|
1 (* Author: Andreas Lochbihler, Uni Karlsruhe *) |
|
2 |
|
3 header {* Almost everywhere constant functions *} |
|
4 |
|
5 theory FinFun |
|
6 imports Card_Univ |
|
7 begin |
|
8 |
|
9 text {* |
|
10 This theory defines functions which are constant except for finitely |
|
11 many points (FinFun) and introduces a type finfin along with a |
|
12 number of operators for them. The code generator is set up such that |
|
13 such functions can be represented as data in the generated code and |
|
14 all operators are executable. |
|
15 |
|
16 For details, see Formalising FinFuns - Generating Code for Functions as Data by A. Lochbihler in TPHOLs 2009. |
|
17 *} |
|
18 |
|
19 |
|
20 definition "code_abort" :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a" |
|
21 where [simp, code del]: "code_abort f = f ()" |
|
22 |
|
23 code_abort "code_abort" |
|
24 |
|
25 hide_const (open) "code_abort" |
|
26 |
|
27 subsection {* The @{text "map_default"} operation *} |
|
28 |
|
29 definition map_default :: "'b \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" |
|
30 where "map_default b f a \<equiv> case f a of None \<Rightarrow> b | Some b' \<Rightarrow> b'" |
|
31 |
|
32 lemma map_default_delete [simp]: |
|
33 "map_default b (f(a := None)) = (map_default b f)(a := b)" |
|
34 by(simp add: map_default_def fun_eq_iff) |
|
35 |
|
36 lemma map_default_insert: |
|
37 "map_default b (f(a \<mapsto> b')) = (map_default b f)(a := b')" |
|
38 by(simp add: map_default_def fun_eq_iff) |
|
39 |
|
40 lemma map_default_empty [simp]: "map_default b empty = (\<lambda>a. b)" |
|
41 by(simp add: fun_eq_iff map_default_def) |
|
42 |
|
43 lemma map_default_inject: |
|
44 fixes g g' :: "'a \<rightharpoonup> 'b" |
|
45 assumes infin_eq: "\<not> finite (UNIV :: 'a set) \<or> b = b'" |
|
46 and fin: "finite (dom g)" and b: "b \<notin> ran g" |
|
47 and fin': "finite (dom g')" and b': "b' \<notin> ran g'" |
|
48 and eq': "map_default b g = map_default b' g'" |
|
49 shows "b = b'" "g = g'" |
|
50 proof - |
|
51 from infin_eq show bb': "b = b'" |
|
52 proof |
|
53 assume infin: "\<not> finite (UNIV :: 'a set)" |
|
54 from fin fin' have "finite (dom g \<union> dom g')" by auto |
|
55 with infin have "UNIV - (dom g \<union> dom g') \<noteq> {}" by(auto dest: finite_subset) |
|
56 then obtain a where a: "a \<notin> dom g \<union> dom g'" by auto |
|
57 hence "map_default b g a = b" "map_default b' g' a = b'" by(auto simp add: map_default_def) |
|
58 with eq' show "b = b'" by simp |
|
59 qed |
|
60 |
|
61 show "g = g'" |
|
62 proof |
|
63 fix x |
|
64 show "g x = g' x" |
|
65 proof(cases "g x") |
|
66 case None |
|
67 hence "map_default b g x = b" by(simp add: map_default_def) |
|
68 with bb' eq' have "map_default b' g' x = b'" by simp |
|
69 with b' have "g' x = None" by(simp add: map_default_def ran_def split: option.split_asm) |
|
70 with None show ?thesis by simp |
|
71 next |
|
72 case (Some c) |
|
73 with b have cb: "c \<noteq> b" by(auto simp add: ran_def) |
|
74 moreover from Some have "map_default b g x = c" by(simp add: map_default_def) |
|
75 with eq' have "map_default b' g' x = c" by simp |
|
76 ultimately have "g' x = Some c" using b' bb' by(auto simp add: map_default_def split: option.splits) |
|
77 with Some show ?thesis by simp |
|
78 qed |
|
79 qed |
|
80 qed |
|
81 |
|
82 subsection {* The finfun type *} |
|
83 |
|
84 definition "finfun = {f::'a\<Rightarrow>'b. \<exists>b. finite {a. f a \<noteq> b}}" |
|
85 |
|
86 typedef (open) ('a,'b) finfun ("(_ \<Rightarrow>\<^isub>f /_)" [22, 21] 21) = "finfun :: ('a => 'b) set" |
|
87 proof - |
|
88 have "\<exists>f. finite {x. f x \<noteq> undefined}" |
|
89 proof |
|
90 show "finite {x. (\<lambda>y. undefined) x \<noteq> undefined}" by auto |
|
91 qed |
|
92 then show ?thesis unfolding finfun_def by auto |
|
93 qed |
|
94 |
|
95 setup_lifting type_definition_finfun |
|
96 |
|
97 lemma fun_upd_finfun: "y(a := b) \<in> finfun \<longleftrightarrow> y \<in> finfun" |
|
98 proof - |
|
99 { fix b' |
|
100 have "finite {a'. (y(a := b)) a' \<noteq> b'} = finite {a'. y a' \<noteq> b'}" |
|
101 proof(cases "b = b'") |
|
102 case True |
|
103 hence "{a'. (y(a := b)) a' \<noteq> b'} = {a'. y a' \<noteq> b'} - {a}" by auto |
|
104 thus ?thesis by simp |
|
105 next |
|
106 case False |
|
107 hence "{a'. (y(a := b)) a' \<noteq> b'} = insert a {a'. y a' \<noteq> b'}" by auto |
|
108 thus ?thesis by simp |
|
109 qed } |
|
110 thus ?thesis unfolding finfun_def by blast |
|
111 qed |
|
112 |
|
113 lemma const_finfun: "(\<lambda>x. a) \<in> finfun" |
|
114 by(auto simp add: finfun_def) |
|
115 |
|
116 lemma finfun_left_compose: |
|
117 assumes "y \<in> finfun" |
|
118 shows "g \<circ> y \<in> finfun" |
|
119 proof - |
|
120 from assms obtain b where "finite {a. y a \<noteq> b}" |
|
121 unfolding finfun_def by blast |
|
122 hence "finite {c. g (y c) \<noteq> g b}" |
|
123 proof(induct "{a. y a \<noteq> b}" arbitrary: y) |
|
124 case empty |
|
125 hence "y = (\<lambda>a. b)" by(auto intro: ext) |
|
126 thus ?case by(simp) |
|
127 next |
|
128 case (insert x F) |
|
129 note IH = `\<And>y. F = {a. y a \<noteq> b} \<Longrightarrow> finite {c. g (y c) \<noteq> g b}` |
|
130 from `insert x F = {a. y a \<noteq> b}` `x \<notin> F` |
|
131 have F: "F = {a. (y(x := b)) a \<noteq> b}" by(auto) |
|
132 show ?case |
|
133 proof(cases "g (y x) = g b") |
|
134 case True |
|
135 hence "{c. g ((y(x := b)) c) \<noteq> g b} = {c. g (y c) \<noteq> g b}" by auto |
|
136 with IH[OF F] show ?thesis by simp |
|
137 next |
|
138 case False |
|
139 hence "{c. g (y c) \<noteq> g b} = insert x {c. g ((y(x := b)) c) \<noteq> g b}" by auto |
|
140 with IH[OF F] show ?thesis by(simp) |
|
141 qed |
|
142 qed |
|
143 thus ?thesis unfolding finfun_def by auto |
|
144 qed |
|
145 |
|
146 lemma assumes "y \<in> finfun" |
|
147 shows fst_finfun: "fst \<circ> y \<in> finfun" |
|
148 and snd_finfun: "snd \<circ> y \<in> finfun" |
|
149 proof - |
|
150 from assms obtain b c where bc: "finite {a. y a \<noteq> (b, c)}" |
|
151 unfolding finfun_def by auto |
|
152 have "{a. fst (y a) \<noteq> b} \<subseteq> {a. y a \<noteq> (b, c)}" |
|
153 and "{a. snd (y a) \<noteq> c} \<subseteq> {a. y a \<noteq> (b, c)}" by auto |
|
154 hence "finite {a. fst (y a) \<noteq> b}" |
|
155 and "finite {a. snd (y a) \<noteq> c}" using bc by(auto intro: finite_subset) |
|
156 thus "fst \<circ> y \<in> finfun" "snd \<circ> y \<in> finfun" |
|
157 unfolding finfun_def by auto |
|
158 qed |
|
159 |
|
160 lemma map_of_finfun: "map_of xs \<in> finfun" |
|
161 unfolding finfun_def |
|
162 by(induct xs)(auto simp add: Collect_neg_eq Collect_conj_eq Collect_imp_eq intro: finite_subset) |
|
163 |
|
164 lemma Diag_finfun: "(\<lambda>x. (f x, g x)) \<in> finfun \<longleftrightarrow> f \<in> finfun \<and> g \<in> finfun" |
|
165 by(auto intro: finite_subset simp add: Collect_neg_eq Collect_imp_eq Collect_conj_eq finfun_def) |
|
166 |
|
167 lemma finfun_right_compose: |
|
168 assumes g: "g \<in> finfun" and inj: "inj f" |
|
169 shows "g o f \<in> finfun" |
|
170 proof - |
|
171 from g obtain b where b: "finite {a. g a \<noteq> b}" unfolding finfun_def by blast |
|
172 moreover have "f ` {a. g (f a) \<noteq> b} \<subseteq> {a. g a \<noteq> b}" by auto |
|
173 moreover from inj have "inj_on f {a. g (f a) \<noteq> b}" by(rule subset_inj_on) blast |
|
174 ultimately have "finite {a. g (f a) \<noteq> b}" |
|
175 by(blast intro: finite_imageD[where f=f] finite_subset) |
|
176 thus ?thesis unfolding finfun_def by auto |
|
177 qed |
|
178 |
|
179 lemma finfun_curry: |
|
180 assumes fin: "f \<in> finfun" |
|
181 shows "curry f \<in> finfun" "curry f a \<in> finfun" |
|
182 proof - |
|
183 from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast |
|
184 moreover have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force) |
|
185 hence "{a. curry f a \<noteq> (\<lambda>b. c)} = fst ` {ab. f ab \<noteq> c}" |
|
186 by(auto simp add: curry_def fun_eq_iff) |
|
187 ultimately have "finite {a. curry f a \<noteq> (\<lambda>b. c)}" by simp |
|
188 thus "curry f \<in> finfun" unfolding finfun_def by blast |
|
189 |
|
190 have "snd ` {ab. f ab \<noteq> c} = {b. \<exists>a. f (a, b) \<noteq> c}" by(force) |
|
191 hence "{b. f (a, b) \<noteq> c} \<subseteq> snd ` {ab. f ab \<noteq> c}" by auto |
|
192 hence "finite {b. f (a, b) \<noteq> c}" by(rule finite_subset)(rule finite_imageI[OF c]) |
|
193 thus "curry f a \<in> finfun" unfolding finfun_def by auto |
|
194 qed |
|
195 |
|
196 lemmas finfun_simp = |
|
197 fst_finfun snd_finfun Abs_finfun_inverse Rep_finfun_inverse Abs_finfun_inject Rep_finfun_inject Diag_finfun finfun_curry |
|
198 lemmas finfun_iff = const_finfun fun_upd_finfun Rep_finfun map_of_finfun |
|
199 lemmas finfun_intro = finfun_left_compose fst_finfun snd_finfun |
|
200 |
|
201 lemma Abs_finfun_inject_finite: |
|
202 fixes x y :: "'a \<Rightarrow> 'b" |
|
203 assumes fin: "finite (UNIV :: 'a set)" |
|
204 shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y" |
|
205 proof |
|
206 assume "Abs_finfun x = Abs_finfun y" |
|
207 moreover have "x \<in> finfun" "y \<in> finfun" unfolding finfun_def |
|
208 by(auto intro: finite_subset[OF _ fin]) |
|
209 ultimately show "x = y" by(simp add: Abs_finfun_inject) |
|
210 qed simp |
|
211 |
|
212 lemma Abs_finfun_inject_finite_class: |
|
213 fixes x y :: "('a :: finite) \<Rightarrow> 'b" |
|
214 shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y" |
|
215 using finite_UNIV |
|
216 by(simp add: Abs_finfun_inject_finite) |
|
217 |
|
218 lemma Abs_finfun_inj_finite: |
|
219 assumes fin: "finite (UNIV :: 'a set)" |
|
220 shows "inj (Abs_finfun :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b)" |
|
221 proof(rule inj_onI) |
|
222 fix x y :: "'a \<Rightarrow> 'b" |
|
223 assume "Abs_finfun x = Abs_finfun y" |
|
224 moreover have "x \<in> finfun" "y \<in> finfun" unfolding finfun_def |
|
