author  hoelzl 
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permissions  rwrr 
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(* Title: HOL/Fun.thy 
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Author: Tobias Nipkow, Cambridge University Computer Laboratory 

923  3 
Copyright 1994 University of Cambridge 
18154  4 
*) 
923  5 

18154  6 
header {* Notions about functions *} 
923  7 

15510  8 
theory Fun 
32139  9 
imports Complete_Lattice 
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uses ("Tools/enriched_type.ML") 
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begin 
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text{*As a simplification rule, it replaces all function equalities by 
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firstorder equalities.*} 

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lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" 
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apply (rule iffI) 
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apply (simp (no_asm_simp)) 

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apply (rule ext) 

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apply (simp (no_asm_simp)) 

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done 

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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" 
26147  24 
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 

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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" 

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by (simp add: id_def) 

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49 

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

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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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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) 

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

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by (simp add: comp_def) 

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

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by (simp add: comp_def) 

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

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by (simp add: comp_def) 

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lemma o_eq_dest: 
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"a o b = c o d \<Longrightarrow> a (b v) = c (d v)" 

80 
by (simp only: o_def) (fact fun_cong) 

81 

82 
lemma o_eq_elim: 

83 
"a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R" 

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by (erule meta_mp) (fact o_eq_dest) 

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13585  86 
lemma image_compose: "(f o g) ` r = f`(g`r)" 
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by (simp add: comp_def, blast) 

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33044  89 
lemma vimage_compose: "(g \<circ> f) ` x = f ` (g ` x)" 
90 
by auto 

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13585  92 
lemma UN_o: "UNION A (g o f) = UNION (f`A) g" 
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by (unfold comp_def, blast) 

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subsection {* The Forward Composition Operator @{text fcomp} *} 
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98 
definition 

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fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) 
26357  100 
where 
37751  101 
"f \<circ>> g = (\<lambda>x. g (f x))" 
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lemma fcomp_apply [simp]: "(f \<circ>> g) x = g (f x)" 
26357  104 
by (simp add: fcomp_def) 
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lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)" 
26357  107 
by (simp add: fcomp_def) 
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37751  109 
lemma id_fcomp [simp]: "id \<circ>> g = g" 
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by (simp add: fcomp_def) 
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lemma fcomp_id [simp]: "f \<circ>> id = f" 
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by (simp add: fcomp_def) 
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code_const fcomp 
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(Eval infixl 1 "#>") 
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no_notation fcomp (infixl "\<circ>>" 60) 
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subsection {* Mapping functions *} 
122 

123 
definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where 

124 
"map_fun f g h = g \<circ> h \<circ> f" 

125 

126 
lemma map_fun_apply [simp]: 

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"map_fun f g h x = g (h (f x))" 

128 
by (simp add: map_fun_def) 

129 

130 

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subsection {* Injectivity and Bijectivity *} 
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definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where  "injective" 
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"inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)" 
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definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where  "bijective" 
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"bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B" 
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text{*A common special case: functions injective, surjective or bijective over 
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the entire domain type.*} 

26147  141 

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abbreviation 

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"inj f \<equiv> inj_on f UNIV" 
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abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where  "surjective" 
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"surj f \<equiv> (range f = UNIV)" 

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abbreviation 
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"bij f \<equiv> bij_betw f UNIV UNIV" 
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151 
lemma injI: 

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

153 
shows "inj f" 

154 
using assms unfolding inj_on_def by auto 

13585  155 

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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) 

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13585  159 
lemma injD: "[ inj(f); f(x) = f(y) ] ==> x=y" 
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by (simp add: inj_on_def) 

161 

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lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)" 
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by (force simp add: inj_on_def) 
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lemma inj_on_cong: 
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"(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A" 
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lemma inj_on_strict_subset: 
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"\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B" 
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lemma inj_comp: 
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"inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)" 

175 
by (simp add: inj_on_def) 

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177 
lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)" 

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by (simp add: inj_on_def fun_eq_iff) 
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32988  180 
lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)" 
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by (simp add: inj_on_eq_iff) 

