|
5272 | 507 |
(* eliminate `lists' in favour of `set' *) |
508 |
||
509 |
Goal "(xs : lists A) = (!x : set xs. x : A)"; |
|
5318 | 510 |
by (induct_tac "xs" 1); |
5316 | 511 |
by Auto_tac; |
5272 | 512 |
qed "in_lists_conv_set"; |
513 |
||
514 |
bind_thm("in_listsD",in_lists_conv_set RS iffD1); |
|
515 |
AddSDs [in_listsD]; |
|
516 |
bind_thm("in_listsI",in_lists_conv_set RS iffD2); |
|
517 |
AddSIs [in_listsI]; |
|
1812 | 518 |
|
5518 | 519 |
(** mem **) |
520 |
||
521 |
section "mem"; |
|
522 |
||
523 |
Goal "(x mem xs) = (x: set xs)"; |
|
524 |
by (induct_tac "xs" 1); |
|
525 |
by Auto_tac; |
|
526 |
qed "set_mem_eq"; |
|
527 |
||
528 |
||
923 | 529 |
(** list_all **) |
530 |
||
3467 | 531 |
section "list_all"; |
532 |
||
5518 | 533 |
Goal "list_all P xs = (!x:set xs. P x)"; |
534 |
by (induct_tac "xs" 1); |
|
535 |
by Auto_tac; |
|
536 |
qed "list_all_conv"; |
|
537 |
||
5443
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
538 |
Goal "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
|
539 |
by (induct_tac "xs" 1); |
5316 | 540 |
by Auto_tac; |
2512 | 541 |
qed "list_all_append"; |
542 |
Addsimps [list_all_append]; |
|
923 | 543 |
|
544 |
||
2608 | 545 |
(** filter **) |
923 | 546 |
|
3467 | 547 |
section "filter"; |
548 |
||
4935 | 549 |
Goal "filter P (xs@ys) = filter P xs @ filter P ys"; |
3457 | 550 |
by (induct_tac "xs" 1); |
5316 | 551 |
by Auto_tac; |
2608 | 552 |
qed "filter_append"; |
553 |
Addsimps [filter_append]; |
|
554 |
||
4935 | 555 |
Goal "filter (%x. True) xs = xs"; |
4605 | 556 |
by (induct_tac "xs" 1); |
5316 | 557 |
by Auto_tac; |
4605 | 558 |
qed "filter_True"; |
559 |
Addsimps [filter_True]; |
|
560 |
||
4935 | 561 |
Goal "filter (%x. False) xs = []"; |
4605 | 562 |
by (induct_tac "xs" 1); |
5316 | 563 |
by Auto_tac; |
4605 | 564 |
qed "filter_False"; |
565 |
Addsimps [filter_False]; |
|
566 |
||
4935 | 567 |
Goal "length (filter P xs) <= length xs"; |
3457 | 568 |
by (induct_tac "xs" 1); |
5316 | 569 |
by Auto_tac; |
4605 | 570 |
qed "length_filter"; |
5443
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
571 |
Addsimps[length_filter]; |
2608 | 572 |
|
5443
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
573 |
Goal "set (filter P xs) <= set xs"; |
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
574 |
by Auto_tac; |
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
575 |
qed "filter_is_subset"; |
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
576 |
Addsimps [filter_is_subset]; |
e2459d18ff47
changed constants mem and list_all to mere translations
oheimb
parents:
5427
diff
changeset
|
577 |
|
2608 | 578 |
|
3467 | 579 |
section "concat"; |
580 |
||
4935 | 581 |
Goal "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
|
582 |
by (induct_tac "xs" 1); |
5316 | 583 |
by Auto_tac; |
2608 | 584 |
qed"concat_append"; |
585 |
Addsimps [concat_append]; |
|
2512 | 586 |
|
4935 | 587 |
Goal "(concat xss = []) = (!xs:set xss. xs=[])"; |
4423 | 588 |
by (induct_tac "xss" 1); |
5316 | 589 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
590 |
qed "concat_eq_Nil_conv"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
591 |
AddIffs [concat_eq_Nil_conv]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
592 |
|
4935 | 593 |
Goal "([] = concat xss) = (!xs:set xss. xs=[])"; |
4423 | 594 |
by (induct_tac "xss" 1); |
5316 | 595 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
596 |
qed "Nil_eq_concat_conv"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
597 |
AddIffs [Nil_eq_concat_conv]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
598 |
|
4935 | 599 |
Goal "set(concat xs) = Union(set `` set xs)"; |
3467 | 600 |
by (induct_tac "xs" 1); |
5316 | 601 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
602 |
qed"set_concat"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
603 |
Addsimps [set_concat]; |
3467 | 604 |
|
4935 | 605 |
Goal "map f (concat xs) = concat (map (map f) xs)"; |
3467 | 606 |
by (induct_tac "xs" 1); |
5316 | 607 |
by Auto_tac; |
3467 | 608 |
qed "map_concat"; |
609 |
||
4935 | 610 |
Goal "filter p (concat xs) = concat (map (filter p) xs)"; |
3467 | 611 |
by (induct_tac "xs" 1); |
5316 | 612 |
by Auto_tac; |
3467 | 613 |
qed"filter_concat"; |
614 |
||
4935 | 615 |
Goal "rev(concat xs) = concat (map rev (rev xs))"; |
3467 | 616 |
by (induct_tac "xs" 1); |
5316 | 617 |
by Auto_tac; |
2608 | 618 |
qed "rev_concat"; |
923 | 619 |
|
620 |
(** nth **) |
|
621 |
||
3467 | 622 |
section "nth"; |
623 |
||
6408 | 624 |
Goal "(x#xs)!0 = x"; |
625 |
by Auto_tac; |
|
626 |
qed "nth_Cons_0"; |
|
627 |
Addsimps [nth_Cons_0]; |
|
5644 | 628 |
|
6408 | 629 |
Goal "(x#xs)!(Suc n) = xs!n"; |
630 |
by Auto_tac; |
|
631 |
qed "nth_Cons_Suc"; |
|
632 |
Addsimps [nth_Cons_Suc]; |
|
633 |
||
634 |
Delsimps (thms "nth.simps"); |
|
635 |
||
636 |
Goal "!n. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"; |
