author | blanchet |
Tue, 14 Sep 2010 13:24:18 +0200 | |
changeset 39359 | 6f49c7fbb1b1 |
parent 39302 | d7728f65b353 |
child 39379 | ab1b070aa412 |
permissions | -rw-r--r-- |
3981 | 1 |
(* Title: HOL/Map.thy |
2 |
Author: Tobias Nipkow, based on a theory by David von Oheimb |
|
13908 | 3 |
Copyright 1997-2003 TU Muenchen |
3981 | 4 |
|
5 |
The datatype of `maps' (written ~=>); strongly resembles maps in VDM. |
|
6 |
*) |
|
7 |
||
13914 | 8 |
header {* Maps *} |
9 |
||
15131 | 10 |
theory Map |
15140 | 11 |
imports List |
15131 | 12 |
begin |
3981 | 13 |
|
35565 | 14 |
types ('a,'b) "map" = "'a => 'b option" (infixr "~=>" 0) |
35427 | 15 |
translations (type) "'a ~=> 'b" <= (type) "'a => 'b option" |
3981 | 16 |
|
35427 | 17 |
type_notation (xsymbols) |
35565 | 18 |
"map" (infixr "\<rightharpoonup>" 0) |
19656
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19378
diff
changeset
|
19 |
|
19378 | 20 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
21 |
empty :: "'a ~=> 'b" where |
19378 | 22 |
"empty == %x. None" |
23 |
||
19656
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19378
diff
changeset
|
24 |
definition |
25670 | 25 |
map_comp :: "('b ~=> 'c) => ('a ~=> 'b) => ('a ~=> 'c)" (infixl "o'_m" 55) where |
20800 | 26 |
"f o_m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)" |
19378 | 27 |
|
21210 | 28 |
notation (xsymbols) |
19656
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19378
diff
changeset
|
29 |
map_comp (infixl "\<circ>\<^sub>m" 55) |
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19378
diff
changeset
|
30 |
|
20800 | 31 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
32 |
map_add :: "('a ~=> 'b) => ('a ~=> 'b) => ('a ~=> 'b)" (infixl "++" 100) where |
20800 | 33 |
"m1 ++ m2 = (\<lambda>x. case m2 x of None => m1 x | Some y => Some y)" |
34 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
35 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
36 |
restrict_map :: "('a ~=> 'b) => 'a set => ('a ~=> 'b)" (infixl "|`" 110) where |
20800 | 37 |
"m|`A = (\<lambda>x. if x : A then m x else None)" |
13910 | 38 |
|
21210 | 39 |
notation (latex output) |
19656
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19378
diff
changeset
|
40 |
restrict_map ("_\<restriction>\<^bsub>_\<^esub>" [111,110] 110) |
09be06943252
tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents:
19378
diff
changeset
|
41 |
|
20800 | 42 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
43 |
dom :: "('a ~=> 'b) => 'a set" where |
20800 | 44 |
"dom m = {a. m a ~= None}" |
45 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
46 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
47 |
ran :: "('a ~=> 'b) => 'b set" where |
20800 | 48 |
"ran m = {b. EX a. m a = Some b}" |
49 |
||
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
50 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
51 |
map_le :: "('a ~=> 'b) => ('a ~=> 'b) => bool" (infix "\<subseteq>\<^sub>m" 50) where |
20800 | 52 |
"(m\<^isub>1 \<subseteq>\<^sub>m m\<^isub>2) = (\<forall>a \<in> dom m\<^isub>1. m\<^isub>1 a = m\<^isub>2 a)" |
53 |
||
14180 | 54 |
nonterminals |
55 |
maplets maplet |
|
56 |
||
5300 | 57 |
syntax |
14180 | 58 |
"_maplet" :: "['a, 'a] => maplet" ("_ /|->/ _") |
59 |
"_maplets" :: "['a, 'a] => maplet" ("_ /[|->]/ _") |
|
60 |
"" :: "maplet => maplets" ("_") |
|
61 |
"_Maplets" :: "[maplet, maplets] => maplets" ("_,/ _") |
|
62 |
"_MapUpd" :: "['a ~=> 'b, maplets] => 'a ~=> 'b" ("_/'(_')" [900,0]900) |
|
63 |
"_Map" :: "maplets => 'a ~=> 'b" ("(1[_])") |
|
3981 | 64 |
|
12114
a8e860c86252
eliminated old "symbols" syntax, use "xsymbols" instead;
wenzelm
parents:
10137
diff
changeset
|
65 |
syntax (xsymbols) |
14180 | 66 |
"_maplet" :: "['a, 'a] => maplet" ("_ /\<mapsto>/ _") |
67 |
"_maplets" :: "['a, 'a] => maplet" ("_ /[\<mapsto>]/ _") |
|
68 |
||
5300 | 69 |
translations |
14180 | 70 |
"_MapUpd m (_Maplets xy ms)" == "_MapUpd (_MapUpd m xy) ms" |
35115 | 71 |
"_MapUpd m (_maplet x y)" == "m(x := CONST Some y)" |
19947 | 72 |
"_Map ms" == "_MapUpd (CONST empty) ms" |
14180 | 73 |
"_Map (_Maplets ms1 ms2)" <= "_MapUpd (_Map ms1) ms2" |
74 |
"_Maplets ms1 (_Maplets ms2 ms3)" <= "_Maplets (_Maplets ms1 ms2) ms3" |
|
75 |
||
5183 | 76 |
primrec |
34941 | 77 |
map_of :: "('a \<times> 'b) list \<Rightarrow> 'a \<rightharpoonup> 'b" where |
78 |
"map_of [] = empty" |
|
79 |
| "map_of (p # ps) = (map_of ps)(fst p \<mapsto> snd p)" |
|
5300 | 80 |
|
34941 | 81 |
definition |
82 |
