author  haftmann 
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child 26342  0f65fa163304 
permissions  rwrr 
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(* Title: HOL/Fun.thy 
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ID: $Id$ 
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Author: Tobias Nipkow, Cambridge University Computer Laboratory 
923  4 
Copyright 1994 University of Cambridge 
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*) 
923  6 

18154  7 
header {* Notions about functions *} 
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15510  9 
theory Fun 
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imports Set 
15131  11 
begin 
2912  12 

26147  13 
text{*As a simplification rule, it replaces all function equalities by 
14 
firstorder equalities.*} 

15 
lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" 

16 
apply (rule iffI) 

17 
apply (simp (no_asm_simp)) 

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apply (rule ext) 

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apply (simp (no_asm_simp)) 

20 
done 

5305  21 

26147  22 
lemma apply_inverse: 
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"f x =u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u" 

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by auto 

2912  25 

12258  26 

26147  27 
subsection {* The Identity Function @{text id} *} 
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definition 
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id :: "'a \<Rightarrow> 'a" 
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where 
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"id = (\<lambda>x. x)" 
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26147  34 
lemma id_apply [simp]: "id x = x" 
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by (simp add: id_def) 

36 

37 
lemma image_ident [simp]: "(%x. x) ` Y = Y" 

38 
by blast 

39 

40 
lemma image_id [simp]: "id ` Y = Y" 

41 
by (simp add: id_def) 

42 

43 
lemma vimage_ident [simp]: "(%x. x) ` Y = Y" 

44 
by blast 

45 

46 
lemma vimage_id [simp]: "id ` A = A" 

47 
by (simp add: id_def) 

48 

49 

50 
subsection {* The Composition Operator @{text "f \<circ> g"} *} 

51 

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definition 
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comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) 
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where 
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"f o g = (\<lambda>x. f (g x))" 
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21210  57 
notation (xsymbols) 
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comp (infixl "\<circ>" 55) 
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notation (HTML output) 
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comp (infixl "\<circ>" 55) 
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13585  63 
text{*compatibility*} 
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lemmas o_def = comp_def 

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13585  66 
lemma o_apply [simp]: "(f o g) x = f (g x)" 
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by (simp add: comp_def) 

68 

69 
lemma o_assoc: "f o (g o h) = f o g o h" 

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by (simp add: comp_def) 

71 

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lemma id_o [simp]: "id o g = g" 

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by (simp add: comp_def) 

74 

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lemma o_id [simp]: "f o id = f" 

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by (simp add: comp_def) 

77 

78 
lemma image_compose: "(f o g) ` r = f`(g`r)" 

79 
by (simp add: comp_def, blast) 

80 

81 
lemma UN_o: "UNION A (g o f) = UNION (f`A) g" 

82 
by (unfold comp_def, blast) 

83 

84 

26147  85 
subsection {* Injectivity and Surjectivity *} 
86 

87 
constdefs 

88 
inj_on :: "['a => 'b, 'a set] => bool"  "injective" 

89 
"inj_on f A == ! x:A. ! y:A. f(x)=f(y) > x=y" 

90 

91 
text{*A common special case: functions injective over the entire domain type.*} 

92 

93 
abbreviation 

94 
"inj f == inj_on f UNIV" 

13585  95 

26147  96 
definition 
97 
bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where  "bijective" 

98 
"bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B" 

99 

100 
constdefs 

101 
surj :: "('a => 'b) => bool" (*surjective*) 

102 
"surj f == ! y. ? x. y=f(x)" 

13585  103 

26147  104 
bij :: "('a => 'b) => bool" (*bijective*) 
105 
"bij f == inj f & surj f" 

106 

107 
lemma injI: 

108 
assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" 

109 
shows "inj f" 

110 
using assms unfolding inj_on_def by auto 

13585  111 

112 
text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*} 

113 
lemma datatype_injI: 