|
637 |
by (induct_tac "xs" 1); |
|
3457 | 638 |
by (Asm_simp_tac 1); |
639 |
by (rtac allI 1); |
|
6408 | 640 |
by (exhaust_tac "n" 1); |
5316 | 641 |
by Auto_tac; |
2608 | 642 |
qed_spec_mp "nth_append"; |
643 |
||
4935 | 644 |
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
645 |
by (induct_tac "xs" 1); |
1301 | 646 |
(* case [] *) |
647 |
by (Asm_full_simp_tac 1); |
|
648 |
(* case x#xl *) |
|
649 |
by (rtac allI 1); |
|
5183 | 650 |
by (induct_tac "n" 1); |
5316 | 651 |
by Auto_tac; |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
652 |
qed_spec_mp "nth_map"; |
1301 | 653 |
Addsimps [nth_map]; |
654 |
||
5518 | 655 |
Goal "!n. n < length xs --> Ball (set xs) P --> P(xs!n)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
656 |
by (induct_tac "xs" 1); |
1301 | 657 |
(* case [] *) |
658 |
by (Simp_tac 1); |
|
659 |
(* case x#xl *) |
|
660 |
by (rtac allI 1); |
|
5183 | 661 |
by (induct_tac "n" 1); |
5316 | 662 |
by Auto_tac; |
5518 | 663 |
qed_spec_mp "list_ball_nth"; |
1301 | 664 |
|
5518 | 665 |
Goal "!n. n < length xs --> xs!n : set xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
666 |
by (induct_tac "xs" 1); |
8009 | 667 |
by (Simp_tac 1); |
1301 | 668 |
by (rtac allI 1); |
5183 | 669 |
by (induct_tac "n" 1); |
8009 | 670 |
by (Asm_full_simp_tac 1); |
4686 | 671 |
by (Asm_full_simp_tac 1); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
672 |
qed_spec_mp "nth_mem"; |
1301 | 673 |
Addsimps [nth_mem]; |
674 |
||
5518 | 675 |
|
8009 | 676 |
Goal "(!i. i < length xs --> P(xs!i)) --> (!x : set xs. P x)"; |
677 |
by (induct_tac "xs" 1); |
|
678 |
by (Asm_full_simp_tac 1); |
|
679 |
by (simp_tac (simpset() addsplits [nat.split] addsimps [nth_Cons]) 1); |
|
680 |
by (fast_tac (claset() addss simpset()) 1); |
|
681 |
qed_spec_mp "all_nth_imp_all_set"; |
|
682 |
||
683 |
Goal "(!x : set xs. P x) = (!i. i<length xs --> P (xs ! i))"; |
|
684 |
br iffI 1; |
|
685 |
by (Asm_full_simp_tac 1); |
|
686 |
be all_nth_imp_all_set 1; |
|
687 |
qed_spec_mp "all_set_conv_all_nth"; |
|
688 |
||
689 |
||
5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
690 |
(** list update **) |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
691 |
|
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
692 |
section "list update"; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
693 |
|
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
694 |
Goal "!i. length(xs[i:=x]) = length xs"; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
695 |
by (induct_tac "xs" 1); |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
696 |
by (Simp_tac 1); |
5183 | 697 |
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1); |
5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
698 |
qed_spec_mp "length_list_update"; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
699 |
Addsimps [length_list_update]; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
700 |
|
5644 | 701 |
Goal "!i j. i < length xs --> (xs[i:=x])!j = (if i=j then x else xs!j)"; |
6162 | 702 |
by (induct_tac "xs" 1); |
703 |
by (Simp_tac 1); |
|
704 |
by (auto_tac (claset(), simpset() addsimps [nth_Cons] addsplits [nat.split])); |
|
5644 | 705 |
qed_spec_mp "nth_list_update"; |
706 |
||
6433 | 707 |
Goal "!i. i < size xs --> xs[i:=x, i:=y] = xs[i:=y]"; |
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
708 |
by (induct_tac "xs" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
709 |
by (Simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
710 |
by (asm_simp_tac (simpset() addsplits [nat.split]) 1); |
6433 | 711 |
qed_spec_mp "list_update_overwrite"; |
712 |
Addsimps [list_update_overwrite]; |
|
713 |
||
714 |
Goal "!i < length xs. (xs[i := x] = xs) = (xs!i = x)"; |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
715 |
by (induct_tac "xs" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
716 |
by (Simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
717 |
by (simp_tac (simpset() addsplits [nat.split]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
718 |
by (Blast_tac 1); |
6433 | 719 |
qed_spec_mp "list_update_same_conv"; |
720 |
||
8009 | 721 |
Goal "!i xy xs. length xs = length ys --> \ |
722 |
\ (zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"; |
|
723 |
by (induct_tac "ys" 1); |
|
724 |
by Auto_tac; |
|
725 |
by (exhaust_tac "xs" 1); |
|
726 |
by (auto_tac (claset(), simpset() addsplits [nat.split])); |
|
727 |
qed_spec_mp "update_zip"; |
|
728 |
||
729 |
Goal "!i. set(xs[i:=x]) <= insert x (set xs)"; |
|
730 |
by (induct_tac "xs" 1); |
|
731 |
by (asm_full_simp_tac (simpset() addsimps []) 1); |
|
732 |
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1); |
|
733 |
by (Fast_tac 1); |
|
734 |
qed_spec_mp "set_update_subset"; |
|
735 |
||
5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