225 by(auto intro: finite_subset[OF _ fin]) |
|
226 ultimately show "x = y" by(simp add: Abs_finfun_inject) |
|
227 qed |
|
228 |
|
229 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
230 |
|
231 lemma Abs_finfun_inverse_finite: |
|
232 fixes x :: "'a \<Rightarrow> 'b" |
|
233 assumes fin: "finite (UNIV :: 'a set)" |
|
234 shows "Rep_finfun (Abs_finfun x) = x" |
|
235 proof - |
|
236 from fin have "x \<in> finfun" |
|
237 by(auto simp add: finfun_def intro: finite_subset) |
|
238 thus ?thesis by simp |
|
239 qed |
|
240 |
|
241 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
242 |
|
243 lemma Abs_finfun_inverse_finite_class: |
|
244 fixes x :: "('a :: finite) \<Rightarrow> 'b" |
|
245 shows "Rep_finfun (Abs_finfun x) = x" |
|
246 using finite_UNIV by(simp add: Abs_finfun_inverse_finite) |
|
247 |
|
248 lemma finfun_eq_finite_UNIV: "finite (UNIV :: 'a set) \<Longrightarrow> (finfun :: ('a \<Rightarrow> 'b) set) = UNIV" |
|
249 unfolding finfun_def by(auto intro: finite_subset) |
|
250 |
|
251 lemma finfun_finite_UNIV_class: "finfun = (UNIV :: ('a :: finite \<Rightarrow> 'b) set)" |
|
252 by(simp add: finfun_eq_finite_UNIV) |
|
253 |
|
254 lemma map_default_in_finfun: |
|
255 assumes fin: "finite (dom f)" |
|
256 shows "map_default b f \<in> finfun" |
|
257 unfolding finfun_def |
|
258 proof(intro CollectI exI) |
|
259 from fin show "finite {a. map_default b f a \<noteq> b}" |
|
260 by(auto simp add: map_default_def dom_def Collect_conj_eq split: option.splits) |
|
261 qed |
|
262 |
|
263 lemma finfun_cases_map_default: |
|
264 obtains b g where "f = Abs_finfun (map_default b g)" "finite (dom g)" "b \<notin> ran g" |
|
265 proof - |
|
266 obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by(cases f) |
|
267 from y obtain b where b: "finite {a. y a \<noteq> b}" unfolding finfun_def by auto |
|
268 let ?g = "(\<lambda>a. if y a = b then None else Some (y a))" |
|
269 have "map_default b ?g = y" by(simp add: fun_eq_iff map_default_def) |
|
270 with f have "f = Abs_finfun (map_default b ?g)" by simp |
|
271 moreover from b have "finite (dom ?g)" by(auto simp add: dom_def) |
|
272 moreover have "b \<notin> ran ?g" by(auto simp add: ran_def) |
|
273 ultimately show ?thesis by(rule that) |
|
274 qed |
|
275 |
|
276 |
|
277 subsection {* Kernel functions for type @{typ "'a \<Rightarrow>\<^isub>f 'b"} *} |
|
278 |
|
279 lift_definition finfun_const :: "'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" ("\<lambda>\<^isup>f/ _" [0] 1) |
|
280 is "\<lambda> b x. b" by (rule const_finfun) |
|
281 |
|
282 lift_definition finfun_update :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" ("_'(\<^sup>f/ _ := _')" [1000,0,0] 1000) is "fun_upd" by (simp add: fun_upd_finfun) |
|
283 |
|
284 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
285 |
|
286 lemma finfun_update_twist: "a \<noteq> a' \<Longrightarrow> f(\<^sup>f a := b)(\<^sup>f a' := b') = f(\<^sup>f a' := b')(\<^sup>f a := b)" |
|
287 by transfer (simp add: fun_upd_twist) |
|
288 |
|
289 lemma finfun_update_twice [simp]: |
|
290 "finfun_update (finfun_update f a b) a b' = finfun_update f a b'" |
|
291 by transfer simp |
|
292 |
|
293 lemma finfun_update_const_same: "(\<lambda>\<^isup>f b)(\<^sup>f a := b) = (\<lambda>\<^isup>f b)" |
|
294 by transfer (simp add: fun_eq_iff) |
|
295 |
|
296 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
297 |
|
298 subsection {* Code generator setup *} |
|
299 |
|
300 definition finfun_update_code :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" ("_'(\<^sup>fc/ _ := _')" [1000,0,0] 1000) |
|
301 where [simp, code del]: "finfun_update_code = finfun_update" |
|
302 |
|
303 code_datatype finfun_const finfun_update_code |
|
304 |
|
305 lemma finfun_update_const_code [code]: |
|
306 "(\<lambda>\<^isup>f b)(\<^sup>f a := b') = (if b = b' then (\<lambda>\<^isup>f b) else finfun_update_code (\<lambda>\<^isup>f b) a b')" |
|
307 by(simp add: finfun_update_const_same) |
|
308 |
|
309 lemma finfun_update_update_code [code]: |
|
310 "(finfun_update_code f a b)(\<^sup>f a' := b') = (if a = a' then f(\<^sup>f a := b') else finfun_update_code (f(\<^sup>f a' := b')) a b)" |
|
311 by(simp add: finfun_update_twist) |
|
312 |
|
313 |
|
314 subsection {* Setup for quickcheck *} |
|
315 |
|
316 quickcheck_generator finfun constructors: finfun_update_code, "finfun_const :: 'b => 'a \<Rightarrow>\<^isub>f 'b" |
|
317 |
|
318 subsection {* @{text "finfun_update"} as instance of @{text "comp_fun_commute"} *} |
|
319 |
|
320 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
321 |
|
322 interpretation finfun_update: comp_fun_commute "\<lambda>a f. f(\<^sup>f a :: 'a := b')" |
|
323 proof |
|
324 fix a a' :: 'a |
|
325 show "(\<lambda>f. f(\<^sup>f a := b')) \<circ> (\<lambda>f. f(\<^sup>f a' := b')) = (\<lambda>f. f(\<^sup>f a' := b')) \<circ> (\<lambda>f. f(\<^sup>f a := b'))" |
|
326 proof |
|
327 fix b |
|
328 have "(Rep_finfun b)(a := b', a' := b') = (Rep_finfun b)(a' := b', a := b')" |
|
329 by(cases "a = a'")(auto simp add: fun_upd_twist) |
|
330 then have "b(\<^sup>f a := b')(\<^sup>f a' := b') = b(\<^sup>f a' := b')(\<^sup>f a := b')" |
|
331 by(auto simp add: finfun_update_def fun_upd_twist) |
|
332 then show "((\<lambda>f. f(\<^sup>f a := b')) \<circ> (\<lambda>f. f(\<^sup>f a' := b'))) b = ((\<lambda>f. f(\<^sup>f a' := b')) \<circ> (\<lambda>f. f(\<^sup>f a := b'))) b" |
|
333 by (simp add: fun_eq_iff) |
|
334 qed |
|
335 qed |
|
336 |
|
337 lemma fold_finfun_update_finite_univ: |
|
338 assumes fin: "finite (UNIV :: 'a set)" |
|
339 shows "Finite_Set.fold (\<lambda>a f. f(\<^sup>f a := b')) (\<lambda>\<^isup>f b) (UNIV :: 'a set) = (\<lambda>\<^isup>f b')" |
|
340 proof - |
|
341 { fix A :: "'a set" |
|
342 from fin have "finite A" by(auto intro: finite_subset) |
|
343 hence "Finite_Set.fold (\<lambda>a f. f(\<^sup>f a := b')) (\<lambda>\<^isup>f b) A = Abs_finfun (\<lambda>a. if a \<in> A then b' else b)" |
|
344 proof(induct) |
|
345 case (insert x F) |
|
346 have "(\<lambda>a. if a = x then b' else (if a \<in> F then b' else b)) = (\<lambda>a. if a = x \<or> a \<in> F then b' else b)" |
|
347 by(auto intro: ext) |
|
348 with insert show ?case |
|
349 by(simp add: finfun_const_def fun_upd_def)(simp add: finfun_update_def Abs_finfun_inverse_finite[OF fin] fun_upd_def) |
|
350 qed(simp add: finfun_const_def) } |
|
351 thus ?thesis by(simp add: finfun_const_def) |
|
352 qed |
|
353 |
|
354 |
|
355 subsection {* Default value for FinFuns *} |
|
356 |
|
357 definition finfun_default_aux :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b" |
|
358 where [code del]: "finfun_default_aux f = (if finite (UNIV :: 'a set) then undefined else THE b. finite {a. f a \<noteq> b})" |
|
359 |
|
360 lemma finfun_default_aux_infinite: |
|
361 fixes f :: "'a \<Rightarrow> 'b" |
|
362 assumes infin: "\<not> finite (UNIV :: 'a set)" |
|
363 and fin: "finite {a. f a \<noteq> b}" |
|
364 shows "finfun_default_aux f = b" |
|
365 proof - |
|
366 let ?B = "{a. f a \<noteq> b}" |
|
367 from fin have "(THE b. finite {a. f a \<noteq> b}) = b" |
|
368 proof(rule the_equality) |
|
369 fix b' |
|
370 assume "finite {a. f a \<noteq> b'}" (is "finite ?B'") |
|
371 with infin fin have "UNIV - (?B' \<union> ?B) \<noteq> {}" by(auto dest: finite_subset) |
|
372 then obtain a where a: "a \<notin> ?B' \<union> ?B" by auto |
|
373 thus "b' = b" by auto |
|
374 qed |
|
375 thus ?thesis using infin by(simp add: finfun_default_aux_def) |
|
376 qed |
|
377 |
|
378 |
|
379 lemma finite_finfun_default_aux: |
|
380 fixes f :: "'a \<Rightarrow> 'b" |
|
381 assumes fin: "f \<in> finfun" |
|
382 shows "finite {a. f a \<noteq> finfun_default_aux f}" |
|
383 proof(cases "finite (UNIV :: 'a set)") |
|
384 case True thus ?thesis using fin |
|
385 by(auto simp add: finfun_def finfun_default_aux_def intro: finite_subset) |
|
386 next |
|
387 case False |
|
388 from fin obtain b where b: "finite {a. f a \<noteq> b}" (is "finite ?B") |
|
389 unfolding finfun_def by blast |
|
390 with False show ?thesis by(simp add: finfun_default_aux_infinite) |
|
391 qed |
|
392 |
|
393 lemma finfun_default_aux_update_const: |
|
394 fixes f :: "'a \<Rightarrow> 'b" |
|
395 assumes fin: "f \<in> finfun" |
|
396 shows "finfun_default_aux (f(a := b)) = finfun_default_aux f" |
|
397 proof(cases "finite (UNIV :: 'a set)") |
|
398 case False |
|
399 from fin obtain b' where b': "finite {a. f a \<noteq> b'}" unfolding finfun_def by blast |
|
400 hence "finite {a'. (f(a := b)) a' \<noteq> b'}" |
|
401 proof(cases "b = b' \<and> f a \<noteq> b'") |
|
402 case True |
|
403 hence "{a. f a \<noteq> b'} = insert a {a'. (f(a := b)) a' \<noteq> b'}" by auto |
|
404 thus ?thesis using b' by simp |
|
405 next |
|
406 case False |
|
407 moreover |
|
408 { assume "b \<noteq> b'" |
|
409 hence "{a'. (f(a := b)) a' \<noteq> b'} = insert a {a. f a \<noteq> b'}" by auto |
|
410 hence ?thesis using b' by simp } |
|
411 moreover |
|
412 { assume "b = b'" "f a = b'" |
|
413 hence "{a'. (f(a := b)) a' \<noteq> b'} = {a. f a \<noteq> b'}" by auto |
|
414 hence ?thesis using b' by simp } |
|
415 ultimately show ?thesis by blast |
|
416 qed |
|
417 with False b' show ?thesis by(auto simp del: fun_upd_apply simp add: finfun_default_aux_infinite) |
|
418 next |
|
419 case True thus ?thesis by(simp add: finfun_default_aux_def) |
|
420 qed |
|
421 |
|
422 lift_definition finfun_default :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'b" |
|
423 is "finfun_default_aux" .. |
|
424 |
|
425 lemma finite_finfun_default: "finite {a. Rep_finfun f a \<noteq> finfun_default f}" |
|
426 apply transfer apply (erule finite_finfun_default_aux) |
|
427 unfolding Rel_def fun_rel_def cr_finfun_def by simp |
|
428 |
|
429 lemma finfun_default_const: "finfun_default ((\<lambda>\<^isup>f b) :: 'a \<Rightarrow>\<^isub>f 'b) = (if finite (UNIV :: 'a set) then undefined else b)" |
|
