src/HOL/Fun.thy
author ballarin
Sat Jun 21 10:41:02 2014 +0200 (2014-06-21)
changeset 57282 7da3e398804c
parent 56608 8e3c848008fa
child 58111 82db9ad610b9
permissions -rw-r--r--
Two basic lemmas on bij_betw.
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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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header {* Notions about functions *}
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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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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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code_printing
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  constant id \<rightharpoonup> (Haskell) "id"
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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 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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code_printing
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  constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."
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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_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 {* 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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  "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 inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
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  by (fact inj_on_eq_iff)
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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 `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 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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  -- {* Courtesy of Stephan Merz *}
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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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lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
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  unfolding bij_betw_def by clarify
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lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
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  unfolding bij_betw_def by auto
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lemma bij_betw_empty1:
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  assumes "bij_betw f {} A"
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  shows "A = {}"
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using assms unfolding bij_betw_def by blast
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lemma bij_betw_empty2:
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  assumes "bij_betw f A {}"
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  shows "A = {}"
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using assms unfolding bij_betw_def by blast
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lemma inj_on_imp_bij_betw:
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  "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
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unfolding bij_betw_def by simp
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lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
hoelzl@40702
   323
  unfolding bij_betw_def ..
hoelzl@39074
   324
paulson@13585
   325
lemma bijI: "[| inj f; surj f |] ==> bij f"
paulson@13585
   326
by (simp add: bij_def)
paulson@13585
   327
paulson@13585
   328
lemma bij_is_inj: "bij f ==> inj f"
paulson@13585
   329
by (simp add: bij_def)
paulson@13585
   330
paulson@13585
   331
lemma bij_is_surj: "bij f ==> surj f"
paulson@13585
   332
by (simp add: bij_def)
paulson@13585
   333
nipkow@26105
   334
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105
   335
by (simp add: bij_betw_def)
nipkow@26105
   336
nipkow@31438
   337
lemma bij_betw_trans:
nipkow@31438
   338
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31438
   339
by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31438
   340
hoelzl@40702
   341
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
hoelzl@40702
   342
  by (rule bij_betw_trans)
hoelzl@40702
   343
hoelzl@40703
   344
lemma bij_betw_comp_iff:
hoelzl@40703
   345
  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   346
by(auto simp add: bij_betw_def inj_on_def)
hoelzl@40703
   347
hoelzl@40703
   348
lemma bij_betw_comp_iff2:
hoelzl@40703
   349
  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
hoelzl@40703
   350
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   351
using assms
hoelzl@40703
   352
proof(auto simp add: bij_betw_comp_iff)
hoelzl@40703
   353
  assume *: "bij_betw (f' \<circ> f) A A''"
hoelzl@40703
   354
  thus "bij_betw f A A'"
hoelzl@40703
   355
  using IM
hoelzl@40703
   356
  proof(auto simp add: bij_betw_def)
hoelzl@40703
   357
    assume "inj_on (f' \<circ> f) A"
hoelzl@40703
   358
    thus "inj_on f A" using inj_on_imageI2 by blast
hoelzl@40703
   359
  next
hoelzl@40703
   360
    fix a' assume **: "a' \<in> A'"
hoelzl@40703
   361
    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
hoelzl@40703
   362
    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
hoelzl@40703
   363
    unfolding bij_betw_def by force
hoelzl@40703
   364
    hence "f a \<in> A'" using IM by auto
hoelzl@40703
   365
    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