736 |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
737 |
(** last & butlast **) |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
738 |
|
5644 | 739 |
section "last / butlast"; |
740 |
||
4935 | 741 |
Goal "last(xs@[x]) = x"; |
4423 | 742 |
by (induct_tac "xs" 1); |
5316 | 743 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
744 |
qed "last_snoc"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
745 |
Addsimps [last_snoc]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
746 |
|
4935 | 747 |
Goal "butlast(xs@[x]) = xs"; |
4423 | 748 |
by (induct_tac "xs" 1); |
5316 | 749 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
750 |
qed "butlast_snoc"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
751 |
Addsimps [butlast_snoc]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
752 |
|
4935 | 753 |
Goal "length(butlast xs) = length xs - 1"; |
754 |
by (res_inst_tac [("xs","xs")] rev_induct 1); |
|
5316 | 755 |
by Auto_tac; |
4643 | 756 |
qed "length_butlast"; |
757 |
Addsimps [length_butlast]; |
|
758 |
||
5278 | 759 |
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)"; |
4423 | 760 |
by (induct_tac "xs" 1); |
5316 | 761 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
762 |
qed_spec_mp "butlast_append"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
763 |
|
4935 | 764 |
Goal "x:set(butlast xs) --> x:set xs"; |
4423 | 765 |
by (induct_tac "xs" 1); |
5316 | 766 |
by Auto_tac; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
767 |
qed_spec_mp "in_set_butlastD"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
768 |
|
5448
40a09282ba14
in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents:
5443
diff
changeset
|
769 |
Goal "x:set(butlast xs) | x:set(butlast ys) ==> x:set(butlast(xs@ys))"; |
40a09282ba14
in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents:
5443
diff
changeset
|
770 |
by (auto_tac (claset() addDs [in_set_butlastD], |
40a09282ba14
in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents:
5443
diff
changeset
|
771 |
simpset() addsimps [butlast_append])); |
40a09282ba14
in_set_butlast_appendI supersedes in_set_butlast_appendI1,2
paulson
parents:
5443
diff
changeset
|
772 |
qed "in_set_butlast_appendI"; |
3902 | 773 |
|
2608 | 774 |
(** take & drop **) |
775 |
section "take & drop"; |
|
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
776 |
|
4935 | 777 |
Goal "take 0 xs = []"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
778 |
by (induct_tac "xs" 1); |
5316 | 779 |
by Auto_tac; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
780 |
qed "take_0"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
781 |
|
4935 | 782 |
Goal "drop 0 xs = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
783 |
by (induct_tac "xs" 1); |
5316 | 784 |
by Auto_tac; |
2608 | 785 |
qed "drop_0"; |
786 |
||
4935 | 787 |
Goal "take (Suc n) (x#xs) = x # take n xs"; |
1552 | 788 |
by (Simp_tac 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
789 |
qed "take_Suc_Cons"; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
790 |
|
4935 | 791 |
Goal "drop (Suc n) (x#xs) = drop n xs"; |
2608 | 792 |
by (Simp_tac 1); |
793 |
qed "drop_Suc_Cons"; |
|
794 |
||
795 |
Delsimps [take_Cons,drop_Cons]; |
|
796 |
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons]; |
|
797 |
||
4935 | 798 |
Goal "!xs. length(take n xs) = min (length xs) n"; |
5183 | 799 |
by (induct_tac "n" 1); |
5316 | 800 |
by Auto_tac; |
3457 | 801 |
by (exhaust_tac "xs" 1); |
5316 | 802 |
by Auto_tac; |
2608 | 803 |
qed_spec_mp "length_take"; |
804 |
Addsimps [length_take]; |
|
923 | 805 |
|
4935 | 806 |
Goal "!xs. length(drop n xs) = (length xs - n)"; |
5183 | 807 |
by (induct_tac "n" 1); |
5316 | 808 |
by Auto_tac; |
3457 | 809 |
by (exhaust_tac "xs" 1); |
5316 | 810 |
by Auto_tac; |
2608 | 811 |
qed_spec_mp "length_drop"; |
812 |
Addsimps [length_drop]; |
|
813 |
||
4935 | 814 |
Goal "!xs. length xs <= n --> take n xs = xs"; |
5183 | 815 |
by (induct_tac "n" 1); |
5316 | 816 |
by Auto_tac; |
3457 | 817 |
by (exhaust_tac "xs" 1); |
5316 | 818 |
by Auto_tac; |
2608 | 819 |
qed_spec_mp "take_all"; |
7246 | 820 |
Addsimps [take_all]; |
923 | 821 |
|
4935 | 822 |
Goal "!xs. length xs <= n --> drop n xs = []"; |
5183 | 823 |
by (induct_tac "n" 1); |
5316 | 824 |
by Auto_tac; |
3457 | 825 |
by (exhaust_tac "xs" 1); |
5316 | 826 |
by Auto_tac; |
2608 | 827 |
qed_spec_mp "drop_all"; |
7246 | 828 |
Addsimps [drop_all]; |
2608 | 829 |
|
5278 | 830 |
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; |
5183 | 831 |
by (induct_tac "n" 1); |
5316 | 832 |
by Auto_tac; |
3457 | 833 |
by (exhaust_tac "xs" 1); |
5316 | 834 |
by Auto_tac; |
2608 | 835 |
qed_spec_mp "take_append"; |
836 |
Addsimps [take_append]; |
|
837 |
||
4935 | 838 |
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; |
5183 | 839 |
by (induct_tac "n" 1); |
5316 | 840 |
by Auto_tac; |