430 apply(transfer) |
|
431 apply(auto simp add: finfun_default_aux_infinite) |
|
432 apply(simp add: finfun_default_aux_def) |
|
433 done |
|
434 |
|
435 lemma finfun_default_update_const: |
|
436 "finfun_default (f(\<^sup>f a := b)) = finfun_default f" |
|
437 by transfer (simp add: finfun_default_aux_update_const) |
|
438 |
|
439 lemma finfun_default_const_code [code]: |
|
440 "finfun_default ((\<lambda>\<^isup>f c) :: ('a :: card_UNIV) \<Rightarrow>\<^isub>f 'b) = (if card_UNIV (TYPE('a)) = 0 then c else undefined)" |
|
441 by(simp add: finfun_default_const card_UNIV_eq_0_infinite_UNIV) |
|
442 |
|
443 lemma finfun_default_update_code [code]: |
|
444 "finfun_default (finfun_update_code f a b) = finfun_default f" |
|
445 by(simp add: finfun_default_update_const) |
|
446 |
|
447 subsection {* Recursion combinator and well-formedness conditions *} |
|
448 |
|
449 definition finfun_rec :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow>\<^isub>f 'b) \<Rightarrow> 'c" |
|
450 where [code del]: |
|
451 "finfun_rec cnst upd f \<equiv> |
|
452 let b = finfun_default f; |
|
453 g = THE g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g |
|
454 in Finite_Set.fold (\<lambda>a. upd a (map_default b g a)) (cnst b) (dom g)" |
|
455 |
|
456 locale finfun_rec_wf_aux = |
|
457 fixes cnst :: "'b \<Rightarrow> 'c" |
|
458 and upd :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c" |
|
459 assumes upd_const_same: "upd a b (cnst b) = cnst b" |
|
460 and upd_commute: "a \<noteq> a' \<Longrightarrow> upd a b (upd a' b' c) = upd a' b' (upd a b c)" |
|
461 and upd_idemp: "b \<noteq> b' \<Longrightarrow> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)" |
|
462 begin |
|
463 |
|
464 |
|
465 lemma upd_left_comm: "comp_fun_commute (\<lambda>a. upd a (f a))" |
|
466 by(unfold_locales)(auto intro: upd_commute simp add: fun_eq_iff) |
|
467 |
|
468 lemma upd_upd_twice: "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)" |
|
469 by(cases "b \<noteq> b'")(auto simp add: fun_upd_def upd_const_same upd_idemp) |
|
470 |
|
471 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
472 |
|
473 lemma map_default_update_const: |
|
474 assumes fin: "finite (dom f)" |
|
475 and anf: "a \<notin> dom f" |
|
476 and fg: "f \<subseteq>\<^sub>m g" |
|
477 shows "upd a d (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)) = |
|
478 Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)" |
|
479 proof - |
|
480 let ?upd = "\<lambda>a. upd a (map_default d g a)" |
|
481 let ?fr = "\<lambda>A. Finite_Set.fold ?upd (cnst d) A" |
|
482 interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm) |
|
483 |
|
484 from fin anf fg show ?thesis |
|
485 proof(induct "dom f" arbitrary: f) |
|
486 case empty |
|
487 from `{} = dom f` have "f = empty" by(auto simp add: dom_def intro: ext) |
|
488 thus ?case by(simp add: finfun_const_def upd_const_same) |
|
489 next |
|
490 case (insert a' A) |
|
491 note IH = `\<And>f. \<lbrakk> A = dom f; a \<notin> dom f; f \<subseteq>\<^sub>m g \<rbrakk> \<Longrightarrow> upd a d (?fr (dom f)) = ?fr (dom f)` |
|
492 note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A` |
|
493 note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g` |
|
494 |
|
495 from domf obtain b where b: "f a' = Some b" by auto |
|
496 let ?f' = "f(a' := None)" |
|
497 have "upd a d (?fr (insert a' A)) = upd a d (upd a' (map_default d g a') (?fr A))" |
|
498 by(subst gwf.fold_insert[OF fin a'nA]) rule |
|
499 also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec) |
|
500 hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def) |
|
501 also from anf domf have "a \<noteq> a'" by auto note upd_commute[OF this] |
|
502 also from domf a'nA anf fg have "a \<notin> dom ?f'" "?f' \<subseteq>\<^sub>m g" and A: "A = dom ?f'" by(auto simp add: ran_def map_le_def) |
|
503 note A also note IH[OF A `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g`] |
|
504 also have "upd a' (map_default d f a') (?fr (dom (f(a' := None)))) = ?fr (dom f)" |
|
505 unfolding domf[symmetric] gwf.fold_insert[OF fin a'nA] ga' unfolding A .. |
|
506 also have "insert a' (dom ?f') = dom f" using domf by auto |
|
507 finally show ?case . |
|
508 qed |
|
509 qed |
|
510 |
|
511 lemma map_default_update_twice: |
|
512 assumes fin: "finite (dom f)" |
|
513 and anf: "a \<notin> dom f" |
|
514 and fg: "f \<subseteq>\<^sub>m g" |
|
515 shows "upd a d'' (upd a d' (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))) = |
|
516 upd a d'' (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))" |
|
517 proof - |
|
518 let ?upd = "\<lambda>a. upd a (map_default d g a)" |
|
519 let ?fr = "\<lambda>A. Finite_Set.fold ?upd (cnst d) A" |
|
520 interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm) |
|
521 |
|
522 from fin anf fg show ?thesis |
|
523 proof(induct "dom f" arbitrary: f) |
|
524 case empty |
|
525 from `{} = dom f` have "f = empty" by(auto simp add: dom_def intro: ext) |
|
526 thus ?case by(auto simp add: finfun_const_def finfun_update_def upd_upd_twice) |
|
527 next |
|
528 case (insert a' A) |
|
529 note IH = `\<And>f. \<lbrakk>A = dom f; a \<notin> dom f; f \<subseteq>\<^sub>m g\<rbrakk> \<Longrightarrow> upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (?fr (dom f))` |
|
530 note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A` |
|
531 note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g` |
|
532 |
|
533 from domf obtain b where b: "f a' = Some b" by auto |
|
534 let ?f' = "f(a' := None)" |
|
535 let ?b' = "case f a' of None \<Rightarrow> d | Some b \<Rightarrow> b" |
|
536 from domf have "upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (upd a d' (?fr (insert a' A)))" by simp |
|
537 also note gwf.fold_insert[OF fin a'nA] |
|
538 also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec) |
|
539 hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def) |
|
540 also from anf domf have ana': "a \<noteq> a'" by auto note upd_commute[OF this] |
|
541 also note upd_commute[OF ana'] |
|
542 also from domf a'nA anf fg have "a \<notin> dom ?f'" "?f' \<subseteq>\<^sub>m g" and A: "A = dom ?f'" by(auto simp add: ran_def map_le_def) |
|
543 note A also note IH[OF A `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g`] |
|
544 also note upd_commute[OF ana'[symmetric]] also note ga'[symmetric] also note A[symmetric] |
|
545 also note gwf.fold_insert[symmetric, OF fin a'nA] also note domf |
|
546 finally show ?case . |
|
547 qed |
|
548 qed |
|
549 |
|
550 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
551 |
|
552 lemma map_default_eq_id [simp]: "map_default d ((\<lambda>a. Some (f a)) |` {a. f a \<noteq> d}) = f" |
|
553 by(auto simp add: map_default_def restrict_map_def intro: ext) |
|
554 |
|
555 lemma finite_rec_cong1: |
|
556 assumes f: "comp_fun_commute f" and g: "comp_fun_commute g" |
|
557 and fin: "finite A" |
|
558 and eq: "\<And>a. a \<in> A \<Longrightarrow> f a = g a" |
|
559 shows "Finite_Set.fold f z A = Finite_Set.fold g z A" |
|
560 proof - |
|
561 interpret f: comp_fun_commute f by(rule f) |
|
562 interpret g: comp_fun_commute g by(rule g) |
|
563 { fix B |
|
564 assume BsubA: "B \<subseteq> A" |
|
565 with fin have "finite B" by(blast intro: finite_subset) |
|
566 hence "B \<subseteq> A \<Longrightarrow> Finite_Set.fold f z B = Finite_Set.fold g z B" |
|
567 proof(induct) |
|
568 case empty thus ?case by simp |
|
569 next |
|
570 case (insert a B) |
|
571 note finB = `finite B` note anB = `a \<notin> B` note sub = `insert a B \<subseteq> A` |
|
572 note IH = `B \<subseteq> A \<Longrightarrow> Finite_Set.fold f z B = Finite_Set.fold g z B` |
|
573 from sub anB have BpsubA: "B \<subset> A" and BsubA: "B \<subseteq> A" and aA: "a \<in> A" by auto |
|
574 from IH[OF BsubA] eq[OF aA] finB anB |
|
575 show ?case by(auto) |
|
576 qed |
|
577 with BsubA have "Finite_Set.fold f z B = Finite_Set.fold g z B" by blast } |
|
578 thus ?thesis by blast |
|
579 qed |
|
580 |
|
581 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
582 |
|
583 lemma finfun_rec_upd [simp]: |
|
584 "finfun_rec cnst upd (f(\<^sup>f a' := b')) = upd a' b' (finfun_rec cnst upd f)" |
|
585 proof - |
|
586 obtain b where b: "b = finfun_default f" by auto |
|
587 let ?the = "\<lambda>f g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g" |
|
588 obtain g where g: "g = The (?the f)" by blast |
|
589 obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by (cases f) |
|
590 from f y b have bfin: "finite {a. y a \<noteq> b}" by(simp add: finfun_default_def finite_finfun_default_aux) |
|
591 |
|
592 let ?g = "(\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}" |
|
593 from bfin have fing: "finite (dom ?g)" by auto |
|
594 have bran: "b \<notin> ran ?g" by(auto simp add: ran_def restrict_map_def) |
|
595 have yg: "y = map_default b ?g" by simp |
|
596 have gg: "g = ?g" unfolding g |
|
597 proof(rule the_equality) |
|
598 from f y bfin show "?the f ?g" |
|
599 by(auto)(simp add: restrict_map_def ran_def split: split_if_asm) |
|
600 next |
|
601 fix g' |
|
602 assume "?the f g'" |
|
603 hence fin': "finite (dom g')" and ran': "b \<notin> ran g'" |
|
604 and eq: "Abs_finfun (map_default b ?g) = Abs_finfun (map_default b g')" using f yg by auto |
|
605 from fin' fing have "map_default b ?g \<in> finfun" "map_default b g' \<in> finfun" by(blast intro: map_default_in_finfun)+ |
|
606 with eq have "map_default b ?g = map_default b g'" by simp |
|
607 with fing bran fin' ran' show "g' = ?g" by(rule map_default_inject[OF disjI2[OF refl], THEN sym]) |
|
608 qed |
|
609 |
|
610 show ?thesis |
|
611 proof(cases "b' = b") |
|
612 case True |
|
613 note b'b = True |
|
614 |
|
615 let ?g' = "(\<lambda>a. Some ((y(a' := b)) a)) |` {a. (y(a' := b)) a \<noteq> b}" |
|
616 from bfin b'b have fing': "finite (dom ?g')" |
|
617 by(auto simp add: Collect_conj_eq Collect_imp_eq intro: finite_subset) |
|