map_upds :: "('a \<rightharpoonup> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'a \<rightharpoonup> 'b" where |
|
83 |
"map_upds m xs ys = m ++ map_of (rev (zip xs ys))" |
|
84 |
||
85 |
translations |
|
86 |
"_MapUpd m (_maplets x y)" == "CONST map_upds m x y" |
|
25965 | 87 |
|
88 |
lemma map_of_Cons_code [code]: |
|
89 |
"map_of [] k = None" |
|
90 |
"map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)" |
|
91 |
by simp_all |
|
92 |
||
20800 | 93 |
|
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
94 |
subsection {* @{term [source] empty} *} |
13908 | 95 |
|
20800 | 96 |
lemma empty_upd_none [simp]: "empty(x := None) = empty" |
24331 | 97 |
by (rule ext) simp |
13908 | 98 |
|
99 |
||
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
100 |
subsection {* @{term [source] map_upd} *} |
13908 | 101 |
|
102 |
lemma map_upd_triv: "t k = Some x ==> t(k|->x) = t" |
|
24331 | 103 |
by (rule ext) simp |
13908 | 104 |
|
20800 | 105 |
lemma map_upd_nonempty [simp]: "t(k|->x) ~= empty" |
106 |
proof |
|
107 |
assume "t(k \<mapsto> x) = empty" |
|
108 |
then have "(t(k \<mapsto> x)) k = None" by simp |
|
109 |
then show False by simp |
|
110 |
qed |
|
13908 | 111 |
|
20800 | 112 |
lemma map_upd_eqD1: |
113 |
assumes "m(a\<mapsto>x) = n(a\<mapsto>y)" |
|
114 |
shows "x = y" |
|
115 |
proof - |
|
116 |
from prems have "(m(a\<mapsto>x)) a = (n(a\<mapsto>y)) a" by simp |
|
117 |
then show ?thesis by simp |
|
118 |
qed |
|
14100 | 119 |
|
20800 | 120 |
lemma map_upd_Some_unfold: |
24331 | 121 |
"((m(a|->b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)" |
122 |
by auto |
|
14100 | 123 |
|
20800 | 124 |
lemma image_map_upd [simp]: "x \<notin> A \<Longrightarrow> m(x \<mapsto> y) ` A = m ` A" |
24331 | 125 |
by auto |
15303 | 126 |
|
13908 | 127 |
lemma finite_range_updI: "finite (range f) ==> finite (range (f(a|->b)))" |
24331 | 128 |
unfolding image_def |
129 |
apply (simp (no_asm_use) add:full_SetCompr_eq) |
|
130 |
apply (rule finite_subset) |
|
131 |
prefer 2 apply assumption |
|
132 |
apply (auto) |
|
133 |
done |
|
13908 | 134 |
|
135 |
||
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
136 |
subsection {* @{term [source] map_of} *} |
13908 | 137 |
|
15304 | 138 |
lemma map_of_eq_None_iff: |
24331 | 139 |
"(map_of xys x = None) = (x \<notin> fst ` (set xys))" |
140 |
by (induct xys) simp_all |
|
15304 | 141 |
|
24331 | 142 |
lemma map_of_is_SomeD: "map_of xys x = Some y \<Longrightarrow> (x,y) \<in> set xys" |
143 |
apply (induct xys) |
|
144 |
apply simp |
|
145 |
apply (clarsimp split: if_splits) |
|
146 |
done |
|
15304 | 147 |
|
20800 | 148 |
lemma map_of_eq_Some_iff [simp]: |
24331 | 149 |
"distinct(map fst xys) \<Longrightarrow> (map_of xys x = Some y) = ((x,y) \<in> set xys)" |
150 |
apply (induct xys) |
|
151 |
apply simp |
|
152 |
apply (auto simp: map_of_eq_None_iff [symmetric]) |
|
153 |
done |
|
15304 | 154 |
|
20800 | 155 |
lemma Some_eq_map_of_iff [simp]: |
24331 | 156 |
"distinct(map fst xys) \<Longrightarrow> (Some y = map_of xys x) = ((x,y) \<in> set xys)" |
157 |
by (auto simp del:map_of_eq_Some_iff simp add: map_of_eq_Some_iff [symmetric]) |
|
15304 | 158 |
|
17724 | 159 |
lemma map_of_is_SomeI [simp]: "\<lbrakk> distinct(map fst xys); (x,y) \<in> set xys \<rbrakk> |
20800 | 160 |
\<Longrightarrow> map_of xys x = Some y" |
24331 | 161 |
apply (induct xys) |
162 |
apply simp |
|
163 |
apply force |
|
164 |
done |
|
15304 | 165 |
|
20800 | 166 |
lemma map_of_zip_is_None [simp]: |
24331 | 167 |
"length xs = length ys \<Longrightarrow> (map_of (zip xs ys) x = None) = (x \<notin> set xs)" |
168 |
by (induct rule: list_induct2) simp_all |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
169 |
|
26443 | 170 |
lemma map_of_zip_is_Some: |
171 |
assumes "length xs = length ys" |
|
172 |
shows "x \<in> set xs \<longleftrightarrow> (\<exists>y. map_of (zip xs ys) x = Some y)" |
|
173 |
using assms by (induct rule: list_induct2) simp_all |
|
174 |
||
175 |
lemma map_of_zip_upd: |
|
176 |
fixes x :: 'a and xs :: "'a list" and ys zs :: "'b list" |
|
177 |
assumes "length ys = length xs" |
|
178 |
and "length zs = length xs" |
|
179 |
and "x \<notin> set xs" |
|
180 |
and "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" |
|
181 |
shows "map_of (zip xs ys) = map_of (zip xs zs)" |
|
182 |
proof |
|
183 |
fix x' :: 'a |
|
184 |
show "map_of (zip xs ys) x' = map_of (zip xs zs) x'" |
|
185 |
proof (cases "x = x'") |
|
186 |
case True |
|
187 |
from assms True map_of_zip_is_None [of xs ys x'] |
|
188 |
have "map_of (zip xs ys) x' = None" by simp |
|
189 |
