src/HOL/Fun.thy
author wenzelm
Wed Apr 10 21:20:35 2013 +0200 (2013-04-10)
changeset 51692 ecd34f863242
parent 51598 5dbe537087aa
child 51717 9e7d1c139569
permissions -rw-r--r--
tuned pretty layout: avoid nested Pretty.string_of, which merely happens to work with Isabelle/jEdit since formatting is delegated to Scala side;
declare command "print_case_translations" where it is actually defined;
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(*  Title:      HOL/Fun.thy
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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header {* Notions about functions *}
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theory Fun
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imports Complete_Lattices
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keywords "enriched_type" :: thy_goal
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begin
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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
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subsection {* The Identity Function @{text id} *}
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definition id :: "'a \<Rightarrow> 'a" where
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  "id = (\<lambda>x. x)"
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lemma id_apply [simp]: "id x = x"
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  by (simp add: id_def)
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lemma image_id [simp]: "image id = id"
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  by (simp add: id_def fun_eq_iff)
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lemma vimage_id [simp]: "vimage id = id"
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  by (simp add: id_def fun_eq_iff)
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subsection {* The Composition Operator @{text "f \<circ> g"} *}
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definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) where
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  "f o g = (\<lambda>x. f (g x))"
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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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lemma comp_apply [simp]: "(f o g) x = f (g x)"
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  by (simp add: comp_def)
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lemma comp_assoc: "(f o g) o h = f o (g o h)"
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  by (simp add: fun_eq_iff)
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lemma id_comp [simp]: "id o g = g"
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  by (simp add: fun_eq_iff)
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lemma comp_id [simp]: "f o id = f"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_dest:
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  "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_elim:
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  "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
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  by (simp add: fun_eq_iff) 
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lemma image_comp:
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  "(f o g) ` r = f ` (g ` r)"
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  by auto
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lemma vimage_comp:
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  "(g \<circ> f) -` x = f -` (g -` x)"
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  by auto
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lemma INF_comp:
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  "INFI A (g \<circ> f) = INFI (f ` A) g"
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  by (simp add: INF_def image_comp)
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lemma SUP_comp:
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  "SUPR A (g \<circ> f) = SUPR (f ` A) g"
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  by (simp add: SUP_def image_comp)
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subsection {* The Forward Composition Operator @{text fcomp} *}
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definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
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  "f \<circ>> g = (\<lambda>x. g (f x))"
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lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
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  by (simp add: fcomp_def)
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lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
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  by (simp add: fcomp_def)
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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 *}
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definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
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  "map_fun f g h = g \<circ> h \<circ> f"
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lemma map_fun_apply [simp]:
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  "map_fun f g h x = g (h (f x))"
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  by (simp add: map_fun_def)
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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.*}
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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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text{* The negated case: *}
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translations
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"\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"
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lemma injI:
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  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
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  shows "inj f"
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  using assms unfolding inj_on_def by auto
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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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lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
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by (simp add: inj_on_def)
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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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unfolding inj_on_def by auto
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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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unfolding inj_on_def unfolding image_def by blast
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lemma inj_comp:
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  "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
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  by (simp add: inj_on_def)
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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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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: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"
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unfolding inj_on_def by blast
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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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unfolding inj_on_def by blast
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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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unfolding inj_on_def by blast
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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 -
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  {
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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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    have "x = y"
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    proof -
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      {
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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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      {
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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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  }
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  then show ?thesis by (unfold inj_on_def UNION_eq) auto
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qed
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lemma surj_id: "surj id"
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by simp
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lemma bij_id[simp]: "bij id"
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by (simp add: bij_betw_def)
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lemma inj_onI:
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    "(!! 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)
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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)
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lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
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by (unfold inj_on_def, blast)
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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)
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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)
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apply blast
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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)
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apply blast
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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)
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lemma inj_singleton: "inj (%s. {s})"
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by (simp add: inj_on_def)
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lemma inj_on_empty[iff]: "inj_on f {}"
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by(simp add: inj_on_def)
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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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lemma inj_on_Un:
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 "inj_on f (A Un B) =
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  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
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apply(unfold inj_on_def)
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apply (blast intro:sym)
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done
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lemma inj_on_insert[iff]:
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  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
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apply(unfold inj_on_def)
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apply (blast intro:sym)
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done
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lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
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apply(unfold inj_on_def)
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apply (blast)
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done
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lemma comp_inj_on_iff:
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  "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
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by(auto simp add: comp_inj_on inj_on_def)
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lemma inj_on_imageI2:
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  "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
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by(auto simp add: comp_inj_on inj_on_def)
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lemma inj_img_insertE:
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  assumes "inj_on f A"
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  assumes "x \<notin> B" and "insert x B = f ` A"
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  obtains x' A' where "x' \<notin> A'" and "A = insert x' A'"
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    and "x = f x'" and "B = f ` A'" 
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proof -
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  from assms have "x \<in> f ` A" by auto
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  then obtain x' where *: "x' \<in> A" "x = f x'" by auto
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  then have "A = insert x' (A - {x'})" by auto
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  with assms * have "B = f ` (A - {x'})"
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    by (auto dest: inj_on_contraD)
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  have "x' \<notin> A - {x'}" by simp
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  from `x' \<notin> A - {x'}` `A = insert x' (A - {x'})` `x = f x'` `B = image f (A - {x'})`
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  show ?thesis ..
