src/HOL/Fun.thy
author Andreas Lochbihler
Wed Nov 11 09:48:24 2015 +0100 (2015-11-11)
changeset 61630 608520e0e8e2
parent 61520 8f85bb443d33
child 61699 a81dc5c4d6a9
permissions -rw-r--r--
add various lemmas
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(*  Title:      HOL/Fun.thy
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Author:     Andrei Popescu, TU Muenchen
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    Copyright   1994, 2012
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*)
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section \<open>Notions about functions\<close>
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theory Fun
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imports Set
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keywords "functor" :: 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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text\<open>Uniqueness, so NOT the axiom of choice.\<close>
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lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
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  by (force intro: theI')
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lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
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  by (force intro: theI')
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subsection \<open>The Identity Function @{text id}\<close>
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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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code_printing
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  constant id \<rightharpoonup> (Haskell) "id"
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subsection \<open>The Composition Operator @{text "f \<circ> g"}\<close>
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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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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 comp_eq_dest_lhs: "a o b = c \<Longrightarrow> a (b v) = c v"
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  by clarsimp
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lemma comp_eq_id_dest: "a o b = id o c \<Longrightarrow> a (b v) = c v"
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  by clarsimp
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lemma image_comp:
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  "f ` (g ` r) = (f o g) ` r"
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  by auto
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lemma vimage_comp:
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  "f -` (g -` x) = (g \<circ> f) -` x"
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  by auto
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lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B"
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  by (auto simp: comp_def elim!: equalityE)
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lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)"
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by(auto simp add: Set.bind_def)
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lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
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by(auto simp add: Set.bind_def)
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lemma (in group_add) minus_comp_minus [simp]:
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  "uminus \<circ> uminus = id"
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  by (simp add: fun_eq_iff)
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lemma (in boolean_algebra) minus_comp_minus [simp]:
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  "uminus \<circ> uminus = id"
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  by (simp add: fun_eq_iff)
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code_printing
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  constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."
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subsection \<open>The Forward Composition Operator @{text fcomp}\<close>
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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_printing
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  constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"
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no_notation fcomp (infixl "\<circ>>" 60)
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subsection \<open>Mapping functions\<close>
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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 \<open>Injectivity and Bijectivity\<close>
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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\<open>A common special case: functions injective, surjective or bijective over
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the entire domain type.\<close>
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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\<open>The negated case:\<close>
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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: "\<lbrakk>inj_on f A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> (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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  "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
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  unfolding inj_on_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 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 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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  by (simp add: inj_on_def) blast
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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 \<open>x' \<notin> A - {x'}\<close> \<open>A = insert x' (A - {x'})\<close> \<open>x = f x'\<close> \<open>B = image f (A - {x'})\<close>
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  show ?thesis ..
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qed
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lemma linorder_injI:
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  assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y"
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  shows "inj f"
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  -- \<open>Courtesy of Stephan Merz\<close>
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proof (rule inj_onI)
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  fix x y
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  assume f_eq: "f x = f y"
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  show "x = y" by (rule linorder_cases) (auto dest: hyp simp: f_eq)
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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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apply (drule_tac x = y in spec, clarify)
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apply (drule_tac x = x in spec, blast)
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done
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lemma bij_betw_imageI:
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  "\<lbrakk> inj_on f A; f ` A = B \<rbrakk> \<Longrightarrow> bij_betw f A B"
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unfolding bij_betw_def by clarify
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ballarin@57282
   319
lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
ballarin@57282
   320
  unfolding bij_betw_def by clarify
ballarin@57282
   321
hoelzl@39074
   322
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
hoelzl@40702
   323
  unfolding bij_betw_def by auto
hoelzl@39074
   324
hoelzl@40703
   325
lemma bij_betw_empty1:
hoelzl@40703
   326
  assumes "bij_betw f {} A"
hoelzl@40703
   327
  shows "A = {}"
hoelzl@40703
   328
using assms unfolding bij_betw_def by blast
hoelzl@40703
   329
hoelzl@40703
   330
lemma bij_betw_empty2:
hoelzl@40703
   331
  assumes "bij_betw f A {}"
hoelzl@40703
   332
  shows "A = {}"
hoelzl@40703
   333
using assms unfolding bij_betw_def by blast
hoelzl@40703
   334
hoelzl@40703
   335
lemma inj_on_imp_bij_betw:
hoelzl@40703
   336
  "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
hoelzl@40703
   337
unfolding bij_betw_def by simp
hoelzl@40703
   338
hoelzl@39076
   339
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
hoelzl@40702
   340
  unfolding bij_betw_def ..