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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) 
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lemma inj_on_Int: "\<lbrakk>inj_on f A; inj_on f B\<rbrakk> \<Longrightarrow> inj_on f (A \<inter> B)" 
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lemma inj_on_INTER: 
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"\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)\<rbrakk> \<Longrightarrow> inj_on f (\<Inter> i \<in> I. A i)" 
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lemma inj_on_Inter: 
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"\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)" 
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lemma inj_on_UNION_chain: 
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assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and 
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INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)" 
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shows "inj_on f (\<Union> i \<in> I. A i)" 
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proof(unfold inj_on_def UNION_def, auto) 
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fix i j x y 
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assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j" 
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and ***: "f x = f y" 
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show "x = y" 
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proof 
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{assume "A i \<le> A j" 
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with ** have "x \<in> A j" by auto 
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with INJ * ** *** have ?thesis 
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by(auto simp add: inj_on_def) 
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} 
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moreover 
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{assume "A j \<le> A i" 
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with ** have "y \<in> A i" by auto 
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with INJ * ** *** have ?thesis 
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by(auto simp add: inj_on_def) 
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} 
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ultimately show ?thesis using CH * by blast 
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qed 
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qed 
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40702  225 
lemma surj_id: "surj id" 
226 
by simp 

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lemma bij_id[simp]: "bij id" 
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by (simp add: bij_betw_def) 
13585  230 

231 
lemma inj_onI: 

232 
"(!! x y. [ x:A; y:A; f(x) = f(y) ] ==> x=y) ==> inj_on f A" 

233 
by (simp add: inj_on_def) 

234 

235 
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" 

236 
by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) 

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238 
lemma inj_onD: "[ inj_on f A; f(x)=f(y); x:A; y:A ] ==> x=y" 

239 
by (unfold inj_on_def, blast) 

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241 
lemma inj_on_iff: "[ inj_on f A; x:A; y:A ] ==> (f(x)=f(y)) = (x=y)" 

242 
by (blast dest!: inj_onD) 

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244 
lemma comp_inj_on: 

245 
"[ inj_on f A; inj_on g (f`A) ] ==> inj_on (g o f) A" 

246 
by (simp add: comp_def inj_on_def) 

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15303  248 
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" 
249 
apply(simp add:inj_on_def image_def) 

250 
apply blast 

251 
done 

252 

15439  253 
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); 
254 
inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" 

255 
apply(unfold inj_on_def) 

256 
apply blast 

257 
done 

258 

13585  259 
lemma inj_on_contraD: "[ inj_on f A; ~x=y; x:A; y:A ] ==> ~ f(x)=f(y)" 
260 
by (unfold inj_on_def, blast) 

12258  261 

13585  262 
lemma inj_singleton: "inj (%s. {s})" 
263 
by (simp add: inj_on_def) 

264 

15111  265 
lemma inj_on_empty[iff]: "inj_on f {}" 
266 
by(simp add: inj_on_def) 

267 

15303  268 
lemma subset_inj_on: "[ inj_on f B; A <= B ] ==> inj_on f A" 
13585  269 
by (unfold inj_on_def, blast) 
270 

15111  271 
lemma inj_on_Un: 
272 
"inj_on f (A Un B) = 

273 
(inj_on f A & inj_on f B & f`(AB) Int f`(BA) = {})" 

274 
apply(unfold inj_on_def) 

275 
apply (blast intro:sym) 

276 
done 

277 

278 
lemma inj_on_insert[iff]: 

279 
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A{a}))" 

280 
apply(unfold inj_on_def) 

281 
apply (blast intro:sym) 

282 
done 

283 

284 
lemma inj_on_diff: "inj_on f A ==> inj_on f (AB)" 

285 
apply(unfold inj_on_def) 

286 
apply (blast) 

287 
done 

288 

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289 
lemma comp_inj_on_iff: 
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290 
"inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A" 
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291 
by(auto simp add: comp_inj_on inj_on_def) 
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292 

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293 
lemma inj_on_imageI2: 
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294 
"inj_on (f' o f) A \<Longrightarrow> inj_on f A" 
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295 
by(auto simp add: comp_inj_on inj_on_def) 
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296 

40702  297 
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)" 
298 
by auto 

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299 

40702  300 
lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g" 
301 
using *[symmetric] by auto 

13585  302 

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303 
lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x" 
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304 
by (simp add: surj_def) 
13585  305 

39076
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306 
lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C" 
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307 
by (simp add: surj_def, blast) 
13585  308 

309 
lemma comp_surj: "[ surj f; surj g ] ==> surj (g o f)" 