moreover from assms True map_of_zip_is_None [of xs zs x'] |
|
190 |
have "map_of (zip xs zs) x' = None" by simp |
|
191 |
ultimately show ?thesis by simp |
|
192 |
next |
|
193 |
case False from assms |
|
194 |
have "(map_of (zip xs ys)(x \<mapsto> y)) x' = (map_of (zip xs zs)(x \<mapsto> z)) x'" by auto |
|
195 |
with False show ?thesis by simp |
|
196 |
qed |
|
197 |
qed |
|
198 |
||
199 |
lemma map_of_zip_inject: |
|
200 |
assumes "length ys = length xs" |
|
201 |
and "length zs = length xs" |
|
202 |
and dist: "distinct xs" |
|
203 |
and map_of: "map_of (zip xs ys) = map_of (zip xs zs)" |
|
204 |
shows "ys = zs" |
|
205 |
using assms(1) assms(2)[symmetric] using dist map_of proof (induct ys xs zs rule: list_induct3) |
|
206 |
case Nil show ?case by simp |
|
207 |
next |
|
208 |
case (Cons y ys x xs z zs) |
|
209 |
from `map_of (zip (x#xs) (y#ys)) = map_of (zip (x#xs) (z#zs))` |
|
210 |
have map_of: "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)" by simp |
|
211 |
from Cons have "length ys = length xs" and "length zs = length xs" |
|
212 |
and "x \<notin> set xs" by simp_all |
|
213 |
then have "map_of (zip xs ys) = map_of (zip xs zs)" using map_of by (rule map_of_zip_upd) |
|
214 |
with Cons.hyps `distinct (x # xs)` have "ys = zs" by simp |
|
215 |
moreover from map_of have "y = z" by (rule map_upd_eqD1) |
|
216 |
ultimately show ?case by simp |
|
217 |
qed |
|
218 |
||
33635 | 219 |
lemma map_of_zip_map: |
220 |
"map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
221 |
by (induct xs) (simp_all add: fun_eq_iff) |
33635 | 222 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
223 |
lemma finite_range_map_of: "finite (range (map_of xys))" |
24331 | 224 |
apply (induct xys) |
225 |
apply (simp_all add: image_constant) |
|
226 |
apply (rule finite_subset) |
|
227 |
prefer 2 apply assumption |
|
228 |
apply auto |
|
229 |
done |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
230 |
|
20800 | 231 |
lemma map_of_SomeD: "map_of xs k = Some y \<Longrightarrow> (k, y) \<in> set xs" |
24331 | 232 |
by (induct xs) (simp, atomize (full), auto) |
13908 | 233 |
|
20800 | 234 |
lemma map_of_mapk_SomeI: |
24331 | 235 |
"inj f ==> map_of t k = Some x ==> |
236 |
map_of (map (split (%k. Pair (f k))) t) (f k) = Some x" |
|
237 |
by (induct t) (auto simp add: inj_eq) |
|
13908 | 238 |
|
20800 | 239 |
lemma weak_map_of_SomeI: "(k, x) : set l ==> \<exists>x. map_of l k = Some x" |
24331 | 240 |
by (induct l) auto |
13908 | 241 |
|
20800 | 242 |
lemma map_of_filter_in: |
24331 | 243 |
"map_of xs k = Some z \<Longrightarrow> P k z \<Longrightarrow> map_of (filter (split P) xs) k = Some z" |
244 |
by (induct xs) auto |
|
13908 | 245 |
|
35607 | 246 |
lemma map_of_map: |
247 |
"map_of (map (\<lambda>(k, v). (k, f v)) xs) = Option.map f \<circ> map_of xs" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
248 |
by (induct xs) (auto simp add: fun_eq_iff) |
35607 | 249 |
|
250 |
lemma dom_option_map: |
|
251 |
"dom (\<lambda>k. Option.map (f k) (m k)) = dom m" |
|
252 |
by (simp add: dom_def) |
|
13908 | 253 |
|
254 |
||
30235
58d147683393
Made Option a separate theory and renamed option_map to Option.map
nipkow
parents:
29622
diff
changeset
|
255 |
subsection {* @{const Option.map} related *} |
13908 | 256 |
|
30235
58d147683393
Made Option a separate theory and renamed option_map to Option.map
nipkow
parents:
29622
diff
changeset
|
257 |
lemma option_map_o_empty [simp]: "Option.map f o empty = empty" |
24331 | 258 |
by (rule ext) simp |
13908 | 259 |
|
20800 | 260 |
lemma option_map_o_map_upd [simp]: |
30235
58d147683393
Made Option a separate theory and renamed option_map to Option.map
nipkow
parents:
29622
diff
changeset
|
261 |
"Option.map f o m(a|->b) = (Option.map f o m)(a|->f b)" |
24331 | 262 |
by (rule ext) simp |
20800 | 263 |
|
13908 | 264 |
|
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
265 |
subsection {* @{term [source] map_comp} related *} |
17391 | 266 |
|
20800 | 267 |
lemma map_comp_empty [simp]: |
24331 | 268 |
"m \<circ>\<^sub>m empty = empty" |
269 |
"empty \<circ>\<^sub>m m = empty" |
|
270 |
by (auto simp add: map_comp_def intro: ext split: option.splits) |
|
17391 | 271 |
|
20800 | 272 |
lemma map_comp_simps [simp]: |
24331 | 273 |
"m2 k = None \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = None" |
274 |
"m2 k = Some k' \<Longrightarrow> (m1 \<circ>\<^sub>m m2) k = m1 k'" |
|
275 |
by (auto simp add: map_comp_def) |
|
17391 | 276 |
|
277 |
lemma map_comp_Some_iff: |
|
24331 | 278 |