618 have brang': "b \<notin> ran ?g'" by(auto simp add: ran_def restrict_map_def) |
|
619 |
|
620 let ?b' = "\<lambda>a. case ?g' a of None \<Rightarrow> b | Some b \<Rightarrow> b" |
|
621 let ?b = "map_default b ?g" |
|
622 from upd_left_comm upd_left_comm fing' |
|
623 have "Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g') = Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')" |
|
624 by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b b map_default_def) |
|
625 also interpret gwf: comp_fun_commute "\<lambda>a. upd a (?b a)" by(rule upd_left_comm) |
|
626 have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g') = upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))" |
|
627 proof(cases "y a' = b") |
|
628 case True |
|
629 with b'b have g': "?g' = ?g" by(auto simp add: restrict_map_def intro: ext) |
|
630 from True have a'ndomg: "a' \<notin> dom ?g" by auto |
|
631 from f b'b b show ?thesis unfolding g' |
|
632 by(subst map_default_update_const[OF fing a'ndomg map_le_refl, symmetric]) simp |
|
633 next |
|
634 case False |
|
635 hence domg: "dom ?g = insert a' (dom ?g')" by auto |
|
636 from False b'b have a'ndomg': "a' \<notin> dom ?g'" by auto |
|
637 have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g')) = |
|
638 upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'))" |
|
639 using fing' a'ndomg' unfolding b'b by(rule gwf.fold_insert) |
|
640 hence "upd a' b (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g'))) = |
|
641 upd a' b (upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')))" by simp |
|
642 also from b'b have g'leg: "?g' \<subseteq>\<^sub>m ?g" by(auto simp add: restrict_map_def map_le_def) |
|
643 note map_default_update_twice[OF fing' a'ndomg' this, of b "?b a'" b] |
|
644 also note map_default_update_const[OF fing' a'ndomg' g'leg, of b] |
|
645 finally show ?thesis unfolding b'b domg[unfolded b'b] by(rule sym) |
|
646 qed |
|
647 also have "The (?the (f(\<^sup>f a' := b'))) = ?g'" |
|
648 proof(rule the_equality) |
|
649 from f y b b'b brang' fing' show "?the (f(\<^sup>f a' := b')) ?g'" |
|
650 by(auto simp del: fun_upd_apply simp add: finfun_update_def) |
|
651 next |
|
652 fix g' |
|
653 assume "?the (f(\<^sup>f a' := b')) g'" |
|
654 hence fin': "finite (dom g')" and ran': "b \<notin> ran g'" |
|
655 and eq: "f(\<^sup>f a' := b') = Abs_finfun (map_default b g')" |
|
656 by(auto simp del: fun_upd_apply) |
|
657 from fin' fing' have "map_default b g' \<in> finfun" "map_default b ?g' \<in> finfun" |
|
658 by(blast intro: map_default_in_finfun)+ |
|
659 with eq f b'b b have "map_default b ?g' = map_default b g'" |
|
660 by(simp del: fun_upd_apply add: finfun_update_def) |
|
661 with fing' brang' fin' ran' show "g' = ?g'" |
|
662 by(rule map_default_inject[OF disjI2[OF refl], THEN sym]) |
|
663 qed |
|
664 ultimately show ?thesis unfolding finfun_rec_def Let_def b gg[unfolded g b] using bfin b'b b |
|
665 by(simp only: finfun_default_update_const map_default_def) |
|
666 next |
|
667 case False |
|
668 note b'b = this |
|
669 let ?g' = "?g(a' \<mapsto> b')" |
|
670 let ?b' = "map_default b ?g'" |
|
671 let ?b = "map_default b ?g" |
|
672 from fing have fing': "finite (dom ?g')" by auto |
|
673 from bran b'b have bnrang': "b \<notin> ran ?g'" by(auto simp add: ran_def) |
|
674 have ffmg': "map_default b ?g' = y(a' := b')" by(auto intro: ext simp add: map_default_def restrict_map_def) |
|
675 with f y have f_Abs: "f(\<^sup>f a' := b') = Abs_finfun (map_default b ?g')" by(auto simp add: finfun_update_def) |
|
676 have g': "The (?the (f(\<^sup>f a' := b'))) = ?g'" |
|
677 proof (rule the_equality) |
|
678 from fing' bnrang' f_Abs show "?the (f(\<^sup>f a' := b')) ?g'" by(auto simp add: finfun_update_def restrict_map_def) |
|
679 next |
|
680 fix g' assume "?the (f(\<^sup>f a' := b')) g'" |
|
681 hence f': "f(\<^sup>f a' := b') = Abs_finfun (map_default b g')" |
|
682 and fin': "finite (dom g')" and brang': "b \<notin> ran g'" by auto |
|
683 from fing' fin' have "map_default b ?g' \<in> finfun" "map_default b g' \<in> finfun" |
|
684 by(auto intro: map_default_in_finfun) |
|
685 with f' f_Abs have "map_default b g' = map_default b ?g'" by simp |
|
686 with fin' brang' fing' bnrang' show "g' = ?g'" |
|
687 by(rule map_default_inject[OF disjI2[OF refl]]) |
|
688 qed |
|
689 have dom: "dom (((\<lambda>a. Some (y a)) |` {a. y a \<noteq> b})(a' \<mapsto> b')) = insert a' (dom ((\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}))" |
|
690 by auto |
|
691 show ?thesis |
|
692 proof(cases "y a' = b") |
|
693 case True |
|
694 hence a'ndomg: "a' \<notin> dom ?g" by auto |
|
695 from f y b'b True have yff: "y = map_default b (?g' |` dom ?g)" |
|
696 by(auto simp add: restrict_map_def map_default_def intro!: ext) |
|
697 hence f': "f = Abs_finfun (map_default b (?g' |` dom ?g))" using f by simp |
|
698 interpret g'wf: comp_fun_commute "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm) |
|
699 from upd_left_comm upd_left_comm fing |
|
700 have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g) = Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)" |
|
701 by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b True map_default_def) |
|
702 thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] |
|
703 unfolding g' g[symmetric] gg g'wf.fold_insert[OF fing a'ndomg, of "cnst b", folded dom] |
|
704 by -(rule arg_cong2[where f="upd a'"], simp_all add: map_default_def) |
|
705 next |
|
706 case False |
|
707 hence "insert a' (dom ?g) = dom ?g" by auto |
|
708 moreover { |
|
709 let ?g'' = "?g(a' := None)" |
|
710 let ?b'' = "map_default b ?g''" |
|
711 from False have domg: "dom ?g = insert a' (dom ?g'')" by auto |
|
712 from False have a'ndomg'': "a' \<notin> dom ?g''" by auto |
|
713 have fing'': "finite (dom ?g'')" by(rule finite_subset[OF _ fing]) auto |
|
714 have bnrang'': "b \<notin> ran ?g''" by(auto simp add: ran_def restrict_map_def) |
|
715 interpret gwf: comp_fun_commute "\<lambda>a. upd a (?b a)" by(rule upd_left_comm) |
|
716 interpret g'wf: comp_fun_commute "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm) |
|
717 have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g''))) = |
|
718 upd a' b' (upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')))" |
|
719 unfolding gwf.fold_insert[OF fing'' a'ndomg''] f .. |
|
720 also have g''leg: "?g |` dom ?g'' \<subseteq>\<^sub>m ?g" by(auto simp add: map_le_def) |
|
721 have "dom (?g |` dom ?g'') = dom ?g''" by auto |
|
722 note map_default_update_twice[where d=b and f = "?g |` dom ?g''" and a=a' and d'="?b a'" and d''=b' and g="?g", |
|
723 unfolded this, OF fing'' a'ndomg'' g''leg] |
|
724 also have b': "b' = ?b' a'" by(auto simp add: map_default_def) |
|
725 from upd_left_comm upd_left_comm fing'' |
|
726 have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'') = Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g'')" |
|
727 by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b map_default_def) |
|
728 with b' have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')) = |
|
729 upd a' (?b' a') (Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g''))" by simp |
|
730 also note g'wf.fold_insert[OF fing'' a'ndomg'', symmetric] |
|
731 finally have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g)) = |
|
732 Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)" |
|
733 unfolding domg . } |
|
734 ultimately have "Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (insert a' (dom ?g)) = |
|
735 upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))" by simp |
|
736 thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] g[symmetric] g' dom[symmetric] |
|
737 using b'b gg by(simp add: map_default_insert) |
|
738 qed |
|
739 qed |
|
740 qed |
|
741 |
|
742 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
743 |
|
744 end |
|
745 |
|
746 locale finfun_rec_wf = finfun_rec_wf_aux + |
|
747 assumes const_update_all: |
|
748 "finite (UNIV :: 'a set) \<Longrightarrow> Finite_Set.fold (\<lambda>a. upd a b') (cnst b) (UNIV :: 'a set) = cnst b'" |
|
749 begin |
|
750 |
|
751 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
752 |
|
753 lemma finfun_rec_const [simp]: |
|
754 "finfun_rec cnst upd (\<lambda>\<^isup>f c) = cnst c" |
|
755 proof(cases "finite (UNIV :: 'a set)") |
|
756 case False |
|
757 hence "finfun_default ((\<lambda>\<^isup>f c) :: 'a \<Rightarrow>\<^isub>f 'b) = c" by(simp add: finfun_default_const) |
|
758 moreover have "(THE g :: 'a \<rightharpoonup> 'b. (\<lambda>\<^isup>f c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g) = empty" |
|
759 proof (rule the_equality) |
|
760 show "(\<lambda>\<^isup>f c) = Abs_finfun (map_default c empty) \<and> finite (dom empty) \<and> c \<notin> ran empty" |
|
761 by(auto simp add: finfun_const_def) |
|
762 next |
|
763 fix g :: "'a \<rightharpoonup> 'b" |
|
764 assume "(\<lambda>\<^isup>f c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g" |
|
765 hence g: "(\<lambda>\<^isup>f c) = Abs_finfun (map_default c g)" and fin: "finite (dom g)" and ran: "c \<notin> ran g" by blast+ |
|
766 from g map_default_in_finfun[OF fin, of c] have "map_default c g = (\<lambda>a. c)" |
|
767 by(simp add: finfun_const_def) |
|
768 moreover have "map_default c empty = (\<lambda>a. c)" by simp |
|
769 ultimately show "g = empty" by-(rule map_default_inject[OF disjI2[OF refl] fin ran], auto) |
|
770 qed |
|
771 ultimately show ?thesis by(simp add: finfun_rec_def) |
|
772 next |
|
773 case True |
|
774 hence default: "finfun_default ((\<lambda>\<^isup>f c) :: 'a \<Rightarrow>\<^isub>f 'b) = undefined" by(simp add: finfun_default_const) |
|
775 let ?the = "\<lambda>g :: 'a \<rightharpoonup> 'b. (\<lambda>\<^isup>f c) = Abs_finfun (map_default undefined g) \<and> finite (dom g) \<and> undefined \<notin> ran g" |
|
776 show ?thesis |
|
777 proof(cases "c = undefined") |
|