310 
apply (simp add: comp_def surj_def, clarify) 

311 
apply (drule_tac x = y in spec, clarify) 

312 
apply (drule_tac x = x in spec, blast) 

313 
done 

314 

39074  315 
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f" 
40702  316 
unfolding bij_betw_def by auto 
39074  317 

40703
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318 
lemma bij_betw_empty1: 
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319 
assumes "bij_betw f {} A" 
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320 
shows "A = {}" 
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changeset

321 
using assms unfolding bij_betw_def by blast 
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diff
changeset

322 

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Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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changeset

323 
lemma bij_betw_empty2: 
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changeset

324 
assumes "bij_betw f A {}" 
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325 
shows "A = {}" 
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Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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diff
changeset

326 
using assms unfolding bij_betw_def by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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parents:
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diff
changeset

327 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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changeset

328 
lemma inj_on_imp_bij_betw: 
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329 
"inj_on f A \<Longrightarrow> bij_betw f A (f ` A)" 
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changeset

330 
unfolding bij_betw_def by simp 
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diff
changeset

331 

39076
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diff
changeset

332 
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f" 
40702  333 
unfolding bij_betw_def .. 
39074  334 

13585  335 
lemma bijI: "[ inj f; surj f ] ==> bij f" 
336 
by (simp add: bij_def) 

337 

338 
lemma bij_is_inj: "bij f ==> inj f" 

339 
by (simp add: bij_def) 

340 

341 
lemma bij_is_surj: "bij f ==> surj f" 

342 
by (simp add: bij_def) 

343 

26105
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moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
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344 
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A" 
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moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
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345 
by (simp add: bij_betw_def) 
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moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
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changeset

346 

31438  347 
lemma bij_betw_trans: 
348 
"bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C" 

349 
by(auto simp add:bij_betw_def comp_inj_on) 

350 

40702  351 
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)" 
352 
by (rule bij_betw_trans) 

353 

40703
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Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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changeset

354 
lemma bij_betw_comp_iff: 
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Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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diff
changeset

355 
"bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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parents:
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diff
changeset

356 
by(auto simp add: bij_betw_def inj_on_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
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diff
changeset

357 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
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diff
changeset

358 
lemma bij_betw_comp_iff2: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
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diff
changeset

359 
assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

360 
shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

361 
using assms 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

362 
proof(auto simp add: bij_betw_comp_iff) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

363 
assume *: "bij_betw (f' \<circ> f) A A''" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

364 
thus "bij_betw f A A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

365 
using IM 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

366 
proof(auto simp add: bij_betw_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

367 
assume "inj_on (f' \<circ> f) A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

368 
thus "inj_on f A" using inj_on_imageI2 by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

369 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

370 
fix a' assume **: "a' \<in> A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

371 
hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

372 
then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using * 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

373 
unfolding bij_betw_def by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

374 
hence "f a \<in> A'" using IM by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

375 
hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

376 
thus "a' \<in> f ` A" using 1 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

377 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

378 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

379 

26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

380 
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

381 
proof  
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

382 
have i: "inj_on f A" and s: "f ` A = B" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

383 
using assms by(auto simp:bij_betw_def) 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

384 
let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

385 
{ fix a b assume P: "?P b a" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

386 
hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

387 
hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i]) 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

388 
hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

389 
} note g = this 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

390 
have "inj_on ?g B" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

391 
proof(rule inj_onI) 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

392 
fix x y assume "x:B" "y:B" "?g x = ?g y" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

393 
from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

394 
from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

395 
from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

396 
qed 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

397 
moreover have "?g ` B = A" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

398 
proof(auto simp:image_def) 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

399 
fix b assume "b:B" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

400 
with s obtain a where P: "?P b a" unfolding image_def by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

401 
thus "?g b \<in> A" using g[OF P] by auto 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

402 
next 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

403 
fix a assume "a:A" 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

404 
then obtain b where P: "?P b a" using s unfolding image_def by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

405 
then have "b:B" using s unfolding image_def by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

406 
with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

407 
qed 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

408 
ultimately show ?thesis by(auto simp:bij_betw_def) 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

409 
qed 
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset

410 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

411 
lemma bij_betw_cong: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

412 
"(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

413 
unfolding bij_betw_def inj_on_def by force 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

414 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

415 
lemma bij_betw_id[intro, simp]: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