"((m1 \<circ>\<^sub>m m2) k = Some v) = (\<exists>k'. m2 k = Some k' \<and> m1 k' = Some v)" |
279 |
by (auto simp add: map_comp_def split: option.splits) |
|
17391 | 280 |
|
281 |
lemma map_comp_None_iff: |
|
24331 | 282 |
"((m1 \<circ>\<^sub>m m2) k = None) = (m2 k = None \<or> (\<exists>k'. m2 k = Some k' \<and> m1 k' = None)) " |
283 |
by (auto simp add: map_comp_def split: option.splits) |
|
13908 | 284 |
|
20800 | 285 |
|
14100 | 286 |
subsection {* @{text "++"} *} |
13908 | 287 |
|
14025 | 288 |
lemma map_add_empty[simp]: "m ++ empty = m" |
24331 | 289 |
by(simp add: map_add_def) |
13908 | 290 |
|
14025 | 291 |
lemma empty_map_add[simp]: "empty ++ m = m" |
24331 | 292 |
by (rule ext) (simp add: map_add_def split: option.split) |
13908 | 293 |
|
14025 | 294 |
lemma map_add_assoc[simp]: "m1 ++ (m2 ++ m3) = (m1 ++ m2) ++ m3" |
24331 | 295 |
by (rule ext) (simp add: map_add_def split: option.split) |
20800 | 296 |
|
297 |
lemma map_add_Some_iff: |
|
24331 | 298 |
"((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)" |
299 |
by (simp add: map_add_def split: option.split) |
|
14025 | 300 |
|
20800 | 301 |
lemma map_add_SomeD [dest!]: |
24331 | 302 |
"(m ++ n) k = Some x \<Longrightarrow> n k = Some x \<or> n k = None \<and> m k = Some x" |
303 |
by (rule map_add_Some_iff [THEN iffD1]) |
|
13908 | 304 |
|
20800 | 305 |
lemma map_add_find_right [simp]: "!!xx. n k = Some xx ==> (m ++ n) k = Some xx" |
24331 | 306 |
by (subst map_add_Some_iff) fast |
13908 | 307 |
|
14025 | 308 |
lemma map_add_None [iff]: "((m ++ n) k = None) = (n k = None & m k = None)" |
24331 | 309 |
by (simp add: map_add_def split: option.split) |
13908 | 310 |
|
14025 | 311 |
lemma map_add_upd[simp]: "f ++ g(x|->y) = (f ++ g)(x|->y)" |
24331 | 312 |
by (rule ext) (simp add: map_add_def) |
13908 | 313 |
|
14186 | 314 |
lemma map_add_upds[simp]: "m1 ++ (m2(xs[\<mapsto>]ys)) = (m1++m2)(xs[\<mapsto>]ys)" |
24331 | 315 |
by (simp add: map_upds_def) |
14186 | 316 |
|
32236
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
317 |
lemma map_add_upd_left: "m\<notin>dom e2 \<Longrightarrow> e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)" |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
318 |
by (rule ext) (auto simp: map_add_def dom_def split: option.split) |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
319 |
|
20800 | 320 |
lemma map_of_append[simp]: "map_of (xs @ ys) = map_of ys ++ map_of xs" |
24331 | 321 |
unfolding map_add_def |
322 |
apply (induct xs) |
|
323 |
apply simp |
|
324 |
apply (rule ext) |
|
325 |
apply (simp split add: option.split) |
|
326 |
done |
|
13908 | 327 |
|
14025 | 328 |
lemma finite_range_map_of_map_add: |
20800 | 329 |
"finite (range f) ==> finite (range (f ++ map_of l))" |
24331 | 330 |
apply (induct l) |
331 |
apply (auto simp del: fun_upd_apply) |
|
332 |
apply (erule finite_range_updI) |
|
333 |
done |
|
13908 | 334 |
|
20800 | 335 |
lemma inj_on_map_add_dom [iff]: |
24331 | 336 |
"inj_on (m ++ m') (dom m') = inj_on m' (dom m')" |
337 |
by (fastsimp simp: map_add_def dom_def inj_on_def split: option.splits) |
|
20800 | 338 |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
339 |
lemma map_upds_fold_map_upd: |
35552 | 340 |
"m(ks[\<mapsto>]vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)" |
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
341 |
unfolding map_upds_def proof (rule sym, rule zip_obtain_same_length) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
342 |
fix ks :: "'a list" and vs :: "'b list" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
343 |
assume "length ks = length vs" |
35552 | 344 |
then show "foldl (\<lambda>m (k, v). m(k\<mapsto>v)) m (zip ks vs) = m ++ map_of (rev (zip ks vs))" |
345 |
by(induct arbitrary: m rule: list_induct2) simp_all |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
346 |
qed |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
347 |
|
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
348 |
lemma map_add_map_of_foldr: |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
349 |
"m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
350 |
by (induct ps) (auto simp add: fun_eq_iff map_add_def) |
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
351 |
|
15304 | 352 |
|
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
353 |
subsection {* @{term [source] restrict_map} *} |
14100 | 354 |
|
20800 | 355 |
lemma restrict_map_to_empty [simp]: "m|`{} = empty" |
24331 | 356 |
by (simp add: restrict_map_def) |
14186 | 357 |
|
31380 | 358 |
lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)" |