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qed
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lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
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  by auto
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lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
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  using *[symmetric] by auto
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lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
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  by (simp add: surj_def)
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lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
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  by (simp add: surj_def, blast)
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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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   319
apply (drule_tac x = y in spec, clarify)
paulson@13585
   320
apply (drule_tac x = x in spec, blast)
paulson@13585
   321
done
paulson@13585
   322
hoelzl@39074
   323
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
hoelzl@40702
   324
  unfolding bij_betw_def by auto
hoelzl@39074
   325
hoelzl@40703
   326
lemma bij_betw_empty1:
hoelzl@40703
   327
  assumes "bij_betw f {} A"
hoelzl@40703
   328
  shows "A = {}"
hoelzl@40703
   329
using assms unfolding bij_betw_def by blast
hoelzl@40703
   330
hoelzl@40703
   331
lemma bij_betw_empty2:
hoelzl@40703
   332
  assumes "bij_betw f A {}"
hoelzl@40703
   333
  shows "A = {}"
hoelzl@40703
   334
using assms unfolding bij_betw_def by blast
hoelzl@40703
   335
hoelzl@40703
   336
lemma inj_on_imp_bij_betw:
hoelzl@40703
   337
  "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
hoelzl@40703
   338
unfolding bij_betw_def by simp
hoelzl@40703
   339
hoelzl@39076
   340
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
hoelzl@40702
   341
  unfolding bij_betw_def ..
hoelzl@39074
   342
paulson@13585
   343
lemma bijI: "[| inj f; surj f |] ==> bij f"
paulson@13585
   344
by (simp add: bij_def)
paulson@13585
   345
paulson@13585
   346
lemma bij_is_inj: "bij f ==> inj f"
paulson@13585
   347
by (simp add: bij_def)
paulson@13585
   348
paulson@13585
   349
lemma bij_is_surj: "bij f ==> surj f"
paulson@13585
   350
by (simp add: bij_def)
paulson@13585
   351
nipkow@26105
   352
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105
   353
by (simp add: bij_betw_def)
nipkow@26105
   354
nipkow@31438
   355
lemma bij_betw_trans:
nipkow@31438
   356
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31438
   357
by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31438
   358
hoelzl@40702
   359
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
hoelzl@40702
   360
  by (rule bij_betw_trans)
hoelzl@40702
   361
hoelzl@40703
   362
lemma bij_betw_comp_iff:
hoelzl@40703
   363
  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   364
by(auto simp add: bij_betw_def inj_on_def)
hoelzl@40703
   365
hoelzl@40703
   366
lemma bij_betw_comp_iff2:
hoelzl@40703
   367
  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
hoelzl@40703
   368
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   369
using assms
hoelzl@40703
   370
proof(auto simp add: bij_betw_comp_iff)
hoelzl@40703
   371
  assume *: "bij_betw (f' \<circ> f) A A''"
hoelzl@40703
   372
  thus "bij_betw f A A'"
hoelzl@40703
   373
  using IM
hoelzl@40703
   374
  proof(auto simp add: bij_betw_def)
hoelzl@40703
   375
    assume "inj_on (f' \<circ> f) A"
hoelzl@40703
   376
    thus "inj_on f A" using inj_on_imageI2 by blast
hoelzl@40703
   377
  next
hoelzl@40703
   378
    fix a' assume **: "a' \<in> A'"
hoelzl@40703
   379
    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
hoelzl@40703
   380
    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
hoelzl@40703
   381
    unfolding bij_betw_def by force
hoelzl@40703
   382
    hence "f a \<in> A'" using IM by auto
hoelzl@40703
   383
    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
hoelzl@40703
   384
    thus "a' \<in> f ` A" using 1 by auto
hoelzl@40703
   385
  qed
hoelzl@40703
   386
qed
hoelzl@40703
   387
nipkow@26105
   388
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105
   389
proof -
nipkow@26105
   390
  have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105
   391
    using assms by(auto simp:bij_betw_def)
nipkow@26105
   392
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105
   393
  { fix a b assume P: "?P b a"
nipkow@26105
   394
    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