416 
"bij_betw id A A" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

417 
unfolding bij_betw_def id_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

418 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

419 
lemma bij_betw_id_iff: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

420 
"bij_betw id A B \<longleftrightarrow> A = B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

421 
by(auto simp add: bij_betw_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

422 

39075  423 
lemma bij_betw_combine: 
424 
assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}" 

425 
shows "bij_betw f (A \<union> C) (B \<union> D)" 

426 
using assms unfolding bij_betw_def inj_on_Un image_Un by auto 

427 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

428 
lemma bij_betw_UNION_chain: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

429 
assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

430 
BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

431 
shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

432 
proof(unfold bij_betw_def, auto simp add: image_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

433 
have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

434 
using BIJ bij_betw_def[of f] by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

435 
thus "inj_on f (\<Union> i \<in> I. A i)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

436 
using CH inj_on_UNION_chain[of I A f] by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

437 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

438 
fix i x 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

439 
assume *: "i \<in> I" "x \<in> A i" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

440 
hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

441 
thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

442 
next 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

443 
fix i x' 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

444 
assume *: "i \<in> I" "x' \<in> A' i" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

445 
hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

446 
thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

447 
using * by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

448 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

449 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

450 
lemma bij_betw_Disj_Un: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

451 
assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

452 
B1: "bij_betw f A A'" and B2: "bij_betw f B B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

453 
shows "bij_betw f (A \<union> B) (A' \<union> B')" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

454 
proof 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

455 
have 1: "inj_on f A \<and> inj_on f B" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

456 
using B1 B2 by (auto simp add: bij_betw_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

457 
have 2: "f`A = A' \<and> f`B = B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

458 
using B1 B2 by (auto simp add: bij_betw_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

459 
hence "f`(A  B) \<inter> f`(B  A) = {}" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

460 
using DISJ DISJ' by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

461 
hence "inj_on f (A \<union> B)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

462 
using 1 by (auto simp add: inj_on_Un) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

463 
(* *) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

464 
moreover 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

465 
have "f`(A \<union> B) = A' \<union> B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

466 
using 2 by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

467 
ultimately show ?thesis 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

468 
unfolding bij_betw_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

469 
qed 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

470 

d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

471 
lemma bij_betw_subset: 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

472 
assumes BIJ: "bij_betw f A A'" and 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

473 
SUB: "B \<le> A" and IM: "f ` B = B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

474 
shows "bij_betw f B B'" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

475 
using assms 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

476 
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def) 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

477 

13585  478 
lemma surj_image_vimage_eq: "surj f ==> f ` (f ` A) = A" 
40702  479 
by simp 
13585  480 

42903  481 
lemma surj_vimage_empty: 
482 
assumes "surj f" shows "f ` A = {} \<longleftrightarrow> A = {}" 

483 
using surj_image_vimage_eq[OF `surj f`, of A] 

484 
by (intro iffI) fastsimp+ 

485 

13585  486 
lemma inj_vimage_image_eq: "inj f ==> f ` (f ` A) = A" 
487 
by (simp add: inj_on_def, blast) 

488 

489 
lemma vimage_subsetD: "surj f ==> f ` B <= A ==> B <= f ` A" 

40702  490 
by (blast intro: sym) 
13585  491 

492 
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f ` B <= A" 

493 
by (unfold inj_on_def, blast) 

494 

495 
lemma vimage_subset_eq: "bij f ==> (f ` B <= A) = (B <= f ` A)" 

496 
apply (unfold bij_def) 

497 
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) 

498 
done 

499 

31438  500 
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" 
501 
by(blast dest: inj_onD) 

502 

13585  503 
lemma inj_on_image_Int: 
504 
"[ inj_on f C; A<=C; B<=C ] ==> f`(A Int B) = f`A Int f`B" 

505 
apply (simp add: inj_on_def, blast) 

506 
done 

507 

508 
lemma inj_on_image_set_diff: 

509 
"[ inj_on f C; A<=C; B<=C ] ==> f`(AB) = f`A  f`B" 

510 
apply (simp add: inj_on_def, blast) 

511 
done 

512 

513 
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" 

514 
by (simp add: inj_on_def, blast) 

515 

516 
lemma image_set_diff: "inj f ==> f`(AB) = f`A  f`B" 

517 
by (simp add: inj_on_def, blast) 