114 
"(!! x. ALL y. f(x) = f(y) > x=y) ==> inj(f)" 

115 
by (simp add: inj_on_def) 

116 

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theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)" 
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by (unfold inj_on_def, blast) 

119 

13585  120 
lemma injD: "[ inj(f); f(x) = f(y) ] ==> x=y" 
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by (simp add: inj_on_def) 

122 

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(*Useful with the simplifier*) 

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lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)" 

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by (force simp add: inj_on_def) 

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lemma inj_on_id[simp]: "inj_on id A" 
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by (simp add: inj_on_def) 

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lemma inj_on_id2[simp]: "inj_on (%x. x) A" 
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by (simp add: inj_on_def) 

132 

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lemma surj_id[simp]: "surj id" 

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by (simp add: surj_def) 

135 

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lemma bij_id[simp]: "bij id" 

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by (simp add: bij_def inj_on_id surj_id) 

13585  138 

139 
lemma inj_onI: 

140 
"(!! x y. [ x:A; y:A; f(x) = f(y) ] ==> x=y) ==> inj_on f A" 

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by (simp add: inj_on_def) 

142 

143 
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" 

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by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) 

145 

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lemma inj_onD: "[ inj_on f A; f(x)=f(y); x:A; y:A ] ==> x=y" 

147 
by (unfold inj_on_def, blast) 

148 

149 
lemma inj_on_iff: "[ inj_on f A; x:A; y:A ] ==> (f(x)=f(y)) = (x=y)" 

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by (blast dest!: inj_onD) 

151 

152 
lemma comp_inj_on: 

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"[ inj_on f A; inj_on g (f`A) ] ==> inj_on (g o f) A" 

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by (simp add: comp_def inj_on_def) 

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lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" 
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apply(simp add:inj_on_def image_def) 

158 
apply blast 

159 
done 

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lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); 
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inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" 

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apply(unfold inj_on_def) 

164 
apply blast 

165 
done 

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lemma inj_on_contraD: "[ inj_on f A; ~x=y; x:A; y:A ] ==> ~ f(x)=f(y)" 
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by (unfold inj_on_def, blast) 

12258  169 

13585  170 
lemma inj_singleton: "inj (%s. {s})" 
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by (simp add: inj_on_def) 

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15111  173 
lemma inj_on_empty[iff]: "inj_on f {}" 
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by(simp add: inj_on_def) 

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15303  176 
lemma subset_inj_on: "[ inj_on f B; A <= B ] ==> inj_on f A" 
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by (unfold inj_on_def, blast) 
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15111  179 
lemma inj_on_Un: 
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"inj_on f (A Un B) = 

181 
(inj_on f A & inj_on f B & f`(AB) Int f`(BA) = {})" 

182 
apply(unfold inj_on_def) 

183 
apply (blast intro:sym) 

184 
done 

185 

186 
lemma inj_on_insert[iff]: 

187 
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A{a}))" 

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apply(unfold inj_on_def) 

189 
apply (blast intro:sym) 

190 
done 

191 

192 
lemma inj_on_diff: "inj_on f A ==> inj_on f (AB)" 

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apply(unfold inj_on_def) 

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apply (blast) 

195 
done 

196 

13585  197 
lemma surjI: "(!! x. g(f x) = x) ==> surj g" 
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apply (simp add: surj_def) 

199 
apply (blast intro: sym) 

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done 

201 

202 
lemma surj_range: "surj f ==> range f = UNIV" 

203 
by (auto simp add: surj_def) 

204 

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lemma surjD: "surj f ==> EX x. y = f x" 

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by (simp add: surj_def) 

207 

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lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C" 

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by (simp add: surj_def, blast) 

210 

211 
lemma comp_surj: "[ surj f; surj g ] ==> surj (g o f)" 

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apply (simp add: comp_def surj_def, clarify) 

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apply (drule_tac x = y in spec, clarify) 

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apply (drule_tac x = x in spec, blast) 