nipkow@26105
   395
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105
   396
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105
   397
  } note g = this
nipkow@26105
   398
  have "inj_on ?g B"
nipkow@26105
   399
  proof(rule inj_onI)
nipkow@26105
   400
    fix x y assume "x:B" "y:B" "?g x = ?g y"
nipkow@26105
   401
    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
nipkow@26105
   402
    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
nipkow@26105
   403
    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
nipkow@26105
   404
  qed
nipkow@26105
   405
  moreover have "?g ` B = A"
nipkow@26105
   406
  proof(auto simp:image_def)
nipkow@26105
   407
    fix b assume "b:B"
nipkow@26105
   408
    with s obtain a where P: "?P b a" unfolding image_def by blast
nipkow@26105
   409
    thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105
   410
  next
nipkow@26105
   411
    fix a assume "a:A"
nipkow@26105
   412
    then obtain b where P: "?P b a" using s unfolding image_def by blast
nipkow@26105
   413
    then have "b:B" using s unfolding image_def by blast
nipkow@26105
   414
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105
   415
  qed
nipkow@26105
   416
  ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105
   417
qed
nipkow@26105
   418
hoelzl@40703
   419
lemma bij_betw_cong:
hoelzl@40703
   420
  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
hoelzl@40703
   421
unfolding bij_betw_def inj_on_def by force
hoelzl@40703
   422
hoelzl@40703
   423
lemma bij_betw_id[intro, simp]:
hoelzl@40703
   424
  "bij_betw id A A"
hoelzl@40703
   425
unfolding bij_betw_def id_def by auto
hoelzl@40703
   426
hoelzl@40703
   427
lemma bij_betw_id_iff:
hoelzl@40703
   428
  "bij_betw id A B \<longleftrightarrow> A = B"
hoelzl@40703
   429
by(auto simp add: bij_betw_def)
hoelzl@40703
   430
hoelzl@39075
   431
lemma bij_betw_combine:
hoelzl@39075
   432
  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
hoelzl@39075
   433
  shows "bij_betw f (A \<union> C) (B \<union> D)"
hoelzl@39075
   434
  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
hoelzl@39075
   435
hoelzl@40703
   436
lemma bij_betw_UNION_chain:
hoelzl@40703
   437
  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
hoelzl@40703
   438
         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
hoelzl@40703
   439
  shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
haftmann@49905
   440
proof (unfold bij_betw_def, auto)
hoelzl@40703
   441
  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
hoelzl@40703
   442
  using BIJ bij_betw_def[of f] by auto
hoelzl@40703
   443
  thus "inj_on f (\<Union> i \<in> I. A i)"
hoelzl@40703
   444
  using CH inj_on_UNION_chain[of I A f] by auto
hoelzl@40703
   445
next
hoelzl@40703
   446
  fix i x
hoelzl@40703
   447
  assume *: "i \<in> I" "x \<in> A i"
hoelzl@40703
   448
  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
hoelzl@40703
   449
  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
hoelzl@40703
   450
next
hoelzl@40703
   451
  fix i x'
hoelzl@40703
   452
  assume *: "i \<in> I" "x' \<in> A' i"
hoelzl@40703
   453
  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
haftmann@49905
   454
  then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
haftmann@49905
   455
    using * by blast
haftmann@49905
   456
  then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by (simp add: image_def)
hoelzl@40703
   457
qed
hoelzl@40703
   458
hoelzl@40703
   459
lemma bij_betw_subset:
hoelzl@40703
   460
  assumes BIJ: "bij_betw f A A'" and
hoelzl@40703
   461
          SUB: "B \<le> A" and IM: "f ` B = B'"
hoelzl@40703
   462
  shows "bij_betw f B B'"
hoelzl@40703
   463
using assms
hoelzl@40703
   464
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
hoelzl@40703
   465
paulson@13585
   466
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
hoelzl@40702
   467
by simp
paulson@13585
   468
hoelzl@42903
   469
lemma surj_vimage_empty:
hoelzl@42903
   470
  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
hoelzl@42903
   471
  using surj_image_vimage_eq[OF `surj f`, of A]
nipkow@44890
   472
  by (intro iffI) fastforce+
hoelzl@42903
   473
paulson@13585
   474
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585
   475
by (simp add: inj_on_def, blast)
paulson@13585
   476
paulson@13585
   477
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
hoelzl@40702
   478
by (blast intro: sym)
paulson@13585
   479
paulson@13585
   480
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585
   481