778 case True |
|
779 have the: "The ?the = empty" |
|
780 proof (rule the_equality) |
|
781 from True show "?the empty" by(auto simp add: finfun_const_def) |
|
782 next |
|
783 fix g' |
|
784 assume "?the g'" |
|
785 hence fg: "(\<lambda>\<^isup>f c) = Abs_finfun (map_default undefined g')" |
|
786 and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all |
|
787 from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun) |
|
788 with fg have "map_default undefined g' = (\<lambda>a. c)" |
|
789 by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1]) |
|
790 with True show "g' = empty" |
|
791 by -(rule map_default_inject(2)[OF _ fin g], auto) |
|
792 qed |
|
793 show ?thesis unfolding finfun_rec_def using `finite UNIV` True |
|
794 unfolding Let_def the default by(simp) |
|
795 next |
|
796 case False |
|
797 have the: "The ?the = (\<lambda>a :: 'a. Some c)" |
|
798 proof (rule the_equality) |
|
799 from False True show "?the (\<lambda>a :: 'a. Some c)" |
|
800 by(auto simp add: map_default_def [abs_def] finfun_const_def dom_def ran_def) |
|
801 next |
|
802 fix g' :: "'a \<rightharpoonup> 'b" |
|
803 assume "?the g'" |
|
804 hence fg: "(\<lambda>\<^isup>f c) = Abs_finfun (map_default undefined g')" |
|
805 and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all |
|
806 from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun) |
|
807 with fg have "map_default undefined g' = (\<lambda>a. c)" |
|
808 by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1]) |
|
809 with True False show "g' = (\<lambda>a::'a. Some c)" |
|
810 by - (rule map_default_inject(2)[OF _ fin g], |
|
811 auto simp add: dom_def ran_def map_default_def [abs_def]) |
|
812 qed |
|
813 show ?thesis unfolding finfun_rec_def using True False |
|
814 unfolding Let_def the default by(simp add: dom_def map_default_def const_update_all) |
|
815 qed |
|
816 qed |
|
817 |
|
818 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
819 |
|
820 end |
|
821 |
|
822 subsection {* Weak induction rule and case analysis for FinFuns *} |
|
823 |
|
824 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
825 |
|
826 lemma finfun_weak_induct [consumes 0, case_names const update]: |
|
827 assumes const: "\<And>b. P (\<lambda>\<^isup>f b)" |
|
828 and update: "\<And>f a b. P f \<Longrightarrow> P (f(\<^sup>f a := b))" |
|
829 shows "P x" |
|
830 proof(induct x rule: Abs_finfun_induct) |
|
831 case (Abs_finfun y) |
|
832 then obtain b where "finite {a. y a \<noteq> b}" unfolding finfun_def by blast |
|
833 thus ?case using `y \<in> finfun` |
|
834 proof(induct "{a. y a \<noteq> b}" arbitrary: y rule: finite_induct) |
|
835 case empty |
|
836 hence "\<And>a. y a = b" by blast |
|
837 hence "y = (\<lambda>a. b)" by(auto intro: ext) |
|
838 hence "Abs_finfun y = finfun_const b" unfolding finfun_const_def by simp |
|
839 thus ?case by(simp add: const) |
|
840 next |
|
841 case (insert a A) |
|
842 note IH = `\<And>y. \<lbrakk> A = {a. y a \<noteq> b}; y \<in> finfun \<rbrakk> \<Longrightarrow> P (Abs_finfun y)` |
|
843 note y = `y \<in> finfun` |
|
844 with `insert a A = {a. y a \<noteq> b}` `a \<notin> A` |
|
845 have "A = {a'. (y(a := b)) a' \<noteq> b}" "y(a := b) \<in> finfun" by auto |
|
846 from IH[OF this] have "P (finfun_update (Abs_finfun (y(a := b))) a (y a))" by(rule update) |
|
847 thus ?case using y unfolding finfun_update_def by simp |
|
848 qed |
|
849 qed |
|
850 |
|
851 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
852 |
|
853 lemma finfun_exhaust_disj: "(\<exists>b. x = finfun_const b) \<or> (\<exists>f a b. x = finfun_update f a b)" |
|
854 by(induct x rule: finfun_weak_induct) blast+ |
|
855 |
|
856 lemma finfun_exhaust: |
|
857 obtains b where "x = (\<lambda>\<^isup>f b)" |
|
858 | f a b where "x = f(\<^sup>f a := b)" |
|
859 by(atomize_elim)(rule finfun_exhaust_disj) |
|
860 |
|
861 lemma finfun_rec_unique: |
|
862 fixes f :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'c" |
|
863 assumes c: "\<And>c. f (\<lambda>\<^isup>f c) = cnst c" |
|
864 and u: "\<And>g a b. f (g(\<^sup>f a := b)) = upd g a b (f g)" |
|
865 and c': "\<And>c. f' (\<lambda>\<^isup>f c) = cnst c" |
|
866 and u': "\<And>g a b. f' (g(\<^sup>f a := b)) = upd g a b (f' g)" |
|
867 shows "f = f'" |
|
868 proof |
|
869 fix g :: "'a \<Rightarrow>\<^isub>f 'b" |
|
870 show "f g = f' g" |
|
871 by(induct g rule: finfun_weak_induct)(auto simp add: c u c' u') |
|
872 qed |
|
873 |
|
874 |
|
875 subsection {* Function application *} |
|
876 |
|
877 definition finfun_apply :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow> 'b" ("_\<^sub>f" [1000] 1000) |
|
878 where [code del]: "finfun_apply = (\<lambda>f a. finfun_rec (\<lambda>b. b) (\<lambda>a' b c. if (a = a') then b else c) f)" |
|
879 |
|
880 interpretation finfun_apply_aux: finfun_rec_wf_aux "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c" |
|
881 by(unfold_locales) auto |
|
882 |
|
883 interpretation finfun_apply: finfun_rec_wf "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c" |
|
884 proof(unfold_locales) |
|
885 fix b' b :: 'a |
|
886 assume fin: "finite (UNIV :: 'b set)" |
|
887 { fix A :: "'b set" |
|
888 interpret comp_fun_commute "\<lambda>a'. If (a = a') b'" by(rule finfun_apply_aux.upd_left_comm) |
|
889 from fin have "finite A" by(auto intro: finite_subset) |
|
890 hence "Finite_Set.fold (\<lambda>a'. If (a = a') b') b A = (if a \<in> A then b' else b)" |
|
891 by induct auto } |
|
892 from this[of UNIV] show "Finite_Set.fold (\<lambda>a'. If (a = a') b') b UNIV = b'" by simp |
|
893 qed |
|
894 |
|
895 lemma finfun_const_apply [simp, code]: "(\<lambda>\<^isup>f b)\<^sub>f a = b" |
|
896 by(simp add: finfun_apply_def) |
|
897 |
|
898 lemma finfun_upd_apply: "f(\<^sup>fa := b)\<^sub>f a' = (if a = a' then b else f\<^sub>f a')" |
|
899 and finfun_upd_apply_code [code]: "(finfun_update_code f a b)\<^sub>f a' = (if a = a' then b else f\<^sub>f a')" |
|
900 by(simp_all add: finfun_apply_def) |
|
901 |
|
902 lemma finfun_upd_apply_same [simp]: |
|
903 "f(\<^sup>fa := b)\<^sub>f a = b" |
|
904 by(simp add: finfun_upd_apply) |
|
905 |
|
906 lemma finfun_upd_apply_other [simp]: |
|
907 "a \<noteq> a' \<Longrightarrow> f(\<^sup>fa := b)\<^sub>f a' = f\<^sub>f a'" |
|
908 by(simp add: finfun_upd_apply) |
|
909 |
|
910 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
911 |
|
912 lemma finfun_apply_Rep_finfun: |
|
913 "finfun_apply = Rep_finfun" |
|
914 proof(rule finfun_rec_unique) |
|
915 fix c show "Rep_finfun (\<lambda>\<^isup>f c) = (\<lambda>a. c)" by(auto simp add: finfun_const_def) |
|
916 next |
|
917 fix g a b show "Rep_finfun g(\<^sup>f a := b) = (\<lambda>c. if c = a then b else Rep_finfun g c)" |
|
918 by(auto simp add: finfun_update_def fun_upd_finfun Abs_finfun_inverse Rep_finfun intro: ext) |
|
919 qed(auto intro: ext) |
|
920 |
|
921 lemma finfun_ext: "(\<And>a. f\<^sub>f a = g\<^sub>f a) \<Longrightarrow> f = g" |
|
922 by(auto simp add: finfun_apply_Rep_finfun Rep_finfun_inject[symmetric] simp del: Rep_finfun_inject intro: ext) |
|
923 |
|
924 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
925 |
|
926 lemma expand_finfun_eq: "(f = g) = (f\<^sub>f = g\<^sub>f)" |
|
927 by(auto intro: finfun_ext) |
|
928 |
|
929 lemma finfun_const_inject [simp]: "(\<lambda>\<^isup>f b) = (\<lambda>\<^isup>f b') \<equiv> b = b'" |
|
930 by(simp add: expand_finfun_eq fun_eq_iff) |
|
931 |
|
932 lemma finfun_const_eq_update: |
|
933 "((\<lambda>\<^isup>f b) = f(\<^sup>f a := b')) = (b = b' \<and> (\<forall>a'. a \<noteq> a' \<longrightarrow> f\<^sub>f a' = b))" |
|
934 by(auto simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
|
935 |
|
936 subsection {* Function composition *} |
|
937 |
|
938 definition finfun_comp :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow>\<^isub>f 'a \<Rightarrow> 'c \<Rightarrow>\<^isub>f 'b" (infixr "\<circ>\<^isub>f" 55) |
|
939 where [code del]: "g \<circ>\<^isub>f f = finfun_rec (\<lambda>b. (\<lambda>\<^isup>f g b)) (\<lambda>a b c. c(\<^sup>f a := g b)) f" |
|
940 |
|
941 interpretation finfun_comp_aux: finfun_rec_wf_aux "(\<lambda>b. (\<lambda>\<^isup>f g b))" "(\<lambda>a b c. c(\<^sup>f a := g b))" |
|
942 by(unfold_locales)(auto simp add: finfun_upd_apply intro: finfun_ext) |
|
943 |
|
944 interpretation finfun_comp: finfun_rec_wf "(\<lambda>b. (\<lambda>\<^isup>f g b))" "(\<lambda>a b c. c(\<^sup>f a := g b))" |
|
945 proof |
|
946 fix b' b :: 'a |
|
947 assume fin: "finite (UNIV :: 'c set)" |
|
948 { fix A :: "'c set" |
|
949 from fin have "finite A" by(auto intro: finite_subset) |
|
950 hence "Finite_Set.fold (\<lambda>(a :: 'c) c. c(\<^sup>f a := g b')) (\<lambda>\<^isup>f g b) A = |
|
951 Abs_finfun (\<lambda>a. if a \<in> A then g b' else g b)" |
|
952 by induct (simp_all add: finfun_const_def, auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite fun_eq_iff fin) } |
|
953 from this[of UNIV] show "Finite_Set.fold (\<lambda>(a :: 'c) c. c(\<^sup>f a := g b')) (\<lambda>\<^isup>f g b) UNIV = (\<lambda>\<^isup>f g b')" |
|
954 by(simp add: finfun_const_def) |
|
955 qed |
|
956 |
|
957 lemma finfun_comp_const [simp, code]: |
|
958 "g \<circ>\<^isub>f (\<lambda>\<^isup>f c) = (\<lambda>\<^isup>f g c)" |
|
959 by(simp add: finfun_comp_def) |
|
960 |
|
961 lemma finfun_comp_update [simp]: "g \<circ>\<^isub>f (f(\<^sup>f a := b)) = (g \<circ>\<^isub>f f)(\<^sup>f a := g b)" |
|
962 and finfun_comp_update_code [code]: "g \<circ>\<^isub>f (finfun_update_code f a b) = finfun_update_code (g \<circ>\<^isub>f f) a (g b)" |
|
963 by(simp_all add: finfun_comp_def) |
|
964 |
|
965 lemma finfun_comp_apply [simp]: |
|
966 "(g \<circ>\<^isub>f f)\<^sub>f = g \<circ> f\<^sub>f" |
|
967 by(induct f rule: finfun_weak_induct)(auto simp add: finfun_upd_apply intro: ext) |
|
968 |
|
969 lemma finfun_comp_comp_collapse [simp]: "f \<circ>\<^isub>f g \<circ>\<^isub>f h = (f o g) \<circ>\<^isub>f h" |
|