518 

519 
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" 

520 
by (blast dest: injD) 

521 

522 
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" 

523 
by (simp add: inj_on_def, blast) 

524 

525 
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" 

526 
by (blast dest: injD) 

527 

528 
(*injectivity's required. Lefttoright inclusion holds even if A is empty*) 

529 
lemma image_INT: 

530 
"[ inj_on f C; ALL x:A. B x <= C; j:A ] 

531 
==> f ` (INTER A B) = (INT x:A. f ` B x)" 

532 
apply (simp add: inj_on_def, blast) 

533 
done 

534 

535 
(*Compare with image_INT: no use of inj_on, and if f is surjective then 

536 
it doesn't matter whether A is empty*) 

537 
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" 

538 
apply (simp add: bij_def) 

539 
apply (simp add: inj_on_def surj_def, blast) 

540 
done 

541 

542 
lemma surj_Compl_image_subset: "surj f ==> (f`A) <= f`(A)" 

40702  543 
by auto 
13585  544 

545 
lemma inj_image_Compl_subset: "inj f ==> f`(A) <= (f`A)" 

546 
by (auto simp add: inj_on_def) 

5852  547 

13585  548 
lemma bij_image_Compl_eq: "bij f ==> f`(A) = (f`A)" 
549 
apply (simp add: bij_def) 

550 
apply (rule equalityI) 

551 
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) 

552 
done 

553 

41657  554 
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f ` {a} \<subseteq> {THE x. f x = a}" 
555 
 {* The inverse image of a singleton under an injective function 

556 
is included in a singleton. *} 

557 
apply (auto simp add: inj_on_def) 

558 
apply (blast intro: the_equality [symmetric]) 

559 
done 

560 

35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset

561 
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" 
35580  562 
by (auto intro!: inj_onI) 
13585  563 

35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset

564 
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A" 
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset

565 
by (auto intro!: inj_onI dest: strict_mono_eq) 
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset

566 

41657  567 

13585  568 
subsection{*Function Updating*} 
569 

35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35115
diff
changeset

570 
definition 
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35115
diff
changeset

571 
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where 
26147  572 
"fun_upd f a b == % x. if x=a then b else f x" 
573 

41229
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40969
diff
changeset

574 
nonterminal updbinds and updbind 
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40969
diff
changeset

575 

26147  576 
syntax 
577 
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") 

578 
"" :: "updbind => updbinds" ("_") 

579 
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") 

35115  580 
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900) 
26147  581 

582 
translations 

35115  583 
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" 
584 
"f(x:=y)" == "CONST fun_upd f x y" 

26147  585 

586 
(* Hint: to define the sum of two functions (or maps), use sum_case. 

587 
A nice infix syntax could be defined (in Datatype.thy or below) by 

35115  588 
notation 
589 
sum_case (infixr "'(+')"80) 

26147  590 
*) 
591 

13585  592 
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" 
593 
apply (simp add: fun_upd_def, safe) 

594 
apply (erule subst) 

595 
apply (rule_tac [2] ext, auto) 

596 
done 

597 

598 
(* f x = y ==> f(x:=y) = f *) 

599 
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] 

600 

601 
(* f(x := f x) = f *) 

17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset

602 
lemmas fun_upd_triv = refl [THEN fun_upd_idem] 
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset

603 
declare fun_upd_triv [iff] 
13585  604 

605 
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" 

17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset

606 
by (simp add: fun_upd_def) 
13585  607 

608 
(* fun_upd_apply supersedes these two, but they are useful 

609 
if fun_upd_apply is intentionally removed from the simpset *) 

610 
lemma fun_upd_same: "(f(x:=y)) x = y" 

611 
by simp 

612 

613 
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" 

614 
by simp 

615 

616 
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39213
diff
changeset

617 
by (simp add: fun_eq_iff) 
13585  618 

619 
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" 

620 
by (rule ext, auto) 

621 

15303  622 
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A" 
34209  623 
by (fastsimp simp:inj_on_def image_def) 
15303  624 

15510  625 
lemma fun_upd_image: 
626 
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A{x})) else f ` A)" 

627 
by auto 

628 

31080  629 
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" 
34209  630 
by (auto intro: ext) 
31080  631 

26147  632 

633 
subsection {* @{text override_on} *} 

634 

635 
definition 

636 
override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" 