hoelzl@39074
   341
paulson@13585
   342
lemma bijI: "[| inj f; surj f |] ==> bij f"
paulson@13585
   343
by (simp add: bij_def)
paulson@13585
   344
paulson@13585
   345
lemma bij_is_inj: "bij f ==> inj f"
paulson@13585
   346
by (simp add: bij_def)
paulson@13585
   347
paulson@13585
   348
lemma bij_is_surj: "bij f ==> surj f"
paulson@13585
   349
by (simp add: bij_def)
paulson@13585
   350
nipkow@26105
   351
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105
   352
by (simp add: bij_betw_def)
nipkow@26105
   353
nipkow@31438
   354
lemma bij_betw_trans:
nipkow@31438
   355
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31438
   356
by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31438
   357
hoelzl@40702
   358
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
hoelzl@40702
   359
  by (rule bij_betw_trans)
hoelzl@40702
   360
hoelzl@40703
   361
lemma bij_betw_comp_iff:
hoelzl@40703
   362
  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   363
by(auto simp add: bij_betw_def inj_on_def)
hoelzl@40703
   364
hoelzl@40703
   365
lemma bij_betw_comp_iff2:
hoelzl@40703
   366
  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
hoelzl@40703
   367
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   368
using assms
hoelzl@40703
   369
proof(auto simp add: bij_betw_comp_iff)
hoelzl@40703
   370
  assume *: "bij_betw (f' \<circ> f) A A''"
hoelzl@40703
   371
  thus "bij_betw f A A'"
hoelzl@40703
   372
  using IM
hoelzl@40703
   373
  proof(auto simp add: bij_betw_def)
hoelzl@40703
   374
    assume "inj_on (f' \<circ> f) A"
hoelzl@40703
   375
    thus "inj_on f A" using inj_on_imageI2 by blast
hoelzl@40703
   376
  next
hoelzl@40703
   377
    fix a' assume **: "a' \<in> A'"
hoelzl@40703
   378
    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
hoelzl@40703
   379
    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
hoelzl@40703
   380
    unfolding bij_betw_def by force
hoelzl@40703
   381
    hence "f a \<in> A'" using IM by auto
hoelzl@40703
   382
    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
hoelzl@40703
   383
    thus "a' \<in> f ` A" using 1 by auto
hoelzl@40703
   384
  qed
hoelzl@40703
   385
qed
hoelzl@40703
   386
nipkow@26105
   387
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105
   388
proof -
nipkow@26105
   389
  have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105
   390
    using assms by(auto simp:bij_betw_def)
nipkow@26105
   391
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105
   392
  { fix a b assume P: "?P b a"
haftmann@56077
   393
    hence ex1: "\<exists>a. ?P b a" using s by blast
nipkow@26105
   394
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105
   395
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105
   396
  } note g = this
nipkow@26105
   397
  have "inj_on ?g B"
nipkow@26105
   398
  proof(rule inj_onI)
nipkow@26105
   399
    fix x y assume "x:B" "y:B" "?g x = ?g y"
wenzelm@60758
   400
    from s \<open>x:B\<close> obtain a1 where a1: "?P x a1" by blast
wenzelm@60758
   401
    from s \<open>y:B\<close> obtain a2 where a2: "?P y a2" by blast
wenzelm@60758
   402
    from g[OF a1] a1 g[OF a2] a2 \<open>?g x = ?g y\<close> show "x=y" by simp
nipkow@26105
   403
  qed
nipkow@26105
   404
  moreover have "?g ` B = A"
haftmann@56077
   405
  proof(auto simp: image_def)
nipkow@26105
   406
    fix b assume "b:B"
haftmann@56077
   407
    with s obtain a where P: "?P b a" by blast
nipkow@26105
   408
    thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105
   409