3457 | 841 |
by (exhaust_tac "xs" 1); |
5316 | 842 |
by Auto_tac; |
2608 | 843 |
qed_spec_mp "drop_append"; |
844 |
Addsimps [drop_append]; |
|
845 |
||
4935 | 846 |
Goal "!xs n. take n (take m xs) = take (min n m) xs"; |
5183 | 847 |
by (induct_tac "m" 1); |
5316 | 848 |
by Auto_tac; |
3457 | 849 |
by (exhaust_tac "xs" 1); |
5316 | 850 |
by Auto_tac; |
5183 | 851 |
by (exhaust_tac "na" 1); |
5316 | 852 |
by Auto_tac; |
2608 | 853 |
qed_spec_mp "take_take"; |
7570 | 854 |
Addsimps [take_take]; |
2608 | 855 |
|
4935 | 856 |
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; |
5183 | 857 |
by (induct_tac "m" 1); |
5316 | 858 |
by Auto_tac; |
3457 | 859 |
by (exhaust_tac "xs" 1); |
5316 | 860 |
by Auto_tac; |
2608 | 861 |
qed_spec_mp "drop_drop"; |
7570 | 862 |
Addsimps [drop_drop]; |
923 | 863 |
|
4935 | 864 |
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; |
5183 | 865 |
by (induct_tac "m" 1); |
5316 | 866 |
by Auto_tac; |
3457 | 867 |
by (exhaust_tac "xs" 1); |
5316 | 868 |
by Auto_tac; |
2608 | 869 |
qed_spec_mp "take_drop"; |
870 |
||
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
871 |
Goal "!xs. take n xs @ drop n xs = xs"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
872 |
by (induct_tac "n" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
873 |
by Auto_tac; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
874 |
by (exhaust_tac "xs" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
875 |
by Auto_tac; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
876 |
qed_spec_mp "append_take_drop_id"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
877 |
|
4935 | 878 |
Goal "!xs. take n (map f xs) = map f (take n xs)"; |
5183 | 879 |
by (induct_tac "n" 1); |
5316 | 880 |
by Auto_tac; |
3457 | 881 |
by (exhaust_tac "xs" 1); |
5316 | 882 |
by Auto_tac; |
2608 | 883 |
qed_spec_mp "take_map"; |
884 |
||
4935 | 885 |
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; |
5183 | 886 |
by (induct_tac "n" 1); |
5316 | 887 |
by Auto_tac; |
3457 | 888 |
by (exhaust_tac "xs" 1); |
5316 | 889 |
by Auto_tac; |
2608 | 890 |
qed_spec_mp "drop_map"; |
891 |
||
4935 | 892 |
Goal "!n i. i < n --> (take n xs)!i = xs!i"; |
3457 | 893 |
by (induct_tac "xs" 1); |
5316 | 894 |
by Auto_tac; |
3457 | 895 |
by (exhaust_tac "n" 1); |
896 |
by (Blast_tac 1); |
|
897 |
by (exhaust_tac "i" 1); |
|
5316 | 898 |
by Auto_tac; |
2608 | 899 |
qed_spec_mp "nth_take"; |
900 |
Addsimps [nth_take]; |
|
923 | 901 |
|
4935 | 902 |
Goal "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)"; |
5183 | 903 |
by (induct_tac "n" 1); |
5316 | 904 |
by Auto_tac; |
3457 | 905 |
by (exhaust_tac "xs" 1); |
5316 | 906 |
by Auto_tac; |
2608 | 907 |
qed_spec_mp "nth_drop"; |
908 |
Addsimps [nth_drop]; |
|
909 |
||
910 |
(** takeWhile & dropWhile **) |
|
911 |
||
3467 | 912 |
section "takeWhile & dropWhile"; |
913 |
||
4935 | 914 |
Goal "takeWhile P xs @ dropWhile P xs = xs"; |
3586 | 915 |
by (induct_tac "xs" 1); |
5316 | 916 |
by Auto_tac; |
3586 | 917 |
qed "takeWhile_dropWhile_id"; |
918 |
Addsimps [takeWhile_dropWhile_id]; |
|
919 |
||
4935 | 920 |
Goal "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; |
3457 | 921 |
by (induct_tac "xs" 1); |
5316 | 922 |
by Auto_tac; |
2608 | 923 |
bind_thm("takeWhile_append1", conjI RS (result() RS mp)); |
924 |
Addsimps [takeWhile_append1]; |
|
923 | 925 |
|
4935 | 926 |
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; |
3457 | 927 |
by (induct_tac "xs" 1); |
5316 | 928 |
by Auto_tac; |
2608 | 929 |
bind_thm("takeWhile_append2", ballI RS (result() RS mp)); |
930 |
Addsimps [takeWhile_append2]; |
|
1169 | 931 |
|
4935 | 932 |
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; |
3457 | 933 |
by (induct_tac "xs" 1); |
5316 | 934 |
by Auto_tac; |
2608 | 935 |
bind_thm("dropWhile_append1", conjI RS (result() RS mp)); |
936 |
Addsimps [dropWhile_append1]; |
|
937 |
||
4935 | 938 |
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; |
3457 | 939 |
by (induct_tac "xs" 1); |
5316 | 940 |
by Auto_tac; |
2608 | 941 |
bind_thm("dropWhile_append2", ballI RS (result() RS mp)); |
942 |
Addsimps [dropWhile_append2]; |
|
943 |
||
4935 | 944 |
Goal "x:set(takeWhile P xs) --> x:set xs & P x"; |
3457 | 945 |
by (induct_tac "xs" 1); |
5316 | 946 |
by Auto_tac; |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
947 |
qed_spec_mp"set_take_whileD"; |
2608 | 948 |
|
6306 | 949 |
(** zip **) |
950 |
section "zip"; |
|
951 |
||
952 |
Goal "zip [] ys = []"; |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
953 |
by (induct_tac "ys" 1); |
6306 | 954 |
by Auto_tac; |
955 |
qed "zip_Nil"; |
|
956 |
Addsimps [zip_Nil]; |
|
957 |
||
958 |
Goal "zip (x#xs) (y#ys) = (x,y)#zip xs ys"; |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
959 |
by (Simp_tac 1); |
6306 | 960 |
qed "zip_Cons_Cons"; |
961 |
Addsimps [zip_Cons_Cons]; |