637 
where 

638 
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" 

13910  639 

15691  640 
lemma override_on_emptyset[simp]: "override_on f g {} = f" 
641 
by(simp add:override_on_def) 

13910  642 

15691  643 
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" 
644 
by(simp add:override_on_def) 

13910  645 

15691  646 
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" 
647 
by(simp add:override_on_def) 

13910  648 

26147  649 

650 
subsection {* @{text swap} *} 

15510  651 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset

652 
definition 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset

653 
swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset

654 
where 
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset

655 
"swap a b f = f (a := f b, b:= f a)" 
15510  656 

34101  657 
lemma swap_self [simp]: "swap a a f = f" 
15691  658 
by (simp add: swap_def) 
15510  659 

660 
lemma swap_commute: "swap a b f = swap b a f" 

661 
by (rule ext, simp add: fun_upd_def swap_def) 

662 

663 
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" 

664 
by (rule ext, simp add: fun_upd_def swap_def) 

665 

34145  666 
lemma swap_triple: 
667 
assumes "a \<noteq> c" and "b \<noteq> c" 

668 
shows "swap a b (swap b c (swap a b f)) = swap a c f" 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39213
diff
changeset

669 
using assms by (simp add: fun_eq_iff swap_def) 
34145  670 

34101  671 
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)" 
672 
by (rule ext, simp add: fun_upd_def swap_def) 

673 

39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

674 
lemma swap_image_eq [simp]: 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

675 
assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A" 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

676 
proof  
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

677 
have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A" 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

678 
using assms by (auto simp: image_iff swap_def) 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

679 
then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" . 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

680 
with subset[of f] show ?thesis by auto 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

681 
qed 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

682 

15510  683 
lemma inj_on_imp_inj_on_swap: 
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

684 
"\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A" 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

685 
by (simp add: inj_on_def swap_def, blast) 
15510  686 

687 
lemma inj_on_swap_iff [simp]: 

39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

688 
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A" 
39075  689 
proof 
15510  690 
assume "inj_on (swap a b f) A" 
39075  691 
with A have "inj_on (swap a b (swap a b f)) A" 
692 
by (iprover intro: inj_on_imp_inj_on_swap) 

693 
thus "inj_on f A" by simp 

15510  694 
next 
695 
assume "inj_on f A" 

34209  696 
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) 
15510  697 
qed 
698 

39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

699 
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)" 
40702  700 
by simp 
15510  701 

39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

702 
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f" 
40702  703 
by simp 
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset

704 

39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

705 
lemma bij_betw_swap_iff [simp]: 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

706 
"\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B" 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

707 
by (auto simp: bij_betw_def) 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

708 

b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

709 
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f" 
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset

710 
by simp 
39075  711 

36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact'  frees some popular keywords;
wenzelm
parents:
35584
diff
changeset

712 
hide_const (open) swap 
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset

713 

31949  714 
subsection {* Inversion of injective functions *} 
715 

33057  716 
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where 
717 
"the_inv_into A f == %x. THE y. y : A & f y = x" 

32961  718 

33057  719 
lemma the_inv_into_f_f: 
720 
"[ inj_on f A; x : A ] ==> the_inv_into A f (f x) = x" 

721 
apply (simp add: the_inv_into_def inj_on_def) 

34209  722 
apply blast 
32961  723 
done 
724 

33057  725 
lemma f_the_inv_into_f: 
726 
"inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y" 

727 
apply (simp add: the_inv_into_def) 

32961  728 
apply (rule the1I2) 
729 
apply(blast dest: inj_onD) 

730 
apply blast 

731 
done 

732 

33057  733 
lemma the_inv_into_into: 
734 
"[ inj_on f A; x : f ` A; A <= B ] ==> the_inv_into A f x : B" 

735 
apply (simp add: the_inv_into_def) 

32961  736 
apply (rule the1I2) 
737 
apply(blast dest: inj_onD) 

738 
apply blast 

739 
done 

740 

33057  741 
lemma the_inv_into_onto[simp]: 
742 
"inj_on f A ==> the_inv_into A f ` (f ` A) = A" 

743 
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric]) 

32961  744 

33057  745 
lemma the_inv_into_f_eq: 
746 
"[ inj_on f A; f x = y; x : A ] ==> the_inv_into A f y = x" 