359 |
by (auto simp add: restrict_map_def intro: ext) |
|
360 |
||
20800 | 361 |
lemma restrict_map_empty [simp]: "empty|`D = empty" |
24331 | 362 |
by (simp add: restrict_map_def) |
14186 | 363 |
|
15693 | 364 |
lemma restrict_in [simp]: "x \<in> A \<Longrightarrow> (m|`A) x = m x" |
24331 | 365 |
by (simp add: restrict_map_def) |
14100 | 366 |
|
15693 | 367 |
lemma restrict_out [simp]: "x \<notin> A \<Longrightarrow> (m|`A) x = None" |
24331 | 368 |
by (simp add: restrict_map_def) |
14100 | 369 |
|
15693 | 370 |
lemma ran_restrictD: "y \<in> ran (m|`A) \<Longrightarrow> \<exists>x\<in>A. m x = Some y" |
24331 | 371 |
by (auto simp: restrict_map_def ran_def split: split_if_asm) |
14100 | 372 |
|
15693 | 373 |
lemma dom_restrict [simp]: "dom (m|`A) = dom m \<inter> A" |
24331 | 374 |
by (auto simp: restrict_map_def dom_def split: split_if_asm) |
14100 | 375 |
|
15693 | 376 |
lemma restrict_upd_same [simp]: "m(x\<mapsto>y)|`(-{x}) = m|`(-{x})" |
24331 | 377 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 378 |
|
15693 | 379 |
lemma restrict_restrict [simp]: "m|`A|`B = m|`(A\<inter>B)" |
24331 | 380 |
by (rule ext) (auto simp: restrict_map_def) |
14100 | 381 |
|
20800 | 382 |
lemma restrict_fun_upd [simp]: |
24331 | 383 |
"m(x := y)|`D = (if x \<in> D then (m|`(D-{x}))(x := y) else m|`D)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
384 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 385 |
|
20800 | 386 |
lemma fun_upd_None_restrict [simp]: |
24331 | 387 |
"(m|`D)(x := None) = (if x:D then m|`(D - {x}) else m|`D)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
388 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 389 |
|
20800 | 390 |
lemma fun_upd_restrict: "(m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
391 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 392 |
|
20800 | 393 |
lemma fun_upd_restrict_conv [simp]: |
24331 | 394 |
"x \<in> D \<Longrightarrow> (m|`D)(x := y) = (m|`(D-{x}))(x := y)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
395 |
by (simp add: restrict_map_def fun_eq_iff) |
14186 | 396 |
|
35159
df38e92af926
added lemma map_of_map_restrict; generalized lemma dom_const
haftmann
parents:
35115
diff
changeset
|
397 |
lemma map_of_map_restrict: |
df38e92af926
added lemma map_of_map_restrict; generalized lemma dom_const
haftmann
parents:
35115
diff
changeset
|
398 |
"map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
399 |
by (induct ks) (simp_all add: fun_eq_iff restrict_map_insert) |
35159
df38e92af926
added lemma map_of_map_restrict; generalized lemma dom_const
haftmann
parents:
35115
diff
changeset
|
400 |
|
35619 | 401 |
lemma restrict_complement_singleton_eq: |
402 |
"f |` (- {x}) = f(x := None)" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
403 |
by (simp add: restrict_map_def fun_eq_iff) |
35619 | 404 |
|
14100 | 405 |
|
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
406 |
subsection {* @{term [source] map_upds} *} |
14025 | 407 |
|
20800 | 408 |
lemma map_upds_Nil1 [simp]: "m([] [|->] bs) = m" |
24331 | 409 |
by (simp add: map_upds_def) |
14025 | 410 |
|
20800 | 411 |
lemma map_upds_Nil2 [simp]: "m(as [|->] []) = m" |
24331 | 412 |
by (simp add:map_upds_def) |
20800 | 413 |
|
414 |
lemma map_upds_Cons [simp]: "m(a#as [|->] b#bs) = (m(a|->b))(as[|->]bs)" |
|
24331 | 415 |
by (simp add:map_upds_def) |
14025 | 416 |
|
20800 | 417 |
lemma map_upds_append1 [simp]: "\<And>ys m. size xs < size ys \<Longrightarrow> |
24331 | 418 |
m(xs@[x] [\<mapsto>] ys) = m(xs [\<mapsto>] ys)(x \<mapsto> ys!size xs)" |
419 |
apply(induct xs) |
|
420 |
apply (clarsimp simp add: neq_Nil_conv) |
|
421 |
apply (case_tac ys) |
|
422 |
apply simp |
|
423 |
apply simp |
|
424 |
done |
|
14187 | 425 |
|
20800 | 426 |
lemma map_upds_list_update2_drop [simp]: |
427 |
"\<lbrakk>size xs \<le> i; i < size ys\<rbrakk> |
|
428 |
\<Longrightarrow> m(xs[\<mapsto>]ys[i:=y]) = m(xs[\<mapsto>]ys)" |
|
24331 | 429 |
apply (induct xs arbitrary: m ys i) |
430 |
apply simp |
|
431 |
apply (case_tac ys) |
|
432 |
apply simp |
|
433 |
apply (simp split: nat.split) |
|
434 |
done |
|
14025 | 435 |
|
20800 | 436 |
lemma map_upd_upds_conv_if: |
437 |
"(f(x|->y))(xs [|->] ys) = |
|
438 |
(if x : set(take (length ys) xs) then f(xs [|->] ys) |
|
439 |
else (f(xs [|->] ys))(x|->y))" |
|
24331 | 440 |
apply (induct xs arbitrary: x y ys f) |
441 |
apply simp |
|
442 |
apply (case_tac ys) |
|
443 |
apply (auto split: split_if simp: fun_upd_twist) |
|
444 |
done |
|
14025 | 445 |
|
446 |