by (unfold inj_on_def, blast)
paulson@13585
   482
paulson@13585
   483
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585
   484
apply (unfold bij_def)
paulson@13585
   485
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   486
done
paulson@13585
   487
nipkow@31438
   488
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
nipkow@31438
   489
by(blast dest: inj_onD)
nipkow@31438
   490
paulson@13585
   491
lemma inj_on_image_Int:
paulson@13585
   492
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   493
apply (simp add: inj_on_def, blast)
paulson@13585
   494
done
paulson@13585
   495
paulson@13585
   496
lemma inj_on_image_set_diff:
paulson@13585
   497
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
paulson@13585
   498
apply (simp add: inj_on_def, blast)
paulson@13585
   499
done
paulson@13585
   500
paulson@13585
   501
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   502
by (simp add: inj_on_def, blast)
paulson@13585
   503
paulson@13585
   504
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585
   505
by (simp add: inj_on_def, blast)
paulson@13585
   506
paulson@13585
   507
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
paulson@13585
   508
by (blast dest: injD)
paulson@13585
   509
paulson@13585
   510
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
paulson@13585
   511
by (simp add: inj_on_def, blast)
paulson@13585
   512
paulson@13585
   513
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
paulson@13585
   514
by (blast dest: injD)
paulson@13585
   515
paulson@13585
   516
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
paulson@13585
   517
lemma image_INT:
paulson@13585
   518
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
paulson@13585
   519
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585
   520
apply (simp add: inj_on_def, blast)
paulson@13585
   521
done
paulson@13585
   522
paulson@13585
   523
(*Compare with image_INT: no use of inj_on, and if f is surjective then
paulson@13585
   524
  it doesn't matter whether A is empty*)
paulson@13585
   525
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585
   526
apply (simp add: bij_def)
paulson@13585
   527
apply (simp add: inj_on_def surj_def, blast)
paulson@13585
   528
done
paulson@13585
   529
paulson@13585
   530
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
hoelzl@40702
   531
by auto
paulson@13585
   532
paulson@13585
   533
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585
   534
by (auto simp add: inj_on_def)
paulson@5852
   535
paulson@13585
   536
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585
   537
apply (simp add: bij_def)
paulson@13585
   538
apply (rule equalityI)
paulson@13585
   539
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585
   540
done
paulson@13585
   541
haftmann@41657
   542
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
haftmann@41657
   543
  -- {* The inverse image of a singleton under an injective function
haftmann@41657
   544
         is included in a singleton. *}
haftmann@41657
   545
  apply (auto simp add: inj_on_def)
haftmann@41657
   546
  apply (blast intro: the_equality [symmetric])
haftmann@41657
   547
  done
haftmann@41657
   548
hoelzl@43991
   549
lemma inj_on_vimage_singleton:
hoelzl@43991
   550
  "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
hoelzl@43991
   551
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
hoelzl@43991
   552
hoelzl@35584
   553
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   554
  by (auto intro!: inj_onI)
paulson@13585
   555
hoelzl@35584
   556
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   557
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   558
haftmann@41657
   559
paulson@13585
   560
subsection{*Function Updating*}
paulson@13585
   561
haftmann@44277
   562
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147
   563
  "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147
   564
wenzelm@41229
   565
nonterminal updbinds and updbind
wenzelm@41229
   566
haftmann@26147
   567
syntax
haftmann@26147
   568
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147
   569
  ""         :: "updbind => updbinds"             ("_")
haftmann@26147
   570
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115
   571
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   572
haftmann@26147
   573
translations
wenzelm@35115
   574
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115
   575
  "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147
   576
haftmann@26147
   577
(* Hint: to define the sum of two functions (or maps), use sum_case.