  next
nipkow@26105
   410
    fix a assume "a:A"
haftmann@56077
   411
    then obtain b where P: "?P b a" using s by blast
haftmann@56077
   412
    then have "b:B" using s by blast
nipkow@26105
   413
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105
   414
  qed
nipkow@26105
   415
  ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105
   416
qed
nipkow@26105
   417
hoelzl@40703
   418
lemma bij_betw_cong:
hoelzl@40703
   419
  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
hoelzl@40703
   420
unfolding bij_betw_def inj_on_def by force
hoelzl@40703
   421
hoelzl@40703
   422
lemma bij_betw_id[intro, simp]:
hoelzl@40703
   423
  "bij_betw id A A"
hoelzl@40703
   424
unfolding bij_betw_def id_def by auto
hoelzl@40703
   425
hoelzl@40703
   426
lemma bij_betw_id_iff:
hoelzl@40703
   427
  "bij_betw id A B \<longleftrightarrow> A = B"
hoelzl@40703
   428
by(auto simp add: bij_betw_def)
hoelzl@40703
   429
hoelzl@39075
   430
lemma bij_betw_combine:
hoelzl@39075
   431
  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
hoelzl@39075
   432
  shows "bij_betw f (A \<union> C) (B \<union> D)"
hoelzl@39075
   433
  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
hoelzl@39075
   434
hoelzl@40703
   435
lemma bij_betw_subset:
hoelzl@40703
   436
  assumes BIJ: "bij_betw f A A'" and
hoelzl@40703
   437
          SUB: "B \<le> A" and IM: "f ` B = B'"
hoelzl@40703
   438
  shows "bij_betw f B B'"
hoelzl@40703
   439
using assms
hoelzl@40703
   440
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
hoelzl@40703
   441
haftmann@58195
   442
lemma bij_pointE:
haftmann@58195
   443
  assumes "bij f"
haftmann@58195
   444
  obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
haftmann@58195
   445
proof -
haftmann@58195
   446
  from assms have "inj f" by (rule bij_is_inj)
haftmann@58195
   447
  moreover from assms have "surj f" by (rule bij_is_surj)
haftmann@58195
   448
  then have "y \<in> range f" by simp
haftmann@58195
   449
  ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq)
haftmann@58195
   450
  with that show thesis by blast
haftmann@58195
   451
qed
haftmann@58195
   452
paulson@13585
   453
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
hoelzl@40702
   454
by simp
paulson@13585
   455
hoelzl@42903
   456
lemma surj_vimage_empty:
hoelzl@42903
   457
  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
wenzelm@60758
   458
  using surj_image_vimage_eq[OF \<open>surj f\<close>, of A]
nipkow@44890
   459
  by (intro iffI) fastforce+
hoelzl@42903
   460
paulson@13585
   461
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585
   462
by (simp add: inj_on_def, blast)
paulson@13585
   463
paulson@13585
   464
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
hoelzl@40702
   465
by (blast intro: sym)
paulson@13585
   466
paulson@13585
   467
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585
   468
by (unfold inj_on_def, blast)
paulson@13585
   469
paulson@13585
   470
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585
   471
apply (unfold bij_def)
paulson@13585
   472
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   473
done
paulson@13585
   474
Andreas@53927
   475
lemma inj_on_image_eq_iff: "\<lbrakk> inj_on f C; A \<subseteq> C; B \<subseteq> C \<rbrakk> \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
Andreas@53927
   476
by(fastforce simp add: inj_on_def)
Andreas@53927
   477
nipkow@31438
   478
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
Andreas@53927
   479
by(erule inj_on_image_eq_iff) simp_all
nipkow@31438