hoelzl@40703
   366
    thus "a' \<in> f ` A" using 1 by auto
hoelzl@40703
   367
  qed
hoelzl@40703
   368
qed
hoelzl@40703
   369
nipkow@26105
   370
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105
   371
proof -
nipkow@26105
   372
  have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105
   373
    using assms by(auto simp:bij_betw_def)
nipkow@26105
   374
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105
   375
  { fix a b assume P: "?P b a"
haftmann@56077
   376
    hence ex1: "\<exists>a. ?P b a" using s by blast
nipkow@26105
   377
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105
   378
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105
   379
  } note g = this
nipkow@26105
   380
  have "inj_on ?g B"
nipkow@26105
   381
  proof(rule inj_onI)
nipkow@26105
   382
    fix x y assume "x:B" "y:B" "?g x = ?g y"
haftmann@56077
   383
    from s `x:B` obtain a1 where a1: "?P x a1" by blast
haftmann@56077
   384
    from s `y:B` obtain a2 where a2: "?P y a2" by blast
nipkow@26105
   385
    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
nipkow@26105
   386
  qed
nipkow@26105
   387
  moreover have "?g ` B = A"
haftmann@56077
   388
  proof(auto simp: image_def)
nipkow@26105
   389
    fix b assume "b:B"
haftmann@56077
   390
    with s obtain a where P: "?P b a" by blast
nipkow@26105
   391
    thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105
   392
  next
nipkow@26105
   393
    fix a assume "a:A"
haftmann@56077
   394
    then obtain b where P: "?P b a" using s by blast
haftmann@56077
   395
    then have "b:B" using s by blast
nipkow@26105
   396
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105
   397
  qed
nipkow@26105
   398
  ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105
   399
qed
nipkow@26105
   400
hoelzl@40703
   401
lemma bij_betw_cong:
hoelzl@40703
   402
  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
hoelzl@40703
   403
unfolding bij_betw_def inj_on_def by force
hoelzl@40703
   404
hoelzl@40703
   405
lemma bij_betw_id[intro, simp]:
hoelzl@40703
   406
  "bij_betw id A A"
hoelzl@40703
   407
unfolding bij_betw_def id_def by auto
hoelzl@40703
   408
hoelzl@40703
   409
lemma bij_betw_id_iff:
hoelzl@40703
   410
  "bij_betw id A B \<longleftrightarrow> A = B"
hoelzl@40703
   411
by(auto simp add: bij_betw_def)
hoelzl@40703
   412
hoelzl@39075
   413
lemma bij_betw_combine:
hoelzl@39075
   414
  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
hoelzl@39075
   415
  shows "bij_betw f (A \<union> C) (B \<union> D)"
hoelzl@39075
   416
  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
hoelzl@39075
   417
hoelzl@40703
   418
lemma bij_betw_subset:
hoelzl@40703
   419
  assumes BIJ: "bij_betw f A A'" and
hoelzl@40703
   420
          SUB: "B \<le> A" and IM: "f ` B = B'"
hoelzl@40703
   421
  shows "bij_betw f B B'"
hoelzl@40703
   422
using assms
hoelzl@40703
   423
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
hoelzl@40703
   424
paulson@13585
   425
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
hoelzl@40702
   426
by simp
paulson@13585
   427
hoelzl@42903
   428
lemma surj_vimage_empty:
hoelzl@42903
   429
  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
hoelzl@42903
   430
  using surj_image_vimage_eq[OF `surj f`, of A]
nipkow@44890
   431
  by (intro iffI) fastforce+
hoelzl@42903
   432
paulson@13585
   433
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585
   434
by (simp add: inj_on_def, blast)
paulson@13585
   435
paulson@13585
   436
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
hoelzl@40702
   437
by (blast intro: sym)
paulson@13585
   438
paulson@13585
   439
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585
   440
by (unfold inj_on_def, blast)
paulson@13585
   441
paulson@13585
   442
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585
   443
apply (unfold bij_def)
paulson@13585
   444
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   445
done
paulson@13585
   446
Andreas@53927
   447
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
   448
by(fastforce simp add: inj_on_def)
Andreas@53927
   449
nipkow@31438
   450
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
Andreas@53927
   451
by(erule inj_on_image_eq_iff) simp_all
nipkow@31438
   452
paulson@13585
   453
lemma inj_on_image_Int:
paulson@13585
   454
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   455
apply (simp add: inj_on_def, blast)
paulson@13585
   456
done
paulson@13585
   457
paulson@13585
   458
lemma inj_on_image_set_diff:
paulson@13585
   459
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
paulson@13585
   460
apply (simp add: inj_on_def, blast)
paulson@13585
   461
done
paulson@13585
   462
paulson@13585
   463
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   464
by (simp add: inj_on_def, blast)
paulson@13585
   465
paulson@13585
   466
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585
   467
by (simp add: inj_on_def, blast)
paulson@13585
   468
paulson@13585
   469
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
paulson@13585
   470
by (blast dest: injD)
paulson@13585
   471
paulson@13585
   472
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
paulson@13585
   473
by (simp add: inj_on_def, blast)
paulson@13585
   474
paulson@13585
   475
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
paulson@13585
   476
by (blast dest: injD)
paulson@13585
   477
paulson@13585
   478
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
hoelzl@40702
   479
by auto
paulson@13585
   480