lemma map_upds_twist [simp]: |
|
24331 | 447 |
"a ~: set as ==> m(a|->b)(as[|->]bs) = m(as[|->]bs)(a|->b)" |
448 |
using set_take_subset by (fastsimp simp add: map_upd_upds_conv_if) |
|
14025 | 449 |
|
20800 | 450 |
lemma map_upds_apply_nontin [simp]: |
24331 | 451 |
"x ~: set xs ==> (f(xs[|->]ys)) x = f x" |
452 |
apply (induct xs arbitrary: ys) |
|
453 |
apply simp |
|
454 |
apply (case_tac ys) |
|
455 |
apply (auto simp: map_upd_upds_conv_if) |
|
456 |
done |
|
14025 | 457 |
|
20800 | 458 |
lemma fun_upds_append_drop [simp]: |
24331 | 459 |
"size xs = size ys \<Longrightarrow> m(xs@zs[\<mapsto>]ys) = m(xs[\<mapsto>]ys)" |
460 |
apply (induct xs arbitrary: m ys) |
|
461 |
apply simp |
|
462 |
apply (case_tac ys) |
|
463 |
apply simp_all |
|
464 |
done |
|
14300 | 465 |
|
20800 | 466 |
lemma fun_upds_append2_drop [simp]: |
24331 | 467 |
"size xs = size ys \<Longrightarrow> m(xs[\<mapsto>]ys@zs) = m(xs[\<mapsto>]ys)" |
468 |
apply (induct xs arbitrary: m ys) |
|
469 |
apply simp |
|
470 |
apply (case_tac ys) |
|
471 |
apply simp_all |
|
472 |
done |
|
14300 | 473 |
|
474 |
||
20800 | 475 |
lemma restrict_map_upds[simp]: |
476 |
"\<lbrakk> length xs = length ys; set xs \<subseteq> D \<rbrakk> |
|
477 |
\<Longrightarrow> m(xs [\<mapsto>] ys)|`D = (m|`(D - set xs))(xs [\<mapsto>] ys)" |
|
24331 | 478 |
apply (induct xs arbitrary: m ys) |
479 |
apply simp |
|
480 |
apply (case_tac ys) |
|
481 |
apply simp |
|
482 |
apply (simp add: Diff_insert [symmetric] insert_absorb) |
|
483 |
apply (simp add: map_upd_upds_conv_if) |
|
484 |
done |
|
14186 | 485 |
|
486 |
||
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
487 |
subsection {* @{term [source] dom} *} |
13908 | 488 |
|
31080 | 489 |
lemma dom_eq_empty_conv [simp]: "dom f = {} \<longleftrightarrow> f = empty" |
490 |
by(auto intro!:ext simp: dom_def) |
|
491 |
||
13908 | 492 |
lemma domI: "m a = Some b ==> a : dom m" |
24331 | 493 |
by(simp add:dom_def) |
14100 | 494 |
(* declare domI [intro]? *) |
13908 | 495 |
|
15369 | 496 |
lemma domD: "a : dom m ==> \<exists>b. m a = Some b" |
24331 | 497 |
by (cases "m a") (auto simp add: dom_def) |
13908 | 498 |
|
20800 | 499 |
lemma domIff [iff, simp del]: "(a : dom m) = (m a ~= None)" |
24331 | 500 |
by(simp add:dom_def) |
13908 | 501 |
|
20800 | 502 |
lemma dom_empty [simp]: "dom empty = {}" |
24331 | 503 |
by(simp add:dom_def) |
13908 | 504 |
|
20800 | 505 |
lemma dom_fun_upd [simp]: |
24331 | 506 |
"dom(f(x := y)) = (if y=None then dom f - {x} else insert x (dom f))" |
507 |
by(auto simp add:dom_def) |
|
13908 | 508 |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
509 |
lemma dom_if: |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
510 |
"dom (\<lambda>x. if P x then f x else g x) = dom f \<inter> {x. P x} \<union> dom g \<inter> {x. \<not> P x}" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
511 |
by (auto split: if_splits) |
13937 | 512 |
|
15304 | 513 |
lemma dom_map_of_conv_image_fst: |
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
514 |
"dom (map_of xys) = fst ` set xys" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
515 |
by (induct xys) (auto simp add: dom_if) |
15304 | 516 |
|
20800 | 517 |
lemma dom_map_of_zip [simp]: "[| length xs = length ys; distinct xs |] ==> |
24331 | 518 |
dom(map_of(zip xs ys)) = set xs" |
519 |
by (induct rule: list_induct2) simp_all |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
14739
diff
changeset
|
520 |
|
13908 | 521 |
lemma finite_dom_map_of: "finite (dom (map_of l))" |
24331 | 522 |
by (induct l) (auto simp add: dom_def insert_Collect [symmetric]) |
13908 | 523 |
|
20800 | 524 |
lemma dom_map_upds [simp]: |
24331 | 525 |
"dom(m(xs[|->]ys)) = set(take (length ys) xs) Un dom m" |
526 |
apply (induct xs arbitrary: m ys) |
|
527 |
apply simp |
|
528 |
apply (case_tac ys) |
|
529 |
apply auto |
|
530 |
done |
|
13910 | 531 |
|
20800 | 532 |
lemma dom_map_add [simp]: "dom(m++n) = dom n Un dom m" |
24331 | 533 |
by(auto simp:dom_def) |
13910 | 534 |
|
20800 | 535 |
lemma dom_override_on [simp]: |
536 |
"dom(override_on f g A) = |
|
537 |
(dom f - {a. a : A - dom g}) Un {a. a : A Int dom g}" |
|
24331 | 538 |
by(auto simp: dom_def override_on_def) |
13908 | 539 |
|
14027 | 540 |
lemma map_add_comm: "dom m1 \<inter> dom m2 = {} \<Longrightarrow> m1++m2 = m2++m1" |
24331 | 541 |
by (rule ext) (force simp: map_add_def dom_def split: option.split) |
20800 | 542 |
|
32236
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
543 |
lemma map_add_dom_app_simps: |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