   480
paulson@13585
   481
lemma inj_on_image_Int:
paulson@13585
   482
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@60303
   483
  by (simp add: inj_on_def, blast)
paulson@13585
   484
paulson@13585
   485
lemma inj_on_image_set_diff:
paulson@60303
   486
   "[| inj_on f C;  A-B \<subseteq> C;  B \<subseteq> C |] ==> f`(A-B) = f`A - f`B"
paulson@60303
   487
  by (simp add: inj_on_def, blast)
paulson@13585
   488
paulson@13585
   489
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@60303
   490
  by (simp add: inj_on_def, blast)
paulson@13585
   491
paulson@13585
   492
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585
   493
by (simp add: inj_on_def, blast)
paulson@13585
   494
lp15@59504
   495
lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
lp15@59504
   496
  by (auto simp: inj_on_def)
lp15@59504
   497
lp15@61520
   498
(*FIXME DELETE*)
lp15@61520
   499
lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
lp15@61520
   500
  by (blast dest: inj_onD)
lp15@61520
   501
lp15@59504
   502
lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
lp15@59504
   503
  by (blast dest: injD)
paulson@13585
   504
paulson@13585
   505
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
lp15@59504
   506
  by (blast dest: injD)
paulson@13585
   507
paulson@13585
   508
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
lp15@59504
   509
  by (blast dest: injD)
paulson@13585
   510
paulson@13585
   511
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
hoelzl@40702
   512
by auto
paulson@13585
   513
paulson@13585
   514
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585
   515
by (auto simp add: inj_on_def)
paulson@5852
   516
paulson@13585
   517
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585
   518
apply (simp add: bij_def)
paulson@13585
   519
apply (rule equalityI)
paulson@13585
   520
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585
   521
done
paulson@13585
   522
haftmann@41657
   523
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
wenzelm@60758
   524
  -- \<open>The inverse image of a singleton under an injective function
wenzelm@60758
   525
         is included in a singleton.\<close>
haftmann@41657
   526
  apply (auto simp add: inj_on_def)
haftmann@41657
   527
  apply (blast intro: the_equality [symmetric])
haftmann@41657
   528
  done
haftmann@41657
   529
hoelzl@43991
   530
lemma inj_on_vimage_singleton:
hoelzl@43991
   531
  "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
hoelzl@43991
   532
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
hoelzl@43991
   533
hoelzl@35584
   534
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   535
  by (auto intro!: inj_onI)
paulson@13585
   536
hoelzl@35584
   537
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   538
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   539
blanchet@55019
   540
lemma bij_betw_byWitness:
blanchet@55019
   541
assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
blanchet@55019
   542
        RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
blanchet@55019
   543
        IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
blanchet@55019
   544
shows "bij_betw f A A'"
blanchet@55019
   545
using assms
blanchet@55019
   546
proof(unfold bij_betw_def inj_on_def, safe)
blanchet@55019
   547
  fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
blanchet@55019
   548
  have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
blanchet@55019
   549
  with ** show "a = b" by simp
blanchet@55019
   550
next
blanchet@55019