970 by(induct h rule: finfun_weak_induct) simp_all |
|
971 |
|
972 lemma finfun_comp_const1 [simp]: "(\<lambda>x. c) \<circ>\<^isub>f f = (\<lambda>\<^isup>f c)" |
|
973 by(induct f rule: finfun_weak_induct)(auto intro: finfun_ext simp add: finfun_upd_apply) |
|
974 |
|
975 lemma finfun_comp_id1 [simp]: "(\<lambda>x. x) \<circ>\<^isub>f f = f" "id \<circ>\<^isub>f f = f" |
|
976 by(induct f rule: finfun_weak_induct) auto |
|
977 |
|
978 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
979 |
|
980 lemma finfun_comp_conv_comp: "g \<circ>\<^isub>f f = Abs_finfun (g \<circ> finfun_apply f)" |
|
981 proof - |
|
982 have "(\<lambda>f. g \<circ>\<^isub>f f) = (\<lambda>f. Abs_finfun (g \<circ> finfun_apply f))" |
|
983 proof(rule finfun_rec_unique) |
|
984 { fix c show "Abs_finfun (g \<circ> (\<lambda>\<^isup>f c)\<^sub>f) = (\<lambda>\<^isup>f g c)" |
|
985 by(simp add: finfun_comp_def o_def)(simp add: finfun_const_def) } |
|
986 { fix g' a b show "Abs_finfun (g \<circ> g'(\<^sup>f a := b)\<^sub>f) = (Abs_finfun (g \<circ> g'\<^sub>f))(\<^sup>f a := g b)" |
|
987 proof - |
|
988 obtain y where y: "y \<in> finfun" and g': "g' = Abs_finfun y" by(cases g') |
|
989 moreover hence "(g \<circ> g'\<^sub>f) \<in> finfun" by(simp add: finfun_apply_Rep_finfun finfun_left_compose) |
|
990 moreover have "g \<circ> y(a := b) = (g \<circ> y)(a := g b)" by(auto intro: ext) |
|
991 ultimately show ?thesis by(simp add: finfun_comp_def finfun_update_def finfun_apply_Rep_finfun) |
|
992 qed } |
|
993 qed auto |
|
994 thus ?thesis by(auto simp add: fun_eq_iff) |
|
995 qed |
|
996 |
|
997 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
998 |
|
999 definition finfun_comp2 :: "'b \<Rightarrow>\<^isub>f 'c \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'c" (infixr "\<^sub>f\<circ>" 55) |
|
1000 where [code del]: "finfun_comp2 g f = Abs_finfun (Rep_finfun g \<circ> f)" |
|
1001 |
|
1002 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
1003 |
|
1004 lemma finfun_comp2_const [code, simp]: "finfun_comp2 (\<lambda>\<^isup>f c) f = (\<lambda>\<^isup>f c)" |
|
1005 by(simp add: finfun_comp2_def finfun_const_def comp_def) |
|
1006 |
|
1007 lemma finfun_comp2_update: |
|
1008 assumes inj: "inj f" |
|
1009 shows "finfun_comp2 (g(\<^sup>f b := c)) f = (if b \<in> range f then (finfun_comp2 g f)(\<^sup>f inv f b := c) else finfun_comp2 g f)" |
|
1010 proof(cases "b \<in> range f") |
|
1011 case True |
|
1012 from inj have "\<And>x. (Rep_finfun g)(f x := c) \<circ> f = (Rep_finfun g \<circ> f)(x := c)" by(auto intro!: ext dest: injD) |
|
1013 with inj True show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def finfun_right_compose) |
|
1014 next |
|
1015 case False |
|
1016 hence "(Rep_finfun g)(b := c) \<circ> f = Rep_finfun g \<circ> f" by(auto simp add: fun_eq_iff) |
|
1017 with False show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def) |
|
1018 qed |
|
1019 |
|
1020 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
1021 |
|
1022 |
|
1023 |
|
1024 subsection {* Universal quantification *} |
|
1025 |
|
1026 definition finfun_All_except :: "'a list \<Rightarrow> 'a \<Rightarrow>\<^isub>f bool \<Rightarrow> bool" |
|
1027 where [code del]: "finfun_All_except A P \<equiv> \<forall>a. a \<in> set A \<or> P\<^sub>f a" |
|
1028 |
|
1029 lemma finfun_All_except_const: "finfun_All_except A (\<lambda>\<^isup>f b) \<longleftrightarrow> b \<or> set A = UNIV" |
|
1030 by(auto simp add: finfun_All_except_def) |
|
1031 |
|
1032 lemma finfun_All_except_const_finfun_UNIV_code [code]: |
|
1033 "finfun_All_except A (\<lambda>\<^isup>f b) = (b \<or> is_list_UNIV A)" |
|
1034 by(simp add: finfun_All_except_const is_list_UNIV_iff) |
|
1035 |
|
1036 lemma finfun_All_except_update: |
|
1037 "finfun_All_except A f(\<^sup>f a := b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)" |
|
1038 by(fastforce simp add: finfun_All_except_def finfun_upd_apply) |
|
1039 |
|
1040 lemma finfun_All_except_update_code [code]: |
|
1041 fixes a :: "'a :: card_UNIV" |
|
1042 shows "finfun_All_except A (finfun_update_code f a b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)" |
|
1043 by(simp add: finfun_All_except_update) |
|
1044 |
|
1045 definition finfun_All :: "'a \<Rightarrow>\<^isub>f bool \<Rightarrow> bool" |
|
1046 where "finfun_All = finfun_All_except []" |
|
1047 |
|
1048 lemma finfun_All_const [simp]: "finfun_All (\<lambda>\<^isup>f b) = b" |
|
1049 by(simp add: finfun_All_def finfun_All_except_def) |
|
1050 |
|
1051 lemma finfun_All_update: "finfun_All f(\<^sup>f a := b) = (b \<and> finfun_All_except [a] f)" |
|
1052 by(simp add: finfun_All_def finfun_All_except_update) |
|
1053 |
|
1054 lemma finfun_All_All: "finfun_All P = All P\<^sub>f" |
|
1055 by(simp add: finfun_All_def finfun_All_except_def) |
|
1056 |
|
1057 |
|
1058 definition finfun_Ex :: "'a \<Rightarrow>\<^isub>f bool \<Rightarrow> bool" |
|
1059 where "finfun_Ex P = Not (finfun_All (Not \<circ>\<^isub>f P))" |
|
1060 |
|
1061 lemma finfun_Ex_Ex: "finfun_Ex P = Ex P\<^sub>f" |
|
1062 unfolding finfun_Ex_def finfun_All_All by simp |
|
1063 |
|
1064 lemma finfun_Ex_const [simp]: "finfun_Ex (\<lambda>\<^isup>f b) = b" |
|
1065 by(simp add: finfun_Ex_def) |
|
1066 |
|
1067 |
|
1068 subsection {* A diagonal operator for FinFuns *} |
|
1069 |
|
1070 definition finfun_Diag :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'c \<Rightarrow> 'a \<Rightarrow>\<^isub>f ('b \<times> 'c)" ("(1'(_,/ _')\<^sup>f)" [0, 0] 1000) |
|
1071 where [code del]: "finfun_Diag f g = finfun_rec (\<lambda>b. Pair b \<circ>\<^isub>f g) (\<lambda>a b c. c(\<^sup>f a := (b, g\<^sub>f a))) f" |
|
1072 |
|
1073 interpretation finfun_Diag_aux: finfun_rec_wf_aux "\<lambda>b. Pair b \<circ>\<^isub>f g" "\<lambda>a b c. c(\<^sup>f a := (b, g\<^sub>f a))" |
|
1074 by(unfold_locales)(simp_all add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
|
1075 |
|
1076 interpretation finfun_Diag: finfun_rec_wf "\<lambda>b. Pair b \<circ>\<^isub>f g" "\<lambda>a b c. c(\<^sup>f a := (b, g\<^sub>f a))" |
|
1077 proof |
|
1078 fix b' b :: 'a |
|
1079 assume fin: "finite (UNIV :: 'c set)" |
|
1080 { fix A :: "'c set" |
|
1081 interpret comp_fun_commute "\<lambda>a c. c(\<^sup>f a := (b', g\<^sub>f a))" by(rule finfun_Diag_aux.upd_left_comm) |
|
1082 from fin have "finite A" by(auto intro: finite_subset) |
|
1083 hence "Finite_Set.fold (\<lambda>a c. c(\<^sup>f a := (b', g\<^sub>f a))) (Pair b \<circ>\<^isub>f g) A = |
|
1084 Abs_finfun (\<lambda>a. (if a \<in> A then b' else b, g\<^sub>f a))" |
|
1085 by(induct)(simp_all add: finfun_const_def finfun_comp_conv_comp o_def, |
|
1086 auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite fun_eq_iff fin) } |
|
1087 from this[of UNIV] show "Finite_Set.fold (\<lambda>a c. c(\<^sup>f a := (b', g\<^sub>f a))) (Pair b \<circ>\<^isub>f g) UNIV = Pair b' \<circ>\<^isub>f g" |
|
1088 by(simp add: finfun_const_def finfun_comp_conv_comp o_def) |
|
1089 qed |
|
1090 |
|
1091 lemma finfun_Diag_const1: "(\<lambda>\<^isup>f b, g)\<^sup>f = Pair b \<circ>\<^isub>f g" |
|
1092 by(simp add: finfun_Diag_def) |
|
1093 |
|
1094 text {* |
|
1095 Do not use @{thm finfun_Diag_const1} for the code generator because @{term "Pair b"} is injective, i.e. if @{term g} is free of redundant updates, there is no need to check for redundant updates as is done for @{text "\<circ>\<^isub>f"}. |
|
1096 *} |
|
1097 |
|
1098 lemma finfun_Diag_const_code [code]: |
|
1099 "(\<lambda>\<^isup>f b, \<lambda>\<^isup>f c)\<^sup>f = (\<lambda>\<^isup>f (b, c))" |
|
1100 "(\<lambda>\<^isup>f b, g(\<^sup>fc a := c))\<^sup>f = (\<lambda>\<^isup>f b, g)\<^sup>f(\<^sup>fc a := (b, c))" |
|
1101 by(simp_all add: finfun_Diag_const1) |
|
1102 |
|
1103 lemma finfun_Diag_update1: "(f(\<^sup>f a := b), g)\<^sup>f = (f, g)\<^sup>f(\<^sup>f a := (b, g\<^sub>f a))" |
|
1104 and finfun_Diag_update1_code [code]: "(finfun_update_code f a b, g)\<^sup>f = (f, g)\<^sup>f(\<^sup>f a := (b, g\<^sub>f a))" |
|
1105 by(simp_all add: finfun_Diag_def) |
|
1106 |
|
1107 lemma finfun_Diag_const2: "(f, \<lambda>\<^isup>f c)\<^sup>f = (\<lambda>b. (b, c)) \<circ>\<^isub>f f" |
|
1108 by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1) |
|
1109 |
|
1110 lemma finfun_Diag_update2: "(f, g(\<^sup>f a := c))\<^sup>f = (f, g)\<^sup>f(\<^sup>f a := (f\<^sub>f a, c))" |
|
1111 by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1) |
|
1112 |
|
1113 lemma finfun_Diag_const_const [simp]: "(\<lambda>\<^isup>f b, \<lambda>\<^isup>f c)\<^sup>f = (\<lambda>\<^isup>f (b, c))" |
|
1114 by(simp add: finfun_Diag_const1) |
|
1115 |
|
1116 lemma finfun_Diag_const_update: |
|
1117 "(\<lambda>\<^isup>f b, g(\<^sup>f a := c))\<^sup>f = (\<lambda>\<^isup>f b, g)\<^sup>f(\<^sup>f a := (b, c))" |
|
1118 by(simp add: finfun_Diag_const1) |
|
1119 |
|
1120 lemma finfun_Diag_update_const: |
|
1121 "(f(\<^sup>f a := b), \<lambda>\<^isup>f c)\<^sup>f = (f, \<lambda>\<^isup>f c)\<^sup>f(\<^sup>f a := (b, c))" |
|
1122 by(simp add: finfun_Diag_def) |
|
1123 |
|
1124 lemma finfun_Diag_update_update: |
|
1125 "(f(\<^sup>f a := b), g(\<^sup>f a' := c))\<^sup>f = (if a = a' then (f, g)\<^sup>f(\<^sup>f a := (b, c)) else (f, g)\<^sup>f(\<^sup>f a := (b, g\<^sub>f a))(\<^sup>f a' := (f\<^sub>f a', c)))" |
|
1126 by(auto simp add: finfun_Diag_update1 finfun_Diag_update2) |
|
1127 |
|
1128 lemma finfun_Diag_apply [simp]: "(f, g)\<^sup>f\<^sub>f = (\<lambda>x. (f\<^sub>f x, g\<^sub>f x))" |
|
1129 by(induct f rule: finfun_weak_induct)(auto simp add: finfun_Diag_const1 finfun_Diag_update1 finfun_upd_apply intro: ext) |
|
1130 |
|
1131 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
1132 |
|
1133 lemma finfun_Diag_conv_Abs_finfun: |
|
1134 "(f, g)\<^sup>f = Abs_finfun ((\<lambda>x. (Rep_finfun f x, Rep_finfun g x)))" |
|
1135 proof - |
|
1136 have "(\<lambda>f :: 'a \<Rightarrow>\<^isub>f 'b. (f, g)\<^sup>f) = (\<lambda>f. Abs_finfun ((\<lambda>x. (Rep_finfun f x, Rep_finfun g x))))" |