32961  747 
apply (erule subst) 
33057  748 
apply (erule the_inv_into_f_f, assumption) 
32961  749 
done 
750 

33057  751 
lemma the_inv_into_comp: 
32961  752 
"[ inj_on f (g ` A); inj_on g A; x : f ` g ` A ] ==> 
33057  753 
the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x" 
754 
apply (rule the_inv_into_f_eq) 

32961  755 
apply (fast intro: comp_inj_on) 
33057  756 
apply (simp add: f_the_inv_into_f the_inv_into_into) 
757 
apply (simp add: the_inv_into_into) 

32961  758 
done 
759 

33057  760 
lemma inj_on_the_inv_into: 
761 
"inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)" 

762 
by (auto intro: inj_onI simp: image_def the_inv_into_f_f) 

32961  763 

33057  764 
lemma bij_betw_the_inv_into: 
765 
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A" 

766 
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) 

32961  767 

32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset

768 
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where 
33057  769 
"the_inv f \<equiv> the_inv_into UNIV f" 
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset

770 

31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset

771 
lemma the_inv_f_f: 
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset

772 
assumes "inj f" 
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset

773 
shows "the_inv f (f x) = x" using assms UNIV_I 
33057  774 
by (rule the_inv_into_f_f) 
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset

775 

40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

776 
subsection {* Cantor's Paradox *} 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

777 

42238  778 
lemma Cantors_paradox [no_atp]: 
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

779 
"\<not>(\<exists>f. f ` A = Pow A)" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

780 
proof clarify 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

781 
fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

782 
let ?X = "{a \<in> A. a \<notin> f a}" 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

783 
have "?X \<in> Pow A" unfolding Pow_def by auto 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

784 
with * obtain x where "x \<in> A \<and> f x = ?X" by blast 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

785 
thus False by best 
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOLimage.
hoelzl
parents:
40702
diff
changeset

786 
qed 
31949  787 

40969  788 
subsection {* Setup *} 
789 

790 
subsubsection {* Proof tools *} 

22845  791 

792 
text {* simplifies terms of the form 

793 
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} 

794 

24017  795 
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => 
22845  796 
let 
797 
fun gen_fun_upd NONE T _ _ = NONE 

24017  798 
 gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) 
22845  799 
fun dest_fun_T1 (Type (_, T :: Ts)) = T 
800 
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = 

801 
let 

802 
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = 

803 
if v aconv x then SOME g else gen_fun_upd (find g) T v w 

804 
 find t = NONE 

805 
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end 

24017  806 

807 
fun proc ss ct = 

808 
let 

809 
val ctxt = Simplifier.the_context ss 

810 
val t = Thm.term_of ct 

811 
in 

812 
case find_double t of 

813 
(T, NONE) => NONE 

814 
 (T, SOME rhs) => 

27330  815 
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) 
24017  816 
(fn _ => 
817 
rtac eq_reflection 1 THEN 

818 
rtac ext 1 THEN 

819 
simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) 

820 
end 

821 
in proc end 

22845  822 
*} 
823 

824 

40969  825 
subsubsection {* Code generator *} 
21870  826 

25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

827 
types_code 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

828 
"fun" ("(_ >/ _)") 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

829 
attach (term_of) {* 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

830 
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT > bT); 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

831 
*} 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

832 
attach (test) {* 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

833 
fun gen_fun_type aF aT bG bT i = 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

834 
let 
32740  835 
val tab = Unsynchronized.ref []; 
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

836 
fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

837 
(aT > bT) > aT > bT > aT > bT) $ t $ aF x $ y () 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

838 
in 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

839 
(fn x => 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

840 
case AList.lookup op = (!tab) x of 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

841 
NONE => 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

842 
let val p as (y, _) = bG i 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

843 
in (tab := (x, p) :: !tab; y) end 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

844 
 SOME (y, _) => y, 
28711  845 
fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT > bT))) 
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

846 
end; 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

847 
*} 
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset

848 

21870  849 
code_const "op \<circ>" 
850 
(SML infixl 5 "o") 

851 
(Haskell infixr 9 ".") 

852 

21906  853 
code_const "id" 
854 
(Haskell "id") 

855 

40969  856 

857 
subsubsection {* Functorial structure of types *} 

858 

41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41229
diff
changeset

859 
use "Tools/enriched_type.ML" 
40969  860 

2912  861 
end 