paulson@13585
   481
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585
   482
by (auto simp add: inj_on_def)
paulson@5852
   483
paulson@13585
   484
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585
   485
apply (simp add: bij_def)
paulson@13585
   486
apply (rule equalityI)
paulson@13585
   487
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585
   488
done
paulson@13585
   489
haftmann@41657
   490
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
haftmann@41657
   491
  -- {* The inverse image of a singleton under an injective function
haftmann@41657
   492
         is included in a singleton. *}
haftmann@41657
   493
  apply (auto simp add: inj_on_def)
haftmann@41657
   494
  apply (blast intro: the_equality [symmetric])
haftmann@41657
   495
  done
haftmann@41657
   496
hoelzl@43991
   497
lemma inj_on_vimage_singleton:
hoelzl@43991
   498
  "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
hoelzl@43991
   499
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
hoelzl@43991
   500
hoelzl@35584
   501
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   502
  by (auto intro!: inj_onI)
paulson@13585
   503
hoelzl@35584
   504
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   505
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   506
blanchet@55019
   507
lemma bij_betw_byWitness:
blanchet@55019
   508
assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
blanchet@55019
   509
        RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
blanchet@55019
   510
        IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
blanchet@55019
   511
shows "bij_betw f A A'"
blanchet@55019
   512
using assms
blanchet@55019
   513
proof(unfold bij_betw_def inj_on_def, safe)
blanchet@55019
   514
  fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
blanchet@55019
   515
  have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
blanchet@55019
   516
  with ** show "a = b" by simp
blanchet@55019
   517
next
blanchet@55019
   518
  fix a' assume *: "a' \<in> A'"
blanchet@55019
   519
  hence "f' a' \<in> A" using IM2 by blast
blanchet@55019
   520
  moreover
blanchet@55019
   521
  have "a' = f(f' a')" using * RIGHT by simp
blanchet@55019
   522
  ultimately show "a' \<in> f ` A" by blast
blanchet@55019
   523
qed
blanchet@55019
   524
blanchet@55019
   525
corollary notIn_Un_bij_betw:
blanchet@55019
   526
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
blanchet@55019
   527
       BIJ: "bij_betw f A A'"
blanchet@55019
   528
shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   529
proof-
blanchet@55019
   530
  have "bij_betw f {b} {f b}"
blanchet@55019
   531
  unfolding bij_betw_def inj_on_def by simp
blanchet@55019
   532
  with assms show ?thesis
blanchet@55019
   533
  using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
blanchet@55019
   534
qed
blanchet@55019
   535
blanchet@55019
   536
lemma notIn_Un_bij_betw3:
blanchet@55019
   537
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
blanchet@55019
   538
shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   539
proof
blanchet@55019
   540
  assume "bij_betw f A A'"
blanchet@55019
   541
  thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   542
  using assms notIn_Un_bij_betw[of b A f A'] by blast
blanchet@55019
   543
next
blanchet@55019
   544
  assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   545
  have "f ` A = A'"
blanchet@55019
   546
  proof(auto)
blanchet@55019
   547
    fix a assume **: "a \<in> A"
blanchet@55019
   548
    hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
blanchet@55019
   549
    moreover
blanchet@55019
   550
    {assume "f a = f b"
blanchet@55019
   551
     hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
blanchet@55019
   552
     with NIN ** have False by blast
blanchet@55019
   553
    }
blanchet@55019
   554
    ultimately show "f a \<in> A'" by blast
blanchet@55019
   555
  next
blanchet@55019
   556
    fix a' assume **: "a' \<in> A'"
blanchet@55019
   557
    hence "a' \<in> f`(A \<union> {b})"
blanchet@55019
   558
    using * by (auto simp add: bij_betw_def)
blanchet@55019
   559
    then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
blanchet@55019
   560
    moreover
blanchet@55019
   561
    {assume "a = b" with 1 ** NIN' have False by blast
blanchet@55019
   562
    }
blanchet@55019
   563
    ultimately have "a \<in> A" by blast
blanchet@55019
   564
    with 1 show "a' \<in> f ` A" by blast
blanchet@55019
   565
  qed
blanchet@55019
   566
  thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
blanchet@55019
   567
qed
blanchet@55019
   568
haftmann@41657
   569
paulson@13585
   570
subsection{*Function Updating*}
paulson@13585
   571
haftmann@44277
   572
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147
   573
  "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147
   574
wenzelm@41229
   575
nonterminal updbinds and updbind
wenzelm@41229
   576
haftmann@26147
   577
syntax
haftmann@26147
   578
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147
   579
  ""         :: "updbind => updbinds"             ("_")
haftmann@26147
   580
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115
   581
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   582
haftmann@26147
   583
translations
wenzelm@35115
   584
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115
   585
  "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147
   586
blanchet@55414
   587
(* Hint: to define the sum of two functions (or maps), use case_sum.