haftmann@26147
   578
         A nice infix syntax could be defined (in Datatype.thy or below) by
wenzelm@35115
   579
notation
wenzelm@35115
   580
  sum_case  (infixr "'(+')"80)
haftmann@26147
   581
*)
haftmann@26147
   582
paulson@13585
   583
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585
   584
apply (simp add: fun_upd_def, safe)
paulson@13585
   585
apply (erule subst)
paulson@13585
   586
apply (rule_tac [2] ext, auto)
paulson@13585
   587
done
paulson@13585
   588
wenzelm@45603
   589
lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
wenzelm@45603
   590
  by (simp only: fun_upd_idem_iff)
paulson@13585
   591
wenzelm@45603
   592
lemma fun_upd_triv [iff]: "f(x := f x) = f"
wenzelm@45603
   593
  by (simp only: fun_upd_idem)
paulson@13585
   594
paulson@13585
   595
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084
   596
by (simp add: fun_upd_def)
paulson@13585
   597
paulson@13585
   598
(* fun_upd_apply supersedes these two,   but they are useful
paulson@13585
   599
   if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585
   600
lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585
   601
by simp
paulson@13585
   602
paulson@13585
   603
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585
   604
by simp
paulson@13585
   605
paulson@13585
   606
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
nipkow@39302
   607
by (simp add: fun_eq_iff)
paulson@13585
   608
paulson@13585
   609
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585
   610
by (rule ext, auto)
paulson@13585
   611
nipkow@15303
   612
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
nipkow@44890
   613
by (fastforce simp:inj_on_def image_def)
nipkow@15303
   614
paulson@15510
   615
lemma fun_upd_image:
paulson@15510
   616
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510
   617
by auto
paulson@15510
   618
nipkow@31080
   619
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
huffman@44921
   620
  by auto
nipkow@31080
   621
nipkow@44744
   622
lemma UNION_fun_upd:
nipkow@44744
   623
  "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
nipkow@44744
   624
by (auto split: if_splits)
nipkow@44744
   625
haftmann@26147
   626
haftmann@26147
   627
subsection {* @{text override_on} *}
haftmann@26147
   628
haftmann@44277
   629
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
haftmann@26147
   630
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   631
nipkow@15691
   632
lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691
   633
by(simp add:override_on_def)
nipkow@13910
   634
nipkow@15691
   635
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691
   636
by(simp add:override_on_def)
nipkow@13910
   637
nipkow@15691
   638
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691
   639
by(simp add:override_on_def)
nipkow@13910
   640
haftmann@26147
   641
haftmann@26147
   642
subsection {* @{text swap} *}
paulson@15510
   643
haftmann@44277
   644
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" where
haftmann@22744
   645
  "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   646
huffman@34101
   647
lemma swap_self [simp]: "swap a a f = f"
nipkow@15691
   648
by (simp add: swap_def)
paulson@15510
   649
paulson@15510
   650
lemma swap_commute: "swap a b f = swap b a f"
paulson@15510
   651
by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510
   652
paulson@15510
   653
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
paulson@15510
   654
by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510
   655
huffman@34145
   656
lemma swap_triple:
huffman@34145
   657
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   658
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
nipkow@39302
   659
  using assms by (simp add: fun_eq_iff swap_def)
huffman@34145
   660
huffman@34101
   661
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
huffman@34101
   662
by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101
   663
hoelzl@39076
   664
lemma swap_image_eq [simp]:
hoelzl@39076
   665
  assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
hoelzl@39076
   666
proof -
hoelzl@39076
   667