|
962 |
||
963 |
Delsimps(tl (thms"zip.simps")); |
|
4605 | 964 |
|
8009 | 965 |
Goal "!xs. length xs = length ys --> length (zip xs ys) = length ys"; |
966 |
by (induct_tac "ys" 1); |
|
967 |
by (Simp_tac 1); |
|
968 |
by (Clarify_tac 1); |
|
969 |
by (exhaust_tac "xs" 1); |
|
970 |
by(Auto_tac); |
|
971 |
qed_spec_mp "length_zip"; |
|
972 |
Addsimps [length_zip]; |
|
973 |
||
974 |
Goal |
|
975 |
"!xs. length xs = length us --> length ys = length vs --> \ |
|
976 |
\ zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"; |
|
977 |
by(induct_tac "us" 1); |
|
978 |
by(Asm_full_simp_tac 1); |
|
979 |
by(Asm_full_simp_tac 1); |
|
980 |
by(Clarify_tac 1); |
|
981 |
by(exhaust_tac "xs" 1); |
|
982 |
by(Auto_tac); |
|
983 |
qed_spec_mp "zip_append"; |
|
984 |
||
985 |
Goal "!xs. length xs = length ys --> zip (rev xs) (rev ys) = rev (zip xs ys)"; |
|
986 |
by(induct_tac "ys" 1); |
|
987 |
by(Asm_full_simp_tac 1); |
|
988 |
by(Asm_full_simp_tac 1); |
|
989 |
by(Clarify_tac 1); |
|
990 |
by(exhaust_tac "xs" 1); |
|
991 |
by(Auto_tac); |
|
992 |
by (asm_full_simp_tac (simpset() addsimps [zip_append]) 1); |
|
993 |
qed_spec_mp "zip_rev"; |
|
994 |
||
995 |
Goal |
|
996 |
"!i xs. i < length xs --> i < length ys --> (zip xs ys)!i = (xs!i, ys!i)"; |
|
997 |
by (induct_tac "ys" 1); |
|
998 |
by (Simp_tac 1); |
|
999 |
by (Clarify_tac 1); |
|
1000 |
by(exhaust_tac "xs" 1); |
|
1001 |
by(Auto_tac); |
|
1002 |
by (asm_full_simp_tac (simpset() addsimps (thms"nth.simps") addsplits [nat.split]) 1); |
|
1003 |
qed_spec_mp "nth_zip"; |
|
1004 |
Addsimps [nth_zip]; |
|
1005 |
||
1006 |
Goal |
|
1007 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"; |
|
1008 |
by(rtac sym 1); |
|
1009 |
by(asm_simp_tac (simpset() addsimps [update_zip]) 1); |
|
1010 |
qed_spec_mp "zip_update"; |
|
1011 |
||
1012 |
Goal "!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"; |
|
1013 |
by (induct_tac "i" 1); |
|
1014 |
by(Auto_tac); |
|
1015 |
by (exhaust_tac "j" 1); |
|
1016 |
by(Auto_tac); |
|
1017 |
qed "zip_replicate"; |
|
1018 |
Addsimps [zip_replicate]; |
|
1019 |
||
5272 | 1020 |
|
1021 |
(** foldl **) |
|
1022 |
section "foldl"; |
|
1023 |
||
1024 |
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"; |
|
5318 | 1025 |
by (induct_tac "xs" 1); |
5316 | 1026 |
by Auto_tac; |
5272 | 1027 |
qed_spec_mp "foldl_append"; |
1028 |
Addsimps [foldl_append]; |
|
1029 |
||
1030 |
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use |
|
1031 |
because it requires an additional transitivity step |
|
1032 |
*) |
|
1033 |
Goal "!n::nat. m <= n --> m <= foldl op+ n ns"; |
|
5318 | 1034 |
by (induct_tac "ns" 1); |
6058 | 1035 |
by Auto_tac; |
5272 | 1036 |
qed_spec_mp "start_le_sum"; |
1037 |
||
1038 |
Goal "n : set ns ==> n <= foldl op+ 0 ns"; |
|
5758
27a2b36efd95
corrected auto_tac (applications of unsafe wrappers)
oheimb
parents:
5644
diff
changeset
|
1039 |
by (force_tac (claset() addIs [start_le_sum], |
27a2b36efd95
corrected auto_tac (applications of unsafe wrappers)
oheimb
parents:
5644
diff
changeset
|
1040 |
simpset() addsimps [in_set_conv_decomp]) 1); |
5272 | 1041 |
qed "elem_le_sum"; |
1042 |
||
1043 |
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))"; |
|
5318 | 1044 |
by (induct_tac "ns" 1); |
5316 | 1045 |
by Auto_tac; |
5272 | 1046 |
qed_spec_mp "sum_eq_0_conv"; |
1047 |
AddIffs [sum_eq_0_conv]; |
|
1048 |
||
5425 | 1049 |
(** upto **) |
1050 |
||
5427 | 1051 |
(* Does not terminate! *) |
1052 |
Goal "[i..j(] = (if i<j then i#[Suc i..j(] else [])"; |
|
6162 | 1053 |
by (induct_tac "j" 1); |
5427 | 1054 |
by Auto_tac; |
1055 |
qed "upt_rec"; |
|
5425 | 1056 |
|
5427 | 1057 |
Goal "j<=i ==> [i..j(] = []"; |
6162 | 1058 |
by (stac upt_rec 1); |
1059 |
by (Asm_simp_tac 1); |
|
5427 | 1060 |
qed "upt_conv_Nil"; |
1061 |
Addsimps [upt_conv_Nil]; |
|
1062 |
||
1063 |
Goal "i<=j ==> [i..(Suc j)(] = [i..j(]@[j]"; |
|
1064 |
by (Asm_simp_tac 1); |
|
1065 |
qed "upt_Suc"; |
|
1066 |
||
1067 |
Goal "i<j ==> [i..j(] = i#[Suc i..j(]"; |
|
6162 | 1068 |
by (rtac trans 1); |
1069 |
by (stac upt_rec 1); |
|
1070 |
by (rtac refl 2); |
|
5427 | 1071 |
by (Asm_simp_tac 1); |
1072 |
qed "upt_conv_Cons"; |
|
1073 |
||
1074 |
Goal "length [i..j(] = j-i"; |
|
6162 | 1075 |
by (induct_tac "j" 1); |
5427 | 1076 |
by (Simp_tac 1); |
6162 | 1077 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1); |
5427 | 1078 |
qed "length_upt"; |
1079 |
Addsimps [length_upt]; |
|
5425 | 1080 |
|
5427 | 1081 |
Goal "i+k < j --> [i..j(] ! k = i+k"; |
6162 | 1082 |
by (induct_tac "j" 1); |
1083 |
by (Simp_tac 1); |
|
1084 |
by (asm_simp_tac (simpset() addsimps [nth_append,less_diff_conv]@add_ac) 1); |
|
1085 |
by (Clarify_tac 1); |
|
1086 |
by (subgoal_tac "n=i+k" 1); |
|
1087 |
by (Asm_simp_tac 2); |
|
1088 |
by (Asm_simp_tac 1); |
|
5427 | 1089 |
qed_spec_mp "nth_upt"; |
1090 |
Addsimps [nth_upt]; |