haftmann@26147
   588
         A nice infix syntax could be defined (in Datatype.thy or below) by
wenzelm@35115
   589
notation
blanchet@55414
   590
  case_sum  (infixr "'(+')"80)
haftmann@26147
   591
*)
haftmann@26147
   592
paulson@13585
   593
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585
   594
apply (simp add: fun_upd_def, safe)
paulson@13585
   595
apply (erule subst)
paulson@13585
   596
apply (rule_tac [2] ext, auto)
paulson@13585
   597
done
paulson@13585
   598
wenzelm@45603
   599
lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
wenzelm@45603
   600
  by (simp only: fun_upd_idem_iff)
paulson@13585
   601
wenzelm@45603
   602
lemma fun_upd_triv [iff]: "f(x := f x) = f"
wenzelm@45603
   603
  by (simp only: fun_upd_idem)
paulson@13585
   604
paulson@13585
   605
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084
   606
by (simp add: fun_upd_def)
paulson@13585
   607
paulson@13585
   608
(* fun_upd_apply supersedes these two,   but they are useful
paulson@13585
   609
   if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585
   610
lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585
   611
by simp
paulson@13585
   612
paulson@13585
   613
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585
   614
by simp
paulson@13585
   615
paulson@13585
   616
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
nipkow@39302
   617
by (simp add: fun_eq_iff)
paulson@13585
   618
paulson@13585
   619
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585
   620
by (rule ext, auto)
paulson@13585
   621
haftmann@56077
   622
lemma inj_on_fun_updI:
haftmann@56077
   623
  "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
haftmann@56077
   624
  by (fastforce simp: inj_on_def)
nipkow@15303
   625
paulson@15510
   626
lemma fun_upd_image:
paulson@15510
   627
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510
   628
by auto
paulson@15510
   629
nipkow@31080
   630
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
huffman@44921
   631
  by auto
nipkow@31080
   632
haftmann@26147
   633
haftmann@26147
   634
subsection {* @{text override_on} *}
haftmann@26147
   635
haftmann@44277
   636
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
haftmann@26147
   637
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   638
nipkow@15691
   639
lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691
   640
by(simp add:override_on_def)
nipkow@13910
   641
nipkow@15691
   642
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691
   643
by(simp add:override_on_def)
nipkow@13910
   644
nipkow@15691
   645
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691
   646
by(simp add:override_on_def)
nipkow@13910
   647
haftmann@26147
   648
haftmann@26147
   649
subsection {* @{text swap} *}
paulson@15510
   650
haftmann@56608
   651
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@56608
   652
where
haftmann@22744
   653
  "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   654
haftmann@56608
   655
lemma swap_apply [simp]:
haftmann@56608
   656
  "swap a b f a = f b"
haftmann@56608
   657
  "swap a b f b = f a"
haftmann@56608
   658
  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
haftmann@56608
   659
  by (simp_all add: swap_def)
haftmann@56608
   660
haftmann@56608
   661
lemma swap_self [simp]:
haftmann@56608
   662
  "swap a a f = f"
haftmann@56608
   663
  by (simp add: swap_def)
paulson@15510
   664
haftmann@56608
   665
lemma swap_commute:
haftmann@56608
   666
  "swap a b f = swap b a f"
haftmann@56608
   667
  by (simp add: fun_upd_def swap_def fun_eq_iff)
paulson@15510
   668
haftmann@56608
   669
lemma swap_nilpotent [simp]:
haftmann@56608
   670
  "swap a b (swap a b f) = f"
haftmann@56608
   671
  by (rule ext, simp add: fun_upd_def swap_def)
haftmann@56608
   672
haftmann@56608
   673
lemma swap_comp_involutory [simp]:
haftmann@56608
   674
  "swap a b \<circ> swap a b = id"