544 |
"\<lbrakk> m\<in>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m" |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
545 |
"\<lbrakk> m\<notin>dom l1 \<rbrakk> \<Longrightarrow> (l1++l2) m = l2 m" |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
546 |
"\<lbrakk> m\<notin>dom l2 \<rbrakk> \<Longrightarrow> (l1++l2) m = l1 m" |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
547 |
by (auto simp add: map_add_def split: option.split_asm) |
0203e1006f1b
some lemmas about maps (contributed by Peter Lammich)
krauss
parents:
31380
diff
changeset
|
548 |
|
29622 | 549 |
lemma dom_const [simp]: |
35159
df38e92af926
added lemma map_of_map_restrict; generalized lemma dom_const
haftmann
parents:
35115
diff
changeset
|
550 |
"dom (\<lambda>x. Some (f x)) = UNIV" |
29622 | 551 |
by auto |
552 |
||
22230 | 553 |
(* Due to John Matthews - could be rephrased with dom *) |
554 |
lemma finite_map_freshness: |
|
555 |
"finite (dom (f :: 'a \<rightharpoonup> 'b)) \<Longrightarrow> \<not> finite (UNIV :: 'a set) \<Longrightarrow> |
|
556 |
\<exists>x. f x = None" |
|
557 |
by(bestsimp dest:ex_new_if_finite) |
|
14027 | 558 |
|
28790 | 559 |
lemma dom_minus: |
560 |
"f x = None \<Longrightarrow> dom f - insert x A = dom f - A" |
|
561 |
unfolding dom_def by simp |
|
562 |
||
563 |
lemma insert_dom: |
|
564 |
"f x = Some y \<Longrightarrow> insert x (dom f) = dom f" |
|
565 |
unfolding dom_def by auto |
|
566 |
||
35607 | 567 |
lemma map_of_map_keys: |
568 |
"set xs = dom m \<Longrightarrow> map_of (map (\<lambda>k. (k, the (m k))) xs) = m" |
|
569 |
by (rule ext) (auto simp add: map_of_map_restrict restrict_map_def) |
|
570 |
||
28790 | 571 |
|
17399
56a3a4affedc
@{term [source] ...} in subsections probably more robust;
wenzelm
parents:
17391
diff
changeset
|
572 |
subsection {* @{term [source] ran} *} |
14100 | 573 |
|
20800 | 574 |
lemma ranI: "m a = Some b ==> b : ran m" |
24331 | 575 |
by(auto simp: ran_def) |
14100 | 576 |
(* declare ranI [intro]? *) |
13908 | 577 |
|
20800 | 578 |
lemma ran_empty [simp]: "ran empty = {}" |
24331 | 579 |
by(auto simp: ran_def) |
13908 | 580 |
|
20800 | 581 |
lemma ran_map_upd [simp]: "m a = None ==> ran(m(a|->b)) = insert b (ran m)" |
24331 | 582 |
unfolding ran_def |
583 |
apply auto |
|
584 |
apply (subgoal_tac "aa ~= a") |
|
585 |
apply auto |
|
586 |
done |
|
20800 | 587 |
|
34979
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
588 |
lemma ran_distinct: |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
589 |
assumes dist: "distinct (map fst al)" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
590 |
shows "ran (map_of al) = snd ` set al" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
591 |
using assms proof (induct al) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
592 |
case Nil then show ?case by simp |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
593 |
next |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
594 |
case (Cons kv al) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
595 |
then have "ran (map_of al) = snd ` set al" by simp |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
596 |
moreover from Cons.prems have "map_of al (fst kv) = None" |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
597 |
by (simp add: map_of_eq_None_iff) |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
598 |
ultimately show ?case by (simp only: map_of.simps ran_map_upd) simp |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
599 |
qed |
8cb6e7a42e9c
more correspondence lemmas between related operations
haftmann
parents:
34941
diff
changeset
|
600 |
|
13910 | 601 |
|
14100 | 602 |
subsection {* @{text "map_le"} *} |
13910 | 603 |
|
13912 | 604 |
lemma map_le_empty [simp]: "empty \<subseteq>\<^sub>m g" |
24331 | 605 |
by (simp add: map_le_def) |
13910 | 606 |
|
17724 | 607 |
lemma upd_None_map_le [simp]: "f(x := None) \<subseteq>\<^sub>m f" |
24331 | 608 |
by (force simp add: map_le_def) |
14187 | 609 |
|
13910 | 610 |
lemma map_le_upd[simp]: "f \<subseteq>\<^sub>m g ==> f(a := b) \<subseteq>\<^sub>m g(a := b)" |
24331 | 611 |
by (fastsimp simp add: map_le_def) |
13910 | 612 |
|
17724 | 613 |
lemma map_le_imp_upd_le [simp]: "m1 \<subseteq>\<^sub>m m2 \<Longrightarrow> m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)" |
24331 | 614 |
by (force simp add: map_le_def) |
14187 | 615 |
|
20800 | 616 |
lemma map_le_upds [simp]: |
24331 | 617 |
"f \<subseteq>\<^sub>m g ==> f(as [|->] bs) \<subseteq>\<^sub>m g(as [|->] bs)" |
618 |