|
5425 | 1091 |
|
6433 | 1092 |
Goal "!i. i+m <= n --> take m [i..n(] = [i..i+m(]"; |
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1093 |
by (induct_tac "m" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1094 |
by (Simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1095 |
by (Clarify_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1096 |
by (stac upt_rec 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1097 |
by (rtac sym 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1098 |
by (stac upt_rec 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1099 |
by (asm_simp_tac (simpset() delsimps (thms"upt.simps")) 1); |
6433 | 1100 |
qed_spec_mp "take_upt"; |
1101 |
Addsimps [take_upt]; |
|
1102 |
||
1103 |
Goal "!m i. i < n-m --> (map f [m..n(]) ! i = f(m+i)"; |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1104 |
by (induct_tac "n" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1105 |
by (Simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1106 |
by (Clarify_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1107 |
by (subgoal_tac "m < Suc n" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1108 |
by (arith_tac 2); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1109 |
by (stac upt_rec 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1110 |
by (asm_simp_tac (simpset() delsplits [split_if]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1111 |
by (split_tac [split_if] 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1112 |
by (rtac conjI 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1113 |
by (simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1114 |
by (simp_tac (simpset() addsimps [nth_append] addsplits [nat.split]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1115 |
by (Clarify_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1116 |
by (rtac conjI 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1117 |
by (Clarify_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1118 |
by (subgoal_tac "Suc(m+nat) < n" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1119 |
by (arith_tac 2); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1120 |
by (Asm_simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1121 |
by (Clarify_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1122 |
by (subgoal_tac "n = Suc(m+nat)" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1123 |
by (arith_tac 2); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1124 |
by (Asm_simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1125 |
by (simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1126 |
by (arith_tac 1); |
6433 | 1127 |
qed_spec_mp "nth_map_upt"; |
1128 |
||
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1129 |
Goal "ALL xs ys. k <= length xs --> k <= length ys --> \ |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1130 |
\ (ALL i. i < k --> xs!i = ys!i) \ |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1131 |
\ --> take k xs = take k ys"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1132 |
by (induct_tac "k" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1133 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq_0_disj, |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1134 |
all_conj_distrib]))); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1135 |
by (Clarify_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1136 |
(*Both lists must be non-empty*) |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1137 |
by (exhaust_tac "xs" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1138 |
by (exhaust_tac "ys" 2); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1139 |
by (ALLGOALS Clarify_tac); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1140 |
(*prenexing's needed, not miniscoping*) |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1141 |
by (ALLGOALS (full_simp_tac (simpset() addsimps (all_simps RL [sym]) |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1142 |
delsimps (all_simps)))); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1143 |
by (Blast_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1144 |
qed_spec_mp "nth_take_lemma"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1145 |
|
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1146 |
Goal "[| length xs = length ys; \ |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1147 |
\ ALL i. i < length xs --> xs!i = ys!i |] \ |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1148 |
\ ==> xs = ys"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1149 |
by (forward_tac [[le_refl, eq_imp_le] MRS nth_take_lemma] 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1150 |
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [take_all]))); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1151 |