haftmann@56608
   675
  by (rule ext) simp
paulson@15510
   676
huffman@34145
   677
lemma swap_triple:
huffman@34145
   678
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   679
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
nipkow@39302
   680
  using assms by (simp add: fun_eq_iff swap_def)
huffman@34145
   681
huffman@34101
   682
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
haftmann@56608
   683
  by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101
   684
hoelzl@39076
   685
lemma swap_image_eq [simp]:
hoelzl@39076
   686
  assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
hoelzl@39076
   687
proof -
hoelzl@39076
   688
  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
hoelzl@39076
   689
    using assms by (auto simp: image_iff swap_def)
hoelzl@39076
   690
  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
hoelzl@39076
   691
  with subset[of f] show ?thesis by auto
hoelzl@39076
   692
qed
hoelzl@39076
   693
paulson@15510
   694
lemma inj_on_imp_inj_on_swap:
hoelzl@39076
   695
  "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
hoelzl@39076
   696
  by (simp add: inj_on_def swap_def, blast)
paulson@15510
   697
paulson@15510
   698
lemma inj_on_swap_iff [simp]:
hoelzl@39076
   699
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
hoelzl@39075
   700
proof
paulson@15510
   701
  assume "inj_on (swap a b f) A"
hoelzl@39075
   702
  with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075
   703
    by (iprover intro: inj_on_imp_inj_on_swap)
hoelzl@39075
   704
  thus "inj_on f A" by simp
paulson@15510
   705
next
paulson@15510
   706
  assume "inj_on f A"
krauss@34209
   707
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   708
qed
paulson@15510
   709
hoelzl@39076
   710
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
hoelzl@40702
   711
  by simp
paulson@15510
   712
hoelzl@39076
   713
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
hoelzl@40702
   714
  by simp
haftmann@21547
   715
hoelzl@39076
   716
lemma bij_betw_swap_iff [simp]:
hoelzl@39076
   717
  "\<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
   718
  by (auto simp: bij_betw_def)
hoelzl@39076
   719
hoelzl@39076
   720
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
hoelzl@39076
   721
  by simp
hoelzl@39075
   722
wenzelm@36176
   723
hide_const (open) swap
haftmann@21547
   724
haftmann@56608
   725
haftmann@31949
   726
subsection {* Inversion of injective functions *}
haftmann@31949
   727
nipkow@33057
   728
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
haftmann@44277
   729
  "the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961
   730
nipkow@33057
   731
lemma the_inv_into_f_f:
nipkow@33057
   732
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057
   733
apply (simp add: the_inv_into_def inj_on_def)
krauss@34209
   734
apply blast
nipkow@32961
   735
done
nipkow@32961
   736
nipkow@33057
   737
lemma f_the_inv_into_f:
nipkow@33057
   738
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057
   739
apply (simp add: the_inv_into_def)
nipkow@32961
   740
apply (rule the1I2)
nipkow@32961
   741
 apply(blast dest: inj_onD)
nipkow@32961
   742
apply blast
nipkow@32961
   743
done
nipkow@32961
   744
nipkow@33057
   745
lemma the_inv_into_into:
nipkow@33057
   746
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057
   747
apply (simp add: the_inv_into_def)
nipkow@32961
   748
apply (rule the1I2)
nipkow@32961
   749
 apply(blast dest: inj_onD)
nipkow@32961
   750
apply blast
nipkow@32961
   751
done
nipkow@32961
   752
nipkow@33057
   753
lemma the_inv_into_onto[simp]:
nipkow@33057
   754
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057
   755
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961
   756
nipkow@33057
   757
lemma the_inv_into_f_eq:
nipkow@33057
   758
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961
   759