215 
done 

216 

217 
lemma bijI: "[ inj f; surj f ] ==> bij f" 

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by (simp add: bij_def) 

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220 
lemma bij_is_inj: "bij f ==> inj f" 

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by (simp add: bij_def) 

222 

223 
lemma bij_is_surj: "bij f ==> surj f" 

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by (simp add: bij_def) 

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lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A" 
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by (simp add: bij_betw_def) 
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lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" 
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proof  
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have i: "inj_on f A" and s: "f ` A = B" 
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using assms by(auto simp:bij_betw_def) 
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let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)" 
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{ fix a b assume P: "?P b a" 
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hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast 
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hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i]) 
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hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp 
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} note g = this 
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have "inj_on ?g B" 
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proof(rule inj_onI) 
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fix x y assume "x:B" "y:B" "?g x = ?g y" 
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from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast 
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from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast 
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from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp 
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qed 
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moreover have "?g ` B = A" 
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proof(auto simp:image_def) 
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fix b assume "b:B" 
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with s obtain a where P: "?P b a" unfolding image_def by blast 
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thus "?g b \<in> A" using g[OF P] by auto 
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next 
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fix a assume "a:A" 
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then obtain b where P: "?P b a" using s unfolding image_def by blast 
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then have "b:B" using s unfolding image_def by blast 
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with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast 
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qed 
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ultimately show ?thesis by(auto simp:bij_betw_def) 
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qed 
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13585  260 
lemma surj_image_vimage_eq: "surj f ==> f ` (f ` A) = A" 
261 
by (simp add: surj_range) 

262 

263 
lemma inj_vimage_image_eq: "inj f ==> f ` (f ` A) = A" 

264 
by (simp add: inj_on_def, blast) 

265 

266 
lemma vimage_subsetD: "surj f ==> f ` B <= A ==> B <= f ` A" 

267 
apply (unfold surj_def) 

268 
apply (blast intro: sym) 

269 
done 

270 

271 
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f ` B <= A" 

272 
by (unfold inj_on_def, blast) 

273 

274 
lemma vimage_subset_eq: "bij f ==> (f ` B <= A) = (B <= f ` A)" 

275 
apply (unfold bij_def) 

276 
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) 

277 
done 

278 

279 
lemma inj_on_image_Int: 

280 
"[ inj_on f C; A<=C; B<=C ] ==> f`(A Int B) = f`A Int f`B" 

281 
apply (simp add: inj_on_def, blast) 

282 
done 

283 

284 
lemma inj_on_image_set_diff: 

285 
"[ inj_on f C; A<=C; B<=C ] ==> f`(AB) = f`A  f`B" 

286 
apply (simp add: inj_on_def, blast) 

287 
done 

288 

289 
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" 

290 
by (simp add: inj_on_def, blast) 

291 

292 
lemma image_set_diff: "inj f ==> f`(AB) = f`A  f`B" 

293 
by (simp add: inj_on_def, blast) 

294 

295 
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" 

296 
by (blast dest: injD) 

297 

298 
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" 

299 
by (simp add: inj_on_def, blast) 

300 

301 
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" 

302 
by (blast dest: injD) 

303 

304 
(*injectivity's required. Lefttoright inclusion holds even if A is empty*) 

305 
lemma image_INT: 

306 
"[ inj_on f C; ALL x:A. B x <= C; j:A ] 

307 
==> f ` (INTER A B) = (INT x:A. f ` B x)" 

308 
apply (simp add: inj_on_def, blast) 

309 
done 

310 

311 
(*Compare with image_INT: no use of inj_on, and if f is surjective then 

312 
it doesn't matter whether A is empty*) 

313 
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" 

314 
apply (simp add: bij_def) 

315 
apply (simp add: inj_on_def surj_def, blast) 

316 
done 

317 

318 
lemma surj_Compl_image_subset: "surj f ==> (f`A) <= f`(A)" 