qed_spec_mp "nth_equalityI"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1152 |
|
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1153 |
(*The famous take-lemma*) |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1154 |
Goal "(ALL i. take i xs = take i ys) ==> xs = ys"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1155 |
by (dres_inst_tac [("x", "max (length xs) (length ys)")] spec 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1156 |
by (full_simp_tac (simpset() addsimps [le_max_iff_disj, take_all]) 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1157 |
qed_spec_mp "take_equalityI"; |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1158 |
|
5272 | 1159 |
|
4605 | 1160 |
(** nodups & remdups **) |
1161 |
section "nodups & remdups"; |
|
1162 |
||
4935 | 1163 |
Goal "set(remdups xs) = set xs"; |
4605 | 1164 |
by (induct_tac "xs" 1); |
1165 |
by (Simp_tac 1); |
|
4686 | 1166 |
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1); |
4605 | 1167 |
qed "set_remdups"; |
1168 |
Addsimps [set_remdups]; |
|
1169 |
||
4935 | 1170 |
Goal "nodups(remdups xs)"; |
4605 | 1171 |
by (induct_tac "xs" 1); |
5316 | 1172 |
by Auto_tac; |
4605 | 1173 |
qed "nodups_remdups"; |
1174 |
||
4935 | 1175 |
Goal "nodups xs --> nodups (filter P xs)"; |
4605 | 1176 |
by (induct_tac "xs" 1); |
5316 | 1177 |
by Auto_tac; |
4605 | 1178 |
qed_spec_mp "nodups_filter"; |
1179 |
||
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1180 |
(** replicate **) |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1181 |
section "replicate"; |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1182 |
|
6794 | 1183 |
Goal "length(replicate n x) = n"; |
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1184 |
by (induct_tac "n" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1185 |
by Auto_tac; |
6794 | 1186 |
qed "length_replicate"; |
1187 |
Addsimps [length_replicate]; |
|
1188 |
||
1189 |
Goal "map f (replicate n x) = replicate n (f x)"; |
|
1190 |
by (induct_tac "n" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1191 |
by Auto_tac; |
6794 | 1192 |
qed "map_replicate"; |
1193 |
Addsimps [map_replicate]; |
|
1194 |
||
1195 |
Goal "(replicate n x) @ (x#xs) = x # replicate n x @ xs"; |
|
1196 |
by (induct_tac "n" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1197 |
by Auto_tac; |
6794 | 1198 |
qed "replicate_app_Cons_same"; |
1199 |
||
1200 |
Goal "rev(replicate n x) = replicate n x"; |
|
1201 |
by (induct_tac "n" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1202 |
by (Simp_tac 1); |
6794 | 1203 |
by (asm_simp_tac (simpset() addsimps [replicate_app_Cons_same]) 1); |
1204 |
qed "rev_replicate"; |
|
1205 |
Addsimps [rev_replicate]; |
|
1206 |
||
8009 | 1207 |
Goal "replicate (n+m) x = replicate n x @ replicate m x"; |
1208 |
by (induct_tac "n" 1); |
|
1209 |
by Auto_tac; |
|
1210 |
qed "replicate_add"; |
|
1211 |
||
6794 | 1212 |
Goal"n ~= 0 --> hd(replicate n x) = x"; |
1213 |
by (induct_tac "n" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1214 |
by Auto_tac; |
6794 | 1215 |
qed_spec_mp "hd_replicate"; |
1216 |
Addsimps [hd_replicate]; |
|
1217 |
||
1218 |
Goal "n ~= 0 --> tl(replicate n x) = replicate (n-1) x"; |
|
1219 |
by (induct_tac "n" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1220 |
by Auto_tac; |
6794 | 1221 |
qed_spec_mp "tl_replicate"; |
1222 |
Addsimps [tl_replicate]; |
|
1223 |
||
1224 |
Goal "n ~= 0 --> last(replicate n x) = x"; |
|
1225 |
by (induct_tac "n" 1); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1226 |
by Auto_tac; |
6794 | 1227 |
qed_spec_mp "last_replicate"; |
1228 |
Addsimps [last_replicate]; |
|
1229 |
||
1230 |
Goal "!i. i<n --> (replicate n x)!i = x"; |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1231 |
by (induct_tac "n" 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1232 |
by (Simp_tac 1); |
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1233 |
by (asm_simp_tac (simpset() addsimps [nth_Cons] addsplits [nat.split]) 1); |
6794 | 1234 |
qed_spec_mp "nth_replicate"; |
1235 |
Addsimps [nth_replicate]; |
|
1236 |
||
4935 | 1237 |
Goal "set(replicate (Suc n) x) = {x}"; |
4423 | 1238 |
by (induct_tac "n" 1); |
5316 | 1239 |
by Auto_tac; |
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1240 |
val lemma = result(); |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1241 |
|
5043 | 1242 |
Goal "n ~= 0 ==> set(replicate n x) = {x}"; |
4423 | 1243 |
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1); |
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1244 |
qed "set_replicate"; |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
1245 |
Addsimps [set_replicate]; |
5162 | 1246 |
|
8009 | 1247 |
Goal "set(replicate n x) = (if n=0 then {} else {x})"; |
1248 |
by(Auto_tac); |
|
1249 |
qed "set_replicate_conv_if"; |
|
1250 |
||
1251 |
Goal "x : set(replicate n y) --> x=y"; |
|
1252 |