   551
  fix a' assume *: "a' \<in> A'"
blanchet@55019
   552
  hence "f' a' \<in> A" using IM2 by blast
blanchet@55019
   553
  moreover
blanchet@55019
   554
  have "a' = f(f' a')" using * RIGHT by simp
blanchet@55019
   555
  ultimately show "a' \<in> f ` A" by blast
blanchet@55019
   556
qed
blanchet@55019
   557
blanchet@55019
   558
corollary notIn_Un_bij_betw:
blanchet@55019
   559
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
blanchet@55019
   560
       BIJ: "bij_betw f A A'"
blanchet@55019
   561
shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   562
proof-
blanchet@55019
   563
  have "bij_betw f {b} {f b}"
blanchet@55019
   564
  unfolding bij_betw_def inj_on_def by simp
blanchet@55019
   565
  with assms show ?thesis
blanchet@55019
   566
  using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
blanchet@55019
   567
qed
blanchet@55019
   568
blanchet@55019
   569
lemma notIn_Un_bij_betw3:
blanchet@55019
   570
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
blanchet@55019
   571
shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   572
proof
blanchet@55019
   573
  assume "bij_betw f A A'"
blanchet@55019
   574
  thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   575
  using assms notIn_Un_bij_betw[of b A f A'] by blast
blanchet@55019
   576
next
blanchet@55019
   577
  assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   578
  have "f ` A = A'"
blanchet@55019
   579
  proof(auto)
blanchet@55019
   580
    fix a assume **: "a \<in> A"
blanchet@55019
   581
    hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
blanchet@55019
   582
    moreover
blanchet@55019
   583
    {assume "f a = f b"
blanchet@55019
   584
     hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
blanchet@55019
   585
     with NIN ** have False by blast
blanchet@55019
   586
    }
blanchet@55019
   587
    ultimately show "f a \<in> A'" by blast
blanchet@55019
   588
  next
blanchet@55019
   589
    fix a' assume **: "a' \<in> A'"
blanchet@55019
   590
    hence "a' \<in> f`(A \<union> {b})"
blanchet@55019
   591
    using * by (auto simp add: bij_betw_def)
blanchet@55019
   592
    then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
blanchet@55019
   593
    moreover
blanchet@55019
   594
    {assume "a = b" with 1 ** NIN' have False by blast
blanchet@55019
   595
    }
blanchet@55019
   596
    ultimately have "a \<in> A" by blast
blanchet@55019
   597
    with 1 show "a' \<in> f ` A" by blast
blanchet@55019
   598
  qed
blanchet@55019
   599
  thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
blanchet@55019
   600
qed
blanchet@55019
   601
haftmann@41657
   602
wenzelm@60758
   603
subsection\<open>Function Updating\<close>
paulson@13585
   604
haftmann@44277
   605
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147
   606
  "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147
   607
wenzelm@41229
   608
nonterminal updbinds and updbind
wenzelm@41229
   609
haftmann@26147
   610
syntax
haftmann@26147
   611
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147
   612
  ""         :: "updbind => updbinds"             ("_")
haftmann@26147
   613
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115
   614
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   615
haftmann@26147
   616
translations
wenzelm@35115
   617
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115
   618
  "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147
   619
blanchet@55414
   620
(* Hint: to define the sum of two functions (or maps), use case_sum.