apply (induct as arbitrary: f g bs) |
|
619 |
apply simp |
|
620 |
apply (case_tac bs) |
|
621 |
apply auto |
|
622 |
done |
|
13908 | 623 |
|
14033 | 624 |
lemma map_le_implies_dom_le: "(f \<subseteq>\<^sub>m g) \<Longrightarrow> (dom f \<subseteq> dom g)" |
24331 | 625 |
by (fastsimp simp add: map_le_def dom_def) |
14033 | 626 |
|
627 |
lemma map_le_refl [simp]: "f \<subseteq>\<^sub>m f" |
|
24331 | 628 |
by (simp add: map_le_def) |
14033 | 629 |
|
14187 | 630 |
lemma map_le_trans[trans]: "\<lbrakk> m1 \<subseteq>\<^sub>m m2; m2 \<subseteq>\<^sub>m m3\<rbrakk> \<Longrightarrow> m1 \<subseteq>\<^sub>m m3" |
24331 | 631 |
by (auto simp add: map_le_def dom_def) |
14033 | 632 |
|
633 |
lemma map_le_antisym: "\<lbrakk> f \<subseteq>\<^sub>m g; g \<subseteq>\<^sub>m f \<rbrakk> \<Longrightarrow> f = g" |
|
24331 | 634 |
unfolding map_le_def |
635 |
apply (rule ext) |
|
636 |
apply (case_tac "x \<in> dom f", simp) |
|
637 |
apply (case_tac "x \<in> dom g", simp, fastsimp) |
|
638 |
done |
|
14033 | 639 |
|
640 |
lemma map_le_map_add [simp]: "f \<subseteq>\<^sub>m (g ++ f)" |
|
24331 | 641 |
by (fastsimp simp add: map_le_def) |
14033 | 642 |
|
15304 | 643 |
lemma map_le_iff_map_add_commute: "(f \<subseteq>\<^sub>m f ++ g) = (f++g = g++f)" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
644 |
by(fastsimp simp: map_add_def map_le_def fun_eq_iff split: option.splits) |
15304 | 645 |
|
15303 | 646 |
lemma map_add_le_mapE: "f++g \<subseteq>\<^sub>m h \<Longrightarrow> g \<subseteq>\<^sub>m h" |
24331 | 647 |
by (fastsimp simp add: map_le_def map_add_def dom_def) |
15303 | 648 |
|
649 |
lemma map_add_le_mapI: "\<lbrakk> f \<subseteq>\<^sub>m h; g \<subseteq>\<^sub>m h; f \<subseteq>\<^sub>m f++g \<rbrakk> \<Longrightarrow> f++g \<subseteq>\<^sub>m h" |
|
24331 | 650 |
by (clarsimp simp add: map_le_def map_add_def dom_def split: option.splits) |
15303 | 651 |
|
31080 | 652 |
lemma dom_eq_singleton_conv: "dom f = {x} \<longleftrightarrow> (\<exists>v. f = [x \<mapsto> v])" |
653 |
proof(rule iffI) |
|
654 |
assume "\<exists>v. f = [x \<mapsto> v]" |
|
655 |
thus "dom f = {x}" by(auto split: split_if_asm) |
|
656 |
next |
|
657 |
assume "dom f = {x}" |
|
658 |
then obtain v where "f x = Some v" by auto |
|
659 |
hence "[x \<mapsto> v] \<subseteq>\<^sub>m f" by(auto simp add: map_le_def) |
|
660 |
moreover have "f \<subseteq>\<^sub>m [x \<mapsto> v]" using `dom f = {x}` `f x = Some v` |
|
661 |
by(auto simp add: map_le_def) |
|
662 |
ultimately have "f = [x \<mapsto> v]" by-(rule map_le_antisym) |
|
663 |
thus "\<exists>v. f = [x \<mapsto> v]" by blast |
|
664 |
qed |
|
665 |
||
35565 | 666 |
|
667 |
subsection {* Various *} |
|
668 |
||
669 |
lemma set_map_of_compr: |
|
670 |
assumes distinct: "distinct (map fst xs)" |
|
671 |
shows "set xs = {(k, v). map_of xs k = Some v}" |
|
672 |
using assms proof (induct xs) |
|
673 |
case Nil then show ?case by simp |
|
674 |
next |
|
675 |
case (Cons x xs) |
|
676 |
obtain k v where "x = (k, v)" by (cases x) blast |
|
677 |
with Cons.prems have "k \<notin> dom (map_of xs)" |
|
678 |
by (simp add: dom_map_of_conv_image_fst) |
|
679 |
then have *: "insert (k, v) {(k, v). map_of xs k = Some v} = |
|
680 |
{(k', v'). (map_of xs(k \<mapsto> v)) k' = Some v'}" |
|
681 |
by (auto split: if_splits) |
|
682 |
from Cons have "set xs = {(k, v). map_of xs k = Some v}" by simp |
|
683 |
with * `x = (k, v)` show ?case by simp |
|
684 |
qed |
|
685 |
||
686 |
lemma map_of_inject_set: |
|
687 |
assumes distinct: "distinct (map fst xs)" "distinct (map fst ys)" |
|
688 |
shows "map_of xs = map_of ys \<longleftrightarrow> set xs = set ys" (is "?lhs \<longleftrightarrow> ?rhs") |
|
689 |
proof |
|
690 |
assume ?lhs |
|
691 |
moreover from `distinct (map fst xs)` have "set xs = {(k, v). map_of xs k = Some v}" |
|
692 |
by (rule set_map_of_compr) |
|
693 |
moreover from `distinct (map fst ys)` have "set ys = {(k, v). map_of ys k = Some v}" |
|
694 |
by (rule set_map_of_compr) |
|
695 |
ultimately show ?rhs by simp |
|
696 |
next |
|
697 |
assume ?rhs show ?lhs proof |
|
698 |
fix k |
|
699 |
show "map_of xs k = map_of ys k" proof (cases "map_of xs k") |
|
700 |
case None |
|
701 |
moreover with `?rhs` have "map_of ys k = None" |
|
702 |
by (simp add: map_of_eq_None_iff) |
|
703 |
ultimately show ?thesis by simp |
|
704 |
next |
|
705 |
case (Some v) |
|
706 |
moreover with distinct `?rhs` have "map_of ys k = Some v" |
|
707 |
by simp |
|
708 |
ultimately show ?thesis by simp |
|
709 |
qed |
|
710 |
qed |
|
711 |
qed |
|
712 |
||
3981 | 713 |
end |