|
1137 proof(rule finfun_rec_unique) |
|
1138 { fix c show "Abs_finfun (\<lambda>x. (Rep_finfun (\<lambda>\<^isup>f c) x, Rep_finfun g x)) = Pair c \<circ>\<^isub>f g" |
|
1139 by(simp add: finfun_comp_conv_comp finfun_apply_Rep_finfun o_def finfun_const_def) } |
|
1140 { fix g' a b |
|
1141 show "Abs_finfun (\<lambda>x. (Rep_finfun g'(\<^sup>f a := b) x, Rep_finfun g x)) = |
|
1142 (Abs_finfun (\<lambda>x. (Rep_finfun g' x, Rep_finfun g x)))(\<^sup>f a := (b, g\<^sub>f a))" |
|
1143 by(auto simp add: finfun_update_def fun_eq_iff finfun_apply_Rep_finfun simp del: fun_upd_apply) simp } |
|
1144 qed(simp_all add: finfun_Diag_const1 finfun_Diag_update1) |
|
1145 thus ?thesis by(auto simp add: fun_eq_iff) |
|
1146 qed |
|
1147 |
|
1148 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
1149 |
|
1150 lemma finfun_Diag_eq: "(f, g)\<^sup>f = (f', g')\<^sup>f \<longleftrightarrow> f = f' \<and> g = g'" |
|
1151 by(auto simp add: expand_finfun_eq fun_eq_iff) |
|
1152 |
|
1153 definition finfun_fst :: "'a \<Rightarrow>\<^isub>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" |
|
1154 where [code]: "finfun_fst f = fst \<circ>\<^isub>f f" |
|
1155 |
|
1156 lemma finfun_fst_const: "finfun_fst (\<lambda>\<^isup>f bc) = (\<lambda>\<^isup>f fst bc)" |
|
1157 by(simp add: finfun_fst_def) |
|
1158 |
|
1159 lemma finfun_fst_update: "finfun_fst (f(\<^sup>f a := bc)) = (finfun_fst f)(\<^sup>f a := fst bc)" |
|
1160 and finfun_fst_update_code: "finfun_fst (finfun_update_code f a bc) = (finfun_fst f)(\<^sup>f a := fst bc)" |
|
1161 by(simp_all add: finfun_fst_def) |
|
1162 |
|
1163 lemma finfun_fst_comp_conv: "finfun_fst (f \<circ>\<^isub>f g) = (fst \<circ> f) \<circ>\<^isub>f g" |
|
1164 by(simp add: finfun_fst_def) |
|
1165 |
|
1166 lemma finfun_fst_conv [simp]: "finfun_fst (f, g)\<^sup>f = f" |
|
1167 by(induct f rule: finfun_weak_induct)(simp_all add: finfun_Diag_const1 finfun_fst_comp_conv o_def finfun_Diag_update1 finfun_fst_update) |
|
1168 |
|
1169 lemma finfun_fst_conv_Abs_finfun: "finfun_fst = (\<lambda>f. Abs_finfun (fst o Rep_finfun f))" |
|
1170 by(simp add: finfun_fst_def [abs_def] finfun_comp_conv_comp finfun_apply_Rep_finfun) |
|
1171 |
|
1172 |
|
1173 definition finfun_snd :: "'a \<Rightarrow>\<^isub>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'c" |
|
1174 where [code]: "finfun_snd f = snd \<circ>\<^isub>f f" |
|
1175 |
|
1176 lemma finfun_snd_const: "finfun_snd (\<lambda>\<^isup>f bc) = (\<lambda>\<^isup>f snd bc)" |
|
1177 by(simp add: finfun_snd_def) |
|
1178 |
|
1179 lemma finfun_snd_update: "finfun_snd (f(\<^sup>f a := bc)) = (finfun_snd f)(\<^sup>f a := snd bc)" |
|
1180 and finfun_snd_update_code [code]: "finfun_snd (finfun_update_code f a bc) = (finfun_snd f)(\<^sup>f a := snd bc)" |
|
1181 by(simp_all add: finfun_snd_def) |
|
1182 |
|
1183 lemma finfun_snd_comp_conv: "finfun_snd (f \<circ>\<^isub>f g) = (snd \<circ> f) \<circ>\<^isub>f g" |
|
1184 by(simp add: finfun_snd_def) |
|
1185 |
|
1186 lemma finfun_snd_conv [simp]: "finfun_snd (f, g)\<^sup>f = g" |
|
1187 apply(induct f rule: finfun_weak_induct) |
|
1188 apply(auto simp add: finfun_Diag_const1 finfun_snd_comp_conv o_def finfun_Diag_update1 finfun_snd_update finfun_upd_apply intro: finfun_ext) |
|
1189 done |
|
1190 |
|
1191 lemma finfun_snd_conv_Abs_finfun: "finfun_snd = (\<lambda>f. Abs_finfun (snd o Rep_finfun f))" |
|
1192 by(simp add: finfun_snd_def [abs_def] finfun_comp_conv_comp finfun_apply_Rep_finfun) |
|
1193 |
|
1194 lemma finfun_Diag_collapse [simp]: "(finfun_fst f, finfun_snd f)\<^sup>f = f" |
|
1195 by(induct f rule: finfun_weak_induct)(simp_all add: finfun_fst_const finfun_snd_const finfun_fst_update finfun_snd_update finfun_Diag_update_update) |
|
1196 |
|
1197 subsection {* Currying for FinFuns *} |
|
1198 |
|
1199 definition finfun_curry :: "('a \<times> 'b) \<Rightarrow>\<^isub>f 'c \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b \<Rightarrow>\<^isub>f 'c" |
|
1200 where [code del]: "finfun_curry = finfun_rec (finfun_const \<circ> finfun_const) (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c)))" |
|
1201 |
|
1202 interpretation finfun_curry_aux: finfun_rec_wf_aux "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))" |
|
1203 apply(unfold_locales) |
|
1204 apply(auto simp add: split_def finfun_update_twist finfun_upd_apply split_paired_all finfun_update_const_same) |
|
1205 done |
|
1206 |
|
1207 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
1208 |
|
1209 interpretation finfun_curry: finfun_rec_wf "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))" |
|
1210 proof(unfold_locales) |
|
1211 fix b' b :: 'b |
|
1212 assume fin: "finite (UNIV :: ('c \<times> 'a) set)" |
|
1213 hence fin1: "finite (UNIV :: 'c set)" and fin2: "finite (UNIV :: 'a set)" |
|
1214 unfolding UNIV_Times_UNIV[symmetric] |
|
1215 by(fastforce dest: finite_cartesian_productD1 finite_cartesian_productD2)+ |
|
1216 note [simp] = Abs_finfun_inverse_finite[OF fin] Abs_finfun_inverse_finite[OF fin1] Abs_finfun_inverse_finite[OF fin2] |
|
1217 { fix A :: "('c \<times> 'a) set" |
|
1218 interpret comp_fun_commute "\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))) a b'" |
|
1219 by(rule finfun_curry_aux.upd_left_comm) |
|
1220 from fin have "finite A" by(auto intro: finite_subset) |
|
1221 hence "Finite_Set.fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))) a b') ((finfun_const \<circ> finfun_const) b) A = Abs_finfun (\<lambda>a. Abs_finfun (\<lambda>b''. if (a, b'') \<in> A then b' else b))" |
|
1222 by induct (simp_all, auto simp add: finfun_update_def finfun_const_def split_def finfun_apply_Rep_finfun intro!: arg_cong[where f="Abs_finfun"] ext) } |
|
1223 from this[of UNIV] |
|
1224 show "Finite_Set.fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))) a b') ((finfun_const \<circ> finfun_const) b) UNIV = (finfun_const \<circ> finfun_const) b'" |
|
1225 by(simp add: finfun_const_def) |
|
1226 qed |
|
1227 |
|
1228 declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del] |
|
1229 |
|
1230 lemma finfun_curry_const [simp, code]: "finfun_curry (\<lambda>\<^isup>f c) = (\<lambda>\<^isup>f \<lambda>\<^isup>f c)" |
|
1231 by(simp add: finfun_curry_def) |
|
1232 |
|
1233 lemma finfun_curry_update [simp]: |
|
1234 "finfun_curry (f(\<^sup>f (a, b) := c)) = (finfun_curry f)(\<^sup>f a := ((finfun_curry f)\<^sub>f a)(\<^sup>f b := c))" |
|
1235 and finfun_curry_update_code [code]: |
|
1236 "finfun_curry (f(\<^sup>fc (a, b) := c)) = (finfun_curry f)(\<^sup>f a := ((finfun_curry f)\<^sub>f a)(\<^sup>f b := c))" |
|
1237 by(simp_all add: finfun_curry_def) |
|
1238 |
|
1239 declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro] |
|
1240 |
|
1241 lemma finfun_Abs_finfun_curry: assumes fin: "f \<in> finfun" |
|
1242 shows "(\<lambda>a. Abs_finfun (curry f a)) \<in> finfun" |
|
1243 proof - |
|
1244 from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast |
|
1245 have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force) |
|
1246 hence "{a. curry f a \<noteq> (\<lambda>x. c)} = fst ` {ab. f ab \<noteq> c}" |
|
1247 by(auto simp add: curry_def fun_eq_iff) |
|
1248 with fin c have "finite {a. Abs_finfun (curry f a) \<noteq> (\<lambda>\<^isup>f c)}" |
|
1249 by(simp add: finfun_const_def finfun_curry) |
|
1250 thus ?thesis unfolding finfun_def by auto |
|
1251 qed |
|
1252 |
|
1253 lemma finfun_curry_conv_curry: |
|
1254 fixes f :: "('a \<times> 'b) \<Rightarrow>\<^isub>f 'c" |
|
1255 shows "finfun_curry f = Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun f) a))" |
|
1256 proof - |
|
1257 have "finfun_curry = (\<lambda>f :: ('a \<times> 'b) \<Rightarrow>\<^isub>f 'c. Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun f) a)))" |
|
1258 proof(rule finfun_rec_unique) |
|
1259 { fix c show "finfun_curry (\<lambda>\<^isup>f c) = (\<lambda>\<^isup>f \<lambda>\<^isup>f c)" by simp } |
|
1260 { fix f a c show "finfun_curry (f(\<^sup>f a := c)) = (finfun_curry f)(\<^sup>f fst a := ((finfun_curry f)\<^sub>f (fst a))(\<^sup>f snd a := c))" |
|
1261 by(cases a) simp } |
|
1262 { fix c show "Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun (\<lambda>\<^isup>f c)) a)) = (\<lambda>\<^isup>f \<lambda>\<^isup>f c)" |
|
1263 by(simp add: finfun_curry_def finfun_const_def curry_def) } |
|
1264 { fix g a b |
|
1265 show "Abs_finfun (\<lambda>aa. Abs_finfun (curry (Rep_finfun g(\<^sup>f a := b)) aa)) = |
|
1266 (Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun g) a)))(\<^sup>f |
|
1267 fst a := ((Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun g) a)))\<^sub>f (fst a))(\<^sup>f snd a := b))" |
|
1268 by(cases a)(auto intro!: ext arg_cong[where f=Abs_finfun] simp add: finfun_curry_def finfun_update_def finfun_apply_Rep_finfun finfun_curry finfun_Abs_finfun_curry) } |
|
1269 qed |
|
1270 thus ?thesis by(auto simp add: fun_eq_iff) |
|
1271 qed |
|
1272 |
|
1273 subsection {* Executable equality for FinFuns *} |
|
1274 |
|
1275 lemma eq_finfun_All_ext: "(f = g) \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>\<^isub>f (f, g)\<^sup>f)" |
|
1276 by(simp add: expand_finfun_eq fun_eq_iff finfun_All_All o_def) |
|
1277 |
|
1278 instantiation finfun :: ("{card_UNIV,equal}",equal) equal begin |
|
1279 definition eq_finfun_def [code]: "HOL.equal f g \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>\<^isub>f (f, g)\<^sup>f)" |
|
1280 instance by(intro_classes)(simp add: eq_finfun_All_ext eq_finfun_def) |
|
1281 end |
|
1282 |
|
1283 lemma [code nbe]: |
|
1284 "HOL.equal (f :: _ \<Rightarrow>\<^isub>f _) f \<longleftrightarrow> True" |
|
1285 by (fact equal_refl) |
|
1286 |
|
1287 subsection {* An operator that explicitly removes all redundant updates in the generated representations *} |
|
1288 |
|
1289 definition finfun_clearjunk :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" |
|