blanchet@58111
   621
         A nice infix syntax could be defined by
wenzelm@35115
   622
notation
blanchet@55414
   623
  case_sum  (infixr "'(+')"80)
haftmann@26147
   624
*)
haftmann@26147
   625
paulson@13585
   626
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585
   627
apply (simp add: fun_upd_def, safe)
paulson@13585
   628
apply (erule subst)
paulson@13585
   629
apply (rule_tac [2] ext, auto)
paulson@13585
   630
done
paulson@13585
   631
wenzelm@45603
   632
lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
wenzelm@45603
   633
  by (simp only: fun_upd_idem_iff)
paulson@13585
   634
wenzelm@45603
   635
lemma fun_upd_triv [iff]: "f(x := f x) = f"
wenzelm@45603
   636
  by (simp only: fun_upd_idem)
paulson@13585
   637
paulson@13585
   638
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084
   639
by (simp add: fun_upd_def)
paulson@13585
   640
paulson@13585
   641
(* fun_upd_apply supersedes these two,   but they are useful
paulson@13585
   642
   if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585
   643
lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585
   644
by simp
paulson@13585
   645
paulson@13585
   646
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585
   647
by simp
paulson@13585
   648
paulson@13585
   649
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
nipkow@39302
   650
by (simp add: fun_eq_iff)
paulson@13585
   651
paulson@13585
   652
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585
   653
by (rule ext, auto)
paulson@13585
   654
haftmann@56077
   655
lemma inj_on_fun_updI:
haftmann@56077
   656
  "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
haftmann@56077
   657
  by (fastforce simp: inj_on_def)
nipkow@15303
   658
paulson@15510
   659
lemma fun_upd_image:
paulson@15510
   660
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510
   661
by auto
paulson@15510
   662
nipkow@31080
   663
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
huffman@44921
   664
  by auto
nipkow@31080
   665
Andreas@61630
   666
lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z"
Andreas@61630
   667
by(simp add: fun_eq_iff split: split_if_asm)
haftmann@26147
   668
wenzelm@60758
   669
subsection \<open>@{text override_on}\<close>
haftmann@26147
   670
haftmann@44277
   671
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
haftmann@26147
   672
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   673
nipkow@15691
   674
lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691
   675
by(simp add:override_on_def)
nipkow@13910
   676
nipkow@15691
   677
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691
   678
by(simp add:override_on_def)
nipkow@13910
   679
nipkow@15691
   680
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691
   681
by(simp add:override_on_def)
nipkow@13910
   682
haftmann@26147
   683
wenzelm@60758
   684
subsection \<open>@{text swap}\<close>
paulson@15510
   685
haftmann@56608
   686
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@56608
   687
where
haftmann@22744
   688
  "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   689
haftmann@56608
   690
lemma swap_apply [simp]:
haftmann@56608
   691
  "swap a b f a = f b"
haftmann@56608
   692
  "swap a b f b = f a"
haftmann@56608
   693
  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
haftmann@56608
   694
  by (simp_all add: swap_def)
haftmann@56608
   695
haftmann@56608
   696
lemma swap_self [simp]:
haftmann@56608
   697
  "swap a a f = f"
haftmann@56608
   698
  by (simp add: swap_def)
paulson@15510
   699
haftmann@56608
   700
lemma swap_commute:
haftmann@56608
   701
  "swap a b f = swap b a f"
haftmann@56608
   702
  by (simp add: fun_upd_def swap_def fun_eq_iff)
paulson@15510
   703
haftmann@56608
   704
lemma swap_nilpotent [simp]:
haftmann@56608
   705
  "swap a b (swap a b f) = f"
haftmann@56608
   706
  by (rule ext, simp add: fun_upd_def swap_def)
haftmann@56608
   707
haftmann@56608
   708
lemma swap_comp_involutory [simp]:
haftmann@56608
   709
  "swap a b \<circ> swap a b = id"
haftmann@56608
   710
  by (rule ext) simp
paulson@15510
   711
huffman@34145
   712
lemma swap_triple:
huffman@34145
   713
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   714