319 
by (auto simp add: surj_def) 

320 

321 
lemma inj_image_Compl_subset: "inj f ==> f`(A) <= (f`A)" 

322 
by (auto simp add: inj_on_def) 

5852  323 

13585  324 
lemma bij_image_Compl_eq: "bij f ==> f`(A) = (f`A)" 
325 
apply (simp add: bij_def) 

326 
apply (rule equalityI) 

327 
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) 

328 
done 

329 

330 

331 
subsection{*Function Updating*} 

332 

26147  333 
constdefs 
334 
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" 

335 
"fun_upd f a b == % x. if x=a then b else f x" 

336 

337 
nonterminals 

338 
updbinds updbind 

339 
syntax 

340 
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") 

341 
"" :: "updbind => updbinds" ("_") 

342 
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") 

343 
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000,0] 900) 

344 

345 
translations 

346 
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" 

347 
"f(x:=y)" == "fun_upd f x y" 

348 

349 
(* Hint: to define the sum of two functions (or maps), use sum_case. 

350 
A nice infix syntax could be defined (in Datatype.thy or below) by 

351 
consts 

352 
fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80) 

353 
translations 

354 
"fun_sum" == sum_case 

355 
*) 

356 

13585  357 
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" 
358 
apply (simp add: fun_upd_def, safe) 

359 
apply (erule subst) 

360 
apply (rule_tac [2] ext, auto) 

361 
done 

362 

363 
(* f x = y ==> f(x:=y) = f *) 

364 
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] 

365 

366 
(* f(x := f x) = f *) 

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lemmas fun_upd_triv = refl [THEN fun_upd_idem] 
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declare fun_upd_triv [iff] 
13585  369 

370 
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" 

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by (simp add: fun_upd_def) 
13585  372 

373 
(* fun_upd_apply supersedes these two, but they are useful 

374 
if fun_upd_apply is intentionally removed from the simpset *) 

375 
lemma fun_upd_same: "(f(x:=y)) x = y" 

376 
by simp 

377 

378 
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" 

379 
by simp 

380 

381 
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" 

382 
by (simp add: expand_fun_eq) 

383 

384 
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" 

385 
by (rule ext, auto) 

386 

15303  387 
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A" 
388 
by(fastsimp simp:inj_on_def image_def) 

389 

15510  390 
lemma fun_upd_image: 
391 
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A{x})) else f ` A)" 

392 
by auto 

393 

26147  394 

395 
subsection {* @{text override_on} *} 

396 

397 
definition 

398 
override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" 

399 
where 

400 
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" 

13910  401 

15691  402 
lemma override_on_emptyset[simp]: "override_on f g {} = f" 
403 
by(simp add:override_on_def) 

13910  404 

15691  405 
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" 
406 
by(simp add:override_on_def) 

13910  407 

15691  408 
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" 
409 
by(simp add:override_on_def) 

13910  410 

26147  411 

412 
subsection {* @{text swap} *} 

15510  413 

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definition 
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swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" 
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where 
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"swap a b f = f (a := f b, b:= f a)" 
15510  418 

419 
lemma swap_self: "swap a a f = f" 

15691  420 
by (simp add: swap_def) 
15510  421 

422 
lemma swap_commute: "swap a b f = swap b a f" 

423 
by (rule ext, simp add: fun_upd_def swap_def) 

424 

425 
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" 

426 
by (rule ext, simp add: fun_upd_def swap_def) 

427 

428 
lemma inj_on_imp_inj_on_swap: 

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"[inj_on f A; a \<in> A; b \<in> A] ==> inj_on (swap a b f) A" 
15510  430 
by (simp add: inj_on_def swap_def, blast) 
431 

432 
lemma inj_on_swap_iff [simp]: 

433 
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A" 

434 
proof 

435 
assume "inj_on (swap a b f) A" 

436 
with A have "inj_on (swap a b (swap a b f)) A" 