1290 where [simp, code del]: "finfun_clearjunk = id" |
|
1291 |
|
1292 lemma finfun_clearjunk_const [code]: "finfun_clearjunk (\<lambda>\<^isup>f b) = (\<lambda>\<^isup>f b)" |
|
1293 by simp |
|
1294 |
|
1295 lemma finfun_clearjunk_update [code]: "finfun_clearjunk (finfun_update_code f a b) = f(\<^sup>f a := b)" |
|
1296 by simp |
|
1297 |
|
1298 subsection {* The domain of a FinFun as a FinFun *} |
|
1299 |
|
1300 definition finfun_dom :: "('a \<Rightarrow>\<^isub>f 'b) \<Rightarrow> ('a \<Rightarrow>\<^isub>f bool)" |
|
1301 where [code del]: "finfun_dom f = Abs_finfun (\<lambda>a. f\<^sub>f a \<noteq> finfun_default f)" |
|
1302 |
|
1303 lemma finfun_dom_const: |
|
1304 "finfun_dom ((\<lambda>\<^isup>f c) :: 'a \<Rightarrow>\<^isub>f 'b) = (\<lambda>\<^isup>f finite (UNIV :: 'a set) \<and> c \<noteq> undefined)" |
|
1305 unfolding finfun_dom_def finfun_default_const |
|
1306 by(auto)(simp_all add: finfun_const_def) |
|
1307 |
|
1308 text {* |
|
1309 @{term "finfun_dom" } raises an exception when called on a FinFun whose domain is a finite type. |
|
1310 For such FinFuns, the default value (and as such the domain) is undefined. |
|
1311 *} |
|
1312 |
|
1313 lemma finfun_dom_const_code [code]: |
|
1314 "finfun_dom ((\<lambda>\<^isup>f c) :: ('a :: card_UNIV) \<Rightarrow>\<^isub>f 'b) = |
|
1315 (if card_UNIV (TYPE('a)) = 0 then (\<lambda>\<^isup>f False) else FinFun.code_abort (\<lambda>_. finfun_dom (\<lambda>\<^isup>f c)))" |
|
1316 unfolding card_UNIV_eq_0_infinite_UNIV |
|
1317 by(simp add: finfun_dom_const) |
|
1318 |
|
1319 lemma finfun_dom_finfunI: "(\<lambda>a. f\<^sub>f a \<noteq> finfun_default f) \<in> finfun" |
|
1320 using finite_finfun_default[of f] |
|
1321 by(simp add: finfun_def finfun_apply_Rep_finfun exI[where x=False]) |
|
1322 |
|
1323 lemma finfun_dom_update [simp]: |
|
1324 "finfun_dom (f(\<^sup>f a := b)) = (finfun_dom f)(\<^sup>f a := (b \<noteq> finfun_default f))" |
|
1325 unfolding finfun_dom_def finfun_update_def |
|
1326 apply(simp add: finfun_default_update_const finfun_upd_apply finfun_dom_finfunI) |
|
1327 apply(fold finfun_update.rep_eq) |
|
1328 apply(simp add: finfun_upd_apply fun_eq_iff finfun_default_update_const) |
|
1329 done |
|
1330 |
|
1331 lemma finfun_dom_update_code [code]: |
|
1332 "finfun_dom (finfun_update_code f a b) = finfun_update_code (finfun_dom f) a (b \<noteq> finfun_default f)" |
|
1333 by(simp) |
|
1334 |
|
1335 lemma finite_finfun_dom: "finite {x. (finfun_dom f)\<^sub>f x}" |
|
1336 proof(induct f rule: finfun_weak_induct) |
|
1337 case (const b) |
|
1338 thus ?case |
|
1339 by (cases "finite (UNIV :: 'a set) \<and> b \<noteq> undefined") |
|
1340 (auto simp add: finfun_dom_const UNIV_def [symmetric] Set.empty_def [symmetric]) |
|
1341 next |
|
1342 case (update f a b) |
|
1343 have "{x. (finfun_dom f(\<^sup>f a := b))\<^sub>f x} = |
|
1344 (if b = finfun_default f then {x. (finfun_dom f)\<^sub>f x} - {a} else insert a {x. (finfun_dom f)\<^sub>f x})" |
|
1345 by (auto simp add: finfun_upd_apply split: split_if_asm) |
|
1346 thus ?case using update by simp |
|
1347 qed |
|
1348 |
|
1349 |
|
1350 subsection {* The domain of a FinFun as a sorted list *} |
|
1351 |
|
1352 definition finfun_to_list :: "('a :: linorder) \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a list" |
|
1353 where |
|
1354 "finfun_to_list f = (THE xs. set xs = {x. (finfun_dom f)\<^sub>f x} \<and> sorted xs \<and> distinct xs)" |
|
1355 |
|
1356 lemma set_finfun_to_list [simp]: "set (finfun_to_list f) = {x. (finfun_dom f)\<^sub>f x}" (is ?thesis1) |
|
1357 and sorted_finfun_to_list: "sorted (finfun_to_list f)" (is ?thesis2) |
|
1358 and distinct_finfun_to_list: "distinct (finfun_to_list f)" (is ?thesis3) |
|
1359 proof - |
|
1360 have "?thesis1 \<and> ?thesis2 \<and> ?thesis3" |
|
1361 unfolding finfun_to_list_def |
|
1362 by(rule theI')(rule finite_sorted_distinct_unique finite_finfun_dom)+ |
|
1363 thus ?thesis1 ?thesis2 ?thesis3 by simp_all |
|
1364 qed |
|
1365 |
|
1366 lemma finfun_const_False_conv_bot: "(\<lambda>\<^isup>f False)\<^sub>f = bot" |
|
1367 by auto |
|
1368 |
|
1369 lemma finfun_const_True_conv_top: "(\<lambda>\<^isup>f True)\<^sub>f = top" |
|
1370 by auto |
|
1371 |
|
1372 lemma finfun_to_list_const: |
|
1373 "finfun_to_list ((\<lambda>\<^isup>f c) :: ('a :: {linorder} \<Rightarrow>\<^isub>f 'b)) = |
|
1374 (if \<not> finite (UNIV :: 'a set) \<or> c = undefined then [] else THE xs. set xs = UNIV \<and> sorted xs \<and> distinct xs)" |
|
1375 by(auto simp add: finfun_to_list_def finfun_const_False_conv_bot finfun_const_True_conv_top finfun_dom_const) |
|
1376 |
|
1377 lemma finfun_to_list_const_code [code]: |
|
1378 "finfun_to_list ((\<lambda>\<^isup>f c) :: ('a :: {linorder, card_UNIV} \<Rightarrow>\<^isub>f 'b)) = |
|
1379 (if card_UNIV (TYPE('a)) = 0 then [] else FinFun.code_abort (\<lambda>_. finfun_to_list ((\<lambda>\<^isup>f c) :: ('a \<Rightarrow>\<^isub>f 'b))))" |
|
1380 unfolding card_UNIV_eq_0_infinite_UNIV |
|
1381 by(auto simp add: finfun_to_list_const) |
|
1382 |
|
1383 lemma remove1_insort_insert_same: |
|
1384 "x \<notin> set xs \<Longrightarrow> remove1 x (insort_insert x xs) = xs" |
|
1385 by (metis insort_insert_insort remove1_insort) |
|
1386 |
|
1387 lemma finfun_dom_conv: |
|
1388 "(finfun_dom f)\<^sub>f x \<longleftrightarrow> f\<^sub>f x \<noteq> finfun_default f" |
|
1389 by(induct f rule: finfun_weak_induct)(auto simp add: finfun_dom_const finfun_default_const finfun_default_update_const finfun_upd_apply) |
|
1390 |
|
1391 lemma finfun_to_list_update: |
|
1392 "finfun_to_list (f(\<^sup>f a := b)) = |
|
1393 (if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))" |
|
1394 proof(subst finfun_to_list_def, rule the_equality) |
|
1395 fix xs |
|
1396 assume "set xs = {x. (finfun_dom f(\<^sup>f a := b))\<^sub>f x} \<and> sorted xs \<and> distinct xs" |
|
1397 hence eq: "set xs = {x. (finfun_dom f(\<^sup>f a := b))\<^sub>f x}" |
|
1398 and [simp]: "sorted xs" "distinct xs" by simp_all |
|
1399 show "xs = (if b = finfun_default f then remove1 a (finfun_to_list f) else insort_insert a (finfun_to_list f))" |
|
1400 proof(cases "b = finfun_default f") |
|
1401 case True [simp] |
|
1402 show ?thesis |
|
1403 proof(cases "(finfun_dom f)\<^sub>f a") |
|
1404 case True |
|
1405 have "finfun_to_list f = insort_insert a xs" |
|
1406 unfolding finfun_to_list_def |
|
1407 proof(rule the_equality) |
|
1408 have "set (insort_insert a xs) = insert a (set xs)" by(simp add: set_insort_insert) |
|
1409 also note eq also |
|
1410 have "insert a {x. (finfun_dom f(\<^sup>f a := b))\<^sub>f x} = {x. (finfun_dom f)\<^sub>f x}" using True |
|
1411 by(auto simp add: finfun_upd_apply split: split_if_asm) |
|
1412 finally show 1: "set (insort_insert a xs) = {x. (finfun_dom f)\<^sub>f x} \<and> sorted (insort_insert a xs) \<and> distinct (insort_insert a xs)" |
|
1413 by(simp add: sorted_insort_insert distinct_insort_insert) |
|
1414 |
|
1415 fix xs' |
|
1416 assume "set xs' = {x. (finfun_dom f)\<^sub>f x} \<and> sorted xs' \<and> distinct xs'" |
|
1417 thus "xs' = insort_insert a xs" using 1 by(auto dest: sorted_distinct_set_unique) |
|
1418 qed |
|
1419 with eq True show ?thesis by(simp add: remove1_insort_insert_same) |
|
1420 next |
|
1421 case False |
|
1422 hence "f\<^sub>f a = b" by(auto simp add: finfun_dom_conv) |
|
1423 hence f: "f(\<^sup>f a := b) = f" by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
|
1424 from eq have "finfun_to_list f = xs" unfolding f finfun_to_list_def |
|
1425 by(auto elim: sorted_distinct_set_unique intro!: the_equality) |
|
1426 with eq False show ?thesis unfolding f by(simp add: remove1_idem) |
|
1427 qed |
|
1428 next |
|
1429 case False |
|
1430 show ?thesis |
|
1431 proof(cases "(finfun_dom f)\<^sub>f a") |
|
1432 case True |
|
1433 have "finfun_to_list f = xs" |
|
1434 unfolding finfun_to_list_def |
|
1435 proof(rule the_equality) |
|
1436 have "finfun_dom f = finfun_dom f(\<^sup>f a := b)" using False True |
|
1437 by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
|
1438 with eq show 1: "set xs = {x. (finfun_dom f)\<^sub>f x} \<and> sorted xs \<and> distinct xs" |
|
1439 by(simp del: finfun_dom_update) |
|
1440 |
|
1441 fix xs' |
|
1442 assume "set xs' = {x. (finfun_dom f)\<^sub>f x} \<and> sorted xs' \<and> distinct xs'" |
|
1443 thus "xs' = xs" using 1 by(auto elim: sorted_distinct_set_unique) |
|
1444 qed |
|
1445 thus ?thesis using False True eq by(simp add: insort_insert_triv) |
|
1446 next |
|
1447 case False |
|
1448 have "finfun_to_list f = remove1 a xs" |
|
1449 unfolding finfun_to_list_def |
|
1450 proof(rule the_equality) |
|
1451 have "set (remove1 a xs) = set xs - {a}" by simp |
|
1452 also note eq also |
|
1453 have "{x. (finfun_dom f(\<^sup>f a := b))\<^sub>f x} - {a} = {x. (finfun_dom f)\<^sub>f x}" using False |
|
1454 by(auto simp add: finfun_upd_apply split: split_if_asm) |
|
1455 finally show 1: "set (remove1 a xs) = {x. (finfun_dom f)\<^sub>f x} \<and> sorted (remove1 a xs) \<and> distinct (remove1 a xs)" |
|
1456 by(simp add: sorted_remove1) |
|
1457 |
|
1458 fix xs' |
|
1459 assume "set xs' = {x. (finfun_dom f)\<^sub>f x} \<and> sorted xs' \<and> distinct xs'" |
|
1460 thus "xs' = remove1 a xs" using 1 by(blast intro: sorted_distinct_set_unique) |
|
1461 qed |
|
1462 thus ?thesis using False eq `b \<noteq> finfun_default f` |
|
1463 by (simp add: insort_insert_insort insort_remove1) |
|
1464 qed |
|
1465 qed |
|
1466 qed (auto simp add: distinct_finfun_to_list sorted_finfun_to_list sorted_remove1 set_insort_insert sorted_insort_insert distinct_insort_insert finfun_upd_apply split: split_if_asm) |
|
1467 |
|
1468 lemma finfun_to_list_update_code [code]: |
|
1469 "finfun_to_list (finfun_update_code f a b) = |
|
1470 (if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))" |
|
1471 by(simp add: finfun_to_list_update) |
|
1472 |
|
1473 end |