  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
hoelzl@39076
   668
    using assms by (auto simp: image_iff swap_def)
hoelzl@39076
   669
  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
hoelzl@39076
   670
  with subset[of f] show ?thesis by auto
hoelzl@39076
   671
qed
hoelzl@39076
   672
paulson@15510
   673
lemma inj_on_imp_inj_on_swap:
hoelzl@39076
   674
  "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
hoelzl@39076
   675
  by (simp add: inj_on_def swap_def, blast)
paulson@15510
   676
paulson@15510
   677
lemma inj_on_swap_iff [simp]:
hoelzl@39076
   678
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
hoelzl@39075
   679
proof
paulson@15510
   680
  assume "inj_on (swap a b f) A"
hoelzl@39075
   681
  with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075
   682
    by (iprover intro: inj_on_imp_inj_on_swap)
hoelzl@39075
   683
  thus "inj_on f A" by simp
paulson@15510
   684
next
paulson@15510
   685
  assume "inj_on f A"
krauss@34209
   686
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   687
qed
paulson@15510
   688
hoelzl@39076
   689
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
hoelzl@40702
   690
  by simp
paulson@15510
   691
hoelzl@39076
   692
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
hoelzl@40702
   693
  by simp
haftmann@21547
   694
hoelzl@39076
   695
lemma bij_betw_swap_iff [simp]:
hoelzl@39076
   696
  "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
hoelzl@39076
   697
  by (auto simp: bij_betw_def)
hoelzl@39076
   698
hoelzl@39076
   699
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
hoelzl@39076
   700
  by simp
hoelzl@39075
   701
wenzelm@36176
   702
hide_const (open) swap
haftmann@21547
   703
haftmann@31949
   704
subsection {* Inversion of injective functions *}
haftmann@31949
   705
nipkow@33057
   706
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
haftmann@44277
   707
  "the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961
   708
nipkow@33057
   709
lemma the_inv_into_f_f:
nipkow@33057
   710
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057
   711
apply (simp add: the_inv_into_def inj_on_def)
krauss@34209
   712
apply blast
nipkow@32961
   713
done
nipkow@32961
   714
nipkow@33057
   715
lemma f_the_inv_into_f:
nipkow@33057
   716
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057
   717
apply (simp add: the_inv_into_def)
nipkow@32961
   718
apply (rule the1I2)
nipkow@32961
   719
 apply(blast dest: inj_onD)
nipkow@32961
   720
apply blast
nipkow@32961
   721
done
nipkow@32961
   722
nipkow@33057
   723
lemma the_inv_into_into:
nipkow@33057
   724
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057
   725
apply (simp add: the_inv_into_def)
nipkow@32961
   726
apply (rule the1I2)
nipkow@32961
   727
 apply(blast dest: inj_onD)
nipkow@32961
   728
apply blast
nipkow@32961
   729
done
nipkow@32961
   730
nipkow@33057
   731
lemma the_inv_into_onto[simp]:
nipkow@33057
   732
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057
   733
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961
   734
nipkow@33057
   735
lemma the_inv_into_f_eq:
nipkow@33057
   736
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961
   737
  apply (erule subst)
nipkow@33057
   738
  apply (erule the_inv_into_f_f, assumption)
nipkow@32961
   739
  done
nipkow@32961
   740
nipkow@33057
   741
lemma the_inv_into_comp:
nipkow@32961
   742
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   743
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057
   744
apply (rule the_inv_into_f_eq)
nipkow@32961
   745
  apply (fast intro: comp_inj_on)
nipkow@33057
   746
 apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057
   747
apply (simp add: the_inv_into_into)
nipkow@32961
   748
done
nipkow@32961
   749
nipkow@33057
   750
lemma inj_on_the_inv_into:
nipkow@33057
   751
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
nipkow@33057
   752
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
nipkow@32961
   753
nipkow@33057
   754
lemma bij_betw_the_inv_into:
nipkow@33057
   755
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057
   756