17589  437 
by (iprover intro: inj_on_imp_inj_on_swap) 
15510  438 
thus "inj_on f A" by simp 
439 
next 

440 
assume "inj_on f A" 

17589  441 
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) 
15510  442 
qed 
443 

444 
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)" 

445 
apply (simp add: surj_def swap_def, clarify) 

446 
apply (rule_tac P = "y = f b" in case_split_thm, blast) 

447 
apply (rule_tac P = "y = f a" in case_split_thm, auto) 

448 
{*We don't yet have @{text case_tac}*} 

449 
done 

450 

451 
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f" 

452 
proof 

453 
assume "surj (swap a b f)" 

454 
hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 

455 
thus "surj f" by simp 

456 
next 

457 
assume "surj f" 

458 
thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 

459 
qed 

460 

461 
lemma bij_swap_iff: "bij (swap a b f) = bij f" 

462 
by (simp add: bij_def) 

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463 

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22845  465 
subsection {* Proof tool setup *} 
466 

467 
text {* simplifies terms of the form 

468 
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} 

469 

24017  470 
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => 
22845  471 
let 
472 
fun gen_fun_upd NONE T _ _ = NONE 

24017  473 
 gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) 
22845  474 
fun dest_fun_T1 (Type (_, T :: Ts)) = T 
475 
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = 

476 
let 

477 
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = 

478 
if v aconv x then SOME g else gen_fun_upd (find g) T v w 

479 
 find t = NONE 

480 
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end 

24017  481 

482 
fun proc ss ct = 

483 
let 

484 
val ctxt = Simplifier.the_context ss 

485 
val t = Thm.term_of ct 

486 
in 

487 
case find_double t of 

488 
(T, NONE) => NONE 

489 
 (T, SOME rhs) => 

490 
SOME (Goal.prove ctxt [] [] (Term.equals T $ t $ rhs) 

491 
(fn _ => 

492 
rtac eq_reflection 1 THEN 

493 
rtac ext 1 THEN 

494 
simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) 

495 
end 

496 
in proc end 

22845  497 
*} 
498 

499 

21870  500 
subsection {* Code generator setup *} 
501 

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types_code 
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"fun" ("(_ >/ _)") 
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attach (term_of) {* 
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fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT > bT); 
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*} 
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attach (test) {* 
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fun gen_fun_type aF aT bG bT i = 
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let 
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val tab = ref []; 
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fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", 
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(aT > bT) > aT > bT > aT > bT) $ t $ aF x $ y () 
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in 
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(fn x => 
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case AList.lookup op = (!tab) x of 
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NONE => 
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let val p as (y, _) = bG i 
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in (tab := (x, p) :: !tab; y) end 
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 SOME (y, _) => y, 
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fn () => Basics.fold mk_upd (!tab) (Const ("arbitrary", aT > bT))) 
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end; 
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*} 
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523 

21870  524 
code_const "op \<circ>" 
525 
(SML infixl 5 "o") 

526 
(Haskell infixr 9 ".") 

527 

21906  528 
code_const "id" 
529 
(Haskell "id") 

530 

21870  531 

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subsection {* ML legacy bindings *} 
15510  533 

22845  534 
ML {* 
535 
val set_cs = claset() delrules [equalityI] 

536 
*} 

5852  537 

22845  538 
ML {* 
539 
val id_apply = @{thm id_apply} 

540 
val id_def = @{thm id_def} 

541 
val o_apply = @{thm o_apply} 

542 
val o_assoc = @{thm o_assoc} 

543 
val o_def = @{thm o_def} 

544 
val injD = @{thm injD} 

545 
val datatype_injI = @{thm datatype_injI} 

546 
val range_ex1_eq = @{thm range_ex1_eq} 

547 
val expand_fun_eq = @{thm expand_fun_eq} 

13585  548 
*} 
5852  549 

2912  550 
end 