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
nipkow@39302
   715
  using assms by (simp add: fun_eq_iff swap_def)
huffman@34145
   716
huffman@34101
   717
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
haftmann@56608
   718
  by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101
   719
hoelzl@39076
   720
lemma swap_image_eq [simp]:
hoelzl@39076
   721
  assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
hoelzl@39076
   722
proof -
hoelzl@39076
   723
  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
hoelzl@39076
   724
    using assms by (auto simp: image_iff swap_def)
hoelzl@39076
   725
  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
hoelzl@39076
   726
  with subset[of f] show ?thesis by auto
hoelzl@39076
   727
qed
hoelzl@39076
   728
paulson@15510
   729
lemma inj_on_imp_inj_on_swap:
hoelzl@39076
   730
  "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
hoelzl@39076
   731
  by (simp add: inj_on_def swap_def, blast)
paulson@15510
   732
paulson@15510
   733
lemma inj_on_swap_iff [simp]:
hoelzl@39076
   734
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
hoelzl@39075
   735
proof
paulson@15510
   736
  assume "inj_on (swap a b f) A"
hoelzl@39075
   737
  with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075
   738
    by (iprover intro: inj_on_imp_inj_on_swap)
hoelzl@39075
   739
  thus "inj_on f A" by simp
paulson@15510
   740
next
paulson@15510
   741
  assume "inj_on f A"
krauss@34209
   742
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   743
qed
paulson@15510
   744
hoelzl@39076
   745
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
hoelzl@40702
   746
  by simp
paulson@15510
   747
hoelzl@39076
   748
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
hoelzl@40702
   749
  by simp
haftmann@21547
   750
hoelzl@39076
   751
lemma bij_betw_swap_iff [simp]:
hoelzl@39076
   752
  "\<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
   753
  by (auto simp: bij_betw_def)
hoelzl@39076
   754
hoelzl@39076
   755
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
hoelzl@39076
   756
  by simp
hoelzl@39075
   757
wenzelm@36176
   758
hide_const (open) swap
haftmann@21547
   759
haftmann@56608
   760
wenzelm@60758
   761
subsection \<open>Inversion of injective functions\<close>
haftmann@31949
   762
nipkow@33057
   763
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
haftmann@44277
   764
  "the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961
   765
nipkow@33057
   766
lemma the_inv_into_f_f:
nipkow@33057
   767
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057
   768
apply (simp add: the_inv_into_def inj_on_def)
krauss@34209
   769
apply blast
nipkow@32961
   770
done
nipkow@32961
   771
nipkow@33057
   772
lemma f_the_inv_into_f:
nipkow@33057
   773
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057
   774
apply (simp add: the_inv_into_def)
nipkow@32961
   775
apply (rule the1I2)
nipkow@32961
   776
 apply(blast dest: inj_onD)
nipkow@32961
   777
apply blast
nipkow@32961
   778
done
nipkow@32961
   779
nipkow@33057
   780
lemma the_inv_into_into:
nipkow@33057
   781
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057
   782
apply (simp add: the_inv_into_def)
nipkow@32961
   783
apply (rule the1I2)
nipkow@32961
   784
 apply(blast dest: inj_onD)
nipkow@32961
   785
apply blast
nipkow@32961
   786
done
nipkow@32961
   787
nipkow@33057
   788
lemma the_inv_into_onto[simp]:
nipkow@33057
   789
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057
   790
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961
   791
nipkow@33057
   792
lemma the_inv_into_f_eq:
nipkow@33057
   793
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961
   794
  apply (erule subst)
nipkow@33057
   795
  apply (erule the_inv_into_f_f, assumption)
nipkow@32961
   796
  done
nipkow@32961
   797
nipkow@33057
   798
lemma the_inv_into_comp:
nipkow@32961
   799
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   800
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057
   801
apply (rule the_inv_into_f_eq)
nipkow@32961
   802
  apply (fast intro: comp_inj_on)
nipkow@33057
   803
 apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057
   804
apply (simp add: the_inv_into_into)
nipkow@32961
   805
done