by(asm_simp_tac (simpset() addsimps [set_replicate_conv_if]) 1); |
|
1253 |
qed_spec_mp "in_set_replicateD"; |
|
1254 |
||
5162 | 1255 |
|
5281 | 1256 |
(*** Lexcicographic orderings on lists ***) |
1257 |
section"Lexcicographic orderings on lists"; |
|
1258 |
||
1259 |
Goal "wf r ==> wf(lexn r n)"; |
|
5318 | 1260 |
by (induct_tac "n" 1); |
1261 |
by (Simp_tac 1); |
|
1262 |
by (Simp_tac 1); |
|
1263 |
by (rtac wf_subset 1); |
|
1264 |
by (rtac Int_lower1 2); |
|
1265 |
by (rtac wf_prod_fun_image 1); |
|
1266 |
by (rtac injI 2); |
|
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1267 |
by Auto_tac; |
5281 | 1268 |
qed "wf_lexn"; |
1269 |
||
1270 |
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n"; |
|
5318 | 1271 |
by (induct_tac "n" 1); |
6813
bf90f86502b2
many new lemmas about take & drop, incl the famous take-lemma
paulson
parents:
6794
diff
changeset
|
1272 |
by Auto_tac; |
5281 | 1273 |
qed_spec_mp "lexn_length"; |
1274 |
||
1275 |
Goalw [lex_def] "wf r ==> wf(lex r)"; |
|
5318 | 1276 |
by (rtac wf_UN 1); |
1277 |
by (blast_tac (claset() addIs [wf_lexn]) 1); |
|
1278 |
by (Clarify_tac 1); |
|
1279 |
by (rename_tac "m n" 1); |
|
1280 |
by (subgoal_tac "m ~= n" 1); |
|
1281 |
by (Blast_tac 2); |
|
1282 |
by (blast_tac (claset() addDs [lexn_length,not_sym]) 1); |
|
5281 | 1283 |
qed "wf_lex"; |
1284 |
AddSIs [wf_lex]; |
|
1285 |
||
1286 |
Goal |
|
1287 |
"lexn r n = \ |
|
1288 |
\ {(xs,ys). length xs = n & length ys = n & \ |
|
1289 |
\ (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"; |
|
5318 | 1290 |
by (induct_tac "n" 1); |
1291 |
by (Simp_tac 1); |
|
1292 |
by (Blast_tac 1); |
|
5641 | 1293 |
by (asm_full_simp_tac (simpset() |
5296 | 1294 |
addsimps [lex_prod_def]) 1); |
5641 | 1295 |
by (auto_tac (claset(), simpset())); |
5318 | 1296 |
by (Blast_tac 1); |
1297 |
by (rename_tac "a xys x xs' y ys'" 1); |
|
1298 |
by (res_inst_tac [("x","a#xys")] exI 1); |
|
1299 |
by (Simp_tac 1); |
|
1300 |
by (exhaust_tac "xys" 1); |
|
5641 | 1301 |
by (ALLGOALS (asm_full_simp_tac (simpset()))); |
5318 | 1302 |
by (Blast_tac 1); |
5281 | 1303 |
qed "lexn_conv"; |
1304 |
||
1305 |
Goalw [lex_def] |
|
1306 |
"lex r = \ |
|
1307 |
\ {(xs,ys). length xs = length ys & \ |
|
1308 |
\ (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"; |
|
5641 | 1309 |
by (force_tac (claset(), simpset() addsimps [lexn_conv]) 1); |
5281 | 1310 |
qed "lex_conv"; |
1311 |
||
1312 |
Goalw [lexico_def] "wf r ==> wf(lexico r)"; |
|
5318 | 1313 |
by (Blast_tac 1); |
5281 | 1314 |
qed "wf_lexico"; |
1315 |
AddSIs [wf_lexico]; |
|
1316 |
||
1317 |
Goalw |
|
1318 |
[lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def] |
|
1319 |
"lexico r = {(xs,ys). length xs < length ys | \ |
|
1320 |
\ length xs = length ys & (xs,ys) : lex r}"; |
|
5318 | 1321 |
by (Simp_tac 1); |
5281 | 1322 |
qed "lexico_conv"; |
1323 |
||
5283 | 1324 |
Goal "([],ys) ~: lex r"; |
5318 | 1325 |
by (simp_tac (simpset() addsimps [lex_conv]) 1); |
5283 | 1326 |
qed "Nil_notin_lex"; |
1327 |
||
1328 |
Goal "(xs,[]) ~: lex r"; |
|
5318 | 1329 |
by (simp_tac (simpset() addsimps [lex_conv]) 1); |
5283 | 1330 |
qed "Nil2_notin_lex"; |
1331 |
||
1332 |
AddIffs [Nil_notin_lex,Nil2_notin_lex]; |
|
1333 |
||
1334 |
Goal "((x#xs,y#ys) : lex r) = \ |
|
1335 |
\ ((x,y) : r & length xs = length ys | x=y & (xs,ys) : lex r)"; |
|
5318 | 1336 |
by (simp_tac (simpset() addsimps [lex_conv]) 1); |
1337 |
by (rtac iffI 1); |
|
1338 |
by (blast_tac (claset() addIs [Cons_eq_appendI]) 2); |
|
1339 |
by (REPEAT(eresolve_tac [conjE, exE] 1)); |
|
1340 |
by (exhaust_tac "xys" 1); |
|
1341 |
by (Asm_full_simp_tac 1); |
|
1342 |
by (Asm_full_simp_tac 1); |
|
1343 |
by (Blast_tac 1); |
|
5283 | 1344 |
qed "Cons_in_lex"; |
1345 |
AddIffs [Cons_in_lex]; |
|
7032 | 1346 |
|
1347 |
||
1348 |
(*** Versions of some theorems above using binary numerals ***) |
|
1349 |
||
1350 |
AddIffs (map (rename_numerals thy) |
|
1351 |
[length_0_conv, zero_length_conv, length_greater_0_conv, |
|
1352 |
sum_eq_0_conv]); |
|
1353 |
||
1354 |
Goal "take n (x#xs) = (if n = #0 then [] else x # take (n-#1) xs)"; |
|
1355 |
by (exhaust_tac "n" 1); |
|
1356 |
by (ALLGOALS |
|
1357 |
(asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]))); |
|
1358 |
qed "take_Cons'"; |
|
1359 |
||
1360 |
Goal "drop n (x#xs) = (if n = #0 then x#xs else drop (n-#1) xs)"; |
|
1361 |
by (exhaust_tac "n" 1); |
|
1362 |
by (ALLGOALS |
|
1363 |
(asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]))); |
|
1364 |
qed "drop_Cons'"; |
|
1365 |
||
1366 |
Goal "(x#xs)!n = (if n = #0 then x else xs!(n-#1))"; |
|
1367 |
by (exhaust_tac "n" 1); |
|
1368 |
by (ALLGOALS |
|
1369 |
(asm_simp_tac (simpset() addsimps [numeral_0_eq_0, numeral_1_eq_1]))); |
|
1370 |
qed "nth_Cons'"; |
|
1371 |
||
1372 |
Addsimps (map (inst "n" "number_of ?v") [take_Cons', drop_Cons', nth_Cons']); |
|
1373 |