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   757
berghofe@32998
   758
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057
   759
  "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   760
berghofe@32998
   761
lemma the_inv_f_f:
berghofe@32998
   762
  assumes "inj f"
berghofe@32998
   763
  shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057
   764
  by (rule the_inv_into_f_f)
berghofe@32998
   765
haftmann@44277
   766
hoelzl@40703
   767
subsection {* Cantor's Paradox *}
hoelzl@40703
   768
blanchet@42238
   769
lemma Cantors_paradox [no_atp]:
hoelzl@40703
   770
  "\<not>(\<exists>f. f ` A = Pow A)"
hoelzl@40703
   771
proof clarify
hoelzl@40703
   772
  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
hoelzl@40703
   773
  let ?X = "{a \<in> A. a \<notin> f a}"
hoelzl@40703
   774
  have "?X \<in> Pow A" unfolding Pow_def by auto
hoelzl@40703
   775
  with * obtain x where "x \<in> A \<and> f x = ?X" by blast
hoelzl@40703
   776
  thus False by best
hoelzl@40703
   777
qed
haftmann@31949
   778
haftmann@40969
   779
subsection {* Setup *} 
haftmann@40969
   780
haftmann@40969
   781
subsubsection {* Proof tools *}
haftmann@22845
   782
haftmann@22845
   783
text {* simplifies terms of the form
haftmann@22845
   784
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
haftmann@22845
   785
wenzelm@24017
   786
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
haftmann@22845
   787
let
haftmann@22845
   788
  fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017
   789
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845
   790
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845
   791
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845
   792
    let
haftmann@22845
   793
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845
   794
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845
   795
        | find t = NONE
haftmann@22845
   796
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   797
wenzelm@24017
   798
  fun proc ss ct =
wenzelm@24017
   799
    let
wenzelm@24017
   800
      val ctxt = Simplifier.the_context ss
wenzelm@24017
   801
      val t = Thm.term_of ct
wenzelm@24017
   802
    in
wenzelm@24017
   803
      case find_double t of
wenzelm@24017
   804
        (T, NONE) => NONE
wenzelm@24017
   805
      | (T, SOME rhs) =>
wenzelm@27330
   806
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017
   807
            (fn _ =>
wenzelm@24017
   808
              rtac eq_reflection 1 THEN
wenzelm@24017
   809
              rtac ext 1 THEN
wenzelm@24017
   810
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
wenzelm@24017
   811
    end
wenzelm@24017
   812
in proc end
haftmann@22845
   813
*}
haftmann@22845
   814
haftmann@22845
   815
haftmann@40969
   816
subsubsection {* Code generator *}
haftmann@21870
   817
haftmann@21870
   818
code_const "op \<circ>"
haftmann@21870
   819
  (SML infixl 5 "o")
haftmann@21870
   820
  (Haskell infixr 9 ".")
haftmann@21870
   821
haftmann@21906
   822
code_const "id"
haftmann@21906
   823
  (Haskell "id")
haftmann@21906
   824
haftmann@40969
   825
haftmann@40969
   826
subsubsection {* Functorial structure of types *}
haftmann@40969
   827
wenzelm@48891
   828
ML_file "Tools/enriched_type.ML"
haftmann@40969
   829
haftmann@47488
   830
enriched_type map_fun: map_fun
haftmann@47488
   831
  by (simp_all add: fun_eq_iff)
haftmann@47488
   832
haftmann@47488
   833
enriched_type vimage
haftmann@49739
   834
  by (simp_all add: fun_eq_iff vimage_comp)
haftmann@49739
   835
haftmann@49739
   836
text {* Legacy theorem names *}
haftmann@49739
   837
haftmann@49739
   838
lemmas o_def = comp_def
haftmann@49739
   839
lemmas o_apply = comp_apply
haftmann@49739
   840
lemmas o_assoc = comp_assoc [symmetric]
haftmann@49739
   841
lemmas id_o = id_comp
haftmann@49739
   842
lemmas o_id = comp_id
haftmann@49739
   843
lemmas o_eq_dest = comp_eq_dest
haftmann@49739
   844
lemmas o_eq_elim = comp_eq_elim
haftmann@49739
   845
lemmas image_compose = image_comp
haftmann@49739
   846
lemmas vimage_compose = vimage_comp
haftmann@47488
   847
nipkow@2912
   848
end
haftmann@47488
   849