nipkow@32961
   806
nipkow@33057
   807
lemma inj_on_the_inv_into:
nipkow@33057
   808
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
haftmann@56077
   809
by (auto intro: inj_onI simp: the_inv_into_f_f)
nipkow@32961
   810
nipkow@33057
   811
lemma bij_betw_the_inv_into:
nipkow@33057
   812
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057
   813
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   814
berghofe@32998
   815
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057
   816
  "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   817
berghofe@32998
   818
lemma the_inv_f_f:
berghofe@32998
   819
  assumes "inj f"
berghofe@32998
   820
  shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057
   821
  by (rule the_inv_into_f_f)
berghofe@32998
   822
haftmann@44277
   823
wenzelm@60758
   824
subsection \<open>Cantor's Paradox\<close>
hoelzl@40703
   825
blanchet@54147
   826
lemma Cantors_paradox:
hoelzl@40703
   827
  "\<not>(\<exists>f. f ` A = Pow A)"
hoelzl@40703
   828
proof clarify
hoelzl@40703
   829
  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
hoelzl@40703
   830
  let ?X = "{a \<in> A. a \<notin> f a}"
hoelzl@40703
   831
  have "?X \<in> Pow A" unfolding Pow_def by auto
hoelzl@40703
   832
  with * obtain x where "x \<in> A \<and> f x = ?X" by blast
hoelzl@40703
   833
  thus False by best
hoelzl@40703
   834
qed
haftmann@31949
   835
paulson@61204
   836
subsection \<open>Setup\<close>
haftmann@40969
   837
wenzelm@60758
   838
subsubsection \<open>Proof tools\<close>
haftmann@22845
   839
wenzelm@60758
   840
text \<open>simplifies terms of the form
wenzelm@60758
   841
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close>
haftmann@22845
   842
wenzelm@60758
   843
simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ =>
haftmann@22845
   844
let
haftmann@22845
   845
  fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017
   846
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845
   847
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845
   848
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845
   849
    let
haftmann@22845
   850
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845
   851
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845
   852
        | find t = NONE
haftmann@22845
   853
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   854
wenzelm@51717
   855
  val ss = simpset_of @{context}
wenzelm@51717
   856
wenzelm@51717
   857
  fun proc ctxt ct =
wenzelm@24017
   858
    let
wenzelm@24017
   859
      val t = Thm.term_of ct
wenzelm@24017
   860
    in
wenzelm@24017
   861
      case find_double t of
wenzelm@24017
   862
        (T, NONE) => NONE
wenzelm@24017
   863
      | (T, SOME rhs) =>
wenzelm@27330
   864
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017
   865
            (fn _ =>
wenzelm@59498
   866
              resolve_tac ctxt [eq_reflection] 1 THEN
wenzelm@59498
   867
              resolve_tac ctxt @{thms ext} 1 THEN
wenzelm@51717
   868
              simp_tac (put_simpset ss ctxt) 1))
wenzelm@24017
   869
    end
wenzelm@24017
   870
in proc end
wenzelm@60758
   871
\<close>
haftmann@22845
   872
haftmann@22845
   873
wenzelm@60758
   874
subsubsection \<open>Functorial structure of types\<close>
haftmann@40969
   875
blanchet@55467
   876
ML_file "Tools/functor.ML"
haftmann@40969
   877
blanchet@55467
   878
functor map_fun: map_fun
haftmann@47488
   879
  by (simp_all add: fun_eq_iff)
haftmann@47488
   880
blanchet@55467
   881
functor vimage
haftmann@49739
   882
  by (simp_all add: fun_eq_iff vimage_comp)
haftmann@49739
   883
wenzelm@60758
   884
text \<open>Legacy theorem names\<close>
haftmann@49739
   885
haftmann@49739
   886
lemmas o_def = comp_def
haftmann@49739
   887
lemmas o_apply = comp_apply
haftmann@49739
   888
lemmas o_assoc = comp_assoc [symmetric]
haftmann@49739
   889
lemmas id_o = id_comp
haftmann@49739
   890
lemmas o_id = comp_id
haftmann@49739
   891
lemmas o_eq_dest = comp_eq_dest
haftmann@49739
   892
lemmas o_eq_elim = comp_eq_elim
blanchet@55066
   893
lemmas o_eq_dest_lhs = comp_eq_dest_lhs
blanchet@55066
   894
lemmas o_eq_id_dest = comp_eq_id_dest
haftmann@47488
   895
nipkow@2912
   896
end
haftmann@56015
   897