  apply (erule subst)
nipkow@33057
   760
  apply (erule the_inv_into_f_f, assumption)
nipkow@32961
   761
  done
nipkow@32961
   762
nipkow@33057
   763
lemma the_inv_into_comp:
nipkow@32961
   764
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   765
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057
   766
apply (rule the_inv_into_f_eq)
nipkow@32961
   767
  apply (fast intro: comp_inj_on)
nipkow@33057
   768
 apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057
   769
apply (simp add: the_inv_into_into)
nipkow@32961
   770
done
nipkow@32961
   771
nipkow@33057
   772
lemma inj_on_the_inv_into:
nipkow@33057
   773
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
haftmann@56077
   774
by (auto intro: inj_onI simp: the_inv_into_f_f)
nipkow@32961
   775
nipkow@33057
   776
lemma bij_betw_the_inv_into:
nipkow@33057
   777
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057
   778
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   779
berghofe@32998
   780
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057
   781
  "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   782
berghofe@32998
   783
lemma the_inv_f_f:
berghofe@32998
   784
  assumes "inj f"
berghofe@32998
   785
  shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057
   786
  by (rule the_inv_into_f_f)
berghofe@32998
   787
haftmann@44277
   788
hoelzl@40703
   789
subsection {* Cantor's Paradox *}
hoelzl@40703
   790
blanchet@54147
   791
lemma Cantors_paradox:
hoelzl@40703
   792
  "\<not>(\<exists>f. f ` A = Pow A)"
hoelzl@40703
   793
proof clarify
hoelzl@40703
   794
  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
hoelzl@40703
   795
  let ?X = "{a \<in> A. a \<notin> f a}"
hoelzl@40703
   796
  have "?X \<in> Pow A" unfolding Pow_def by auto
hoelzl@40703
   797
  with * obtain x where "x \<in> A \<and> f x = ?X" by blast
hoelzl@40703
   798
  thus False by best
hoelzl@40703
   799
qed
haftmann@31949
   800
haftmann@40969
   801
subsection {* Setup *} 
haftmann@40969
   802
haftmann@40969
   803
subsubsection {* Proof tools *}
haftmann@22845
   804
haftmann@22845
   805
text {* simplifies terms of the form
haftmann@22845
   806
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
haftmann@22845
   807
wenzelm@24017
   808
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
haftmann@22845
   809
let
haftmann@22845
   810
  fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017
   811
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845
   812
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845
   813
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845
   814
    let
haftmann@22845
   815
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845
   816
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845
   817
        | find t = NONE
haftmann@22845
   818
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   819
wenzelm@51717
   820
  val ss = simpset_of @{context}
wenzelm@51717
   821
wenzelm@51717
   822
  fun proc ctxt ct =
wenzelm@24017
   823
    let
wenzelm@24017
   824
      val t = Thm.term_of ct
wenzelm@24017
   825
    in
wenzelm@24017
   826
      case find_double t of
wenzelm@24017
   827
        (T, NONE) => NONE
wenzelm@24017
   828
      | (T, SOME rhs) =>
wenzelm@27330
   829
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017
   830
            (fn _ =>
wenzelm@24017
   831
              rtac eq_reflection 1 THEN
wenzelm@55990
   832
              rtac @{thm ext} 1 THEN
wenzelm@51717
   833
              simp_tac (put_simpset ss ctxt) 1))
wenzelm@24017
   834
    end
wenzelm@24017
   835
in proc end
haftmann@22845
   836
*}
haftmann@22845
   837
haftmann@22845
   838
haftmann@40969
   839
subsubsection {* Functorial structure of types *}
haftmann@40969
   840
blanchet@55467
   841
ML_file "Tools/functor.ML"
haftmann@40969
   842
blanchet@55467
   843
functor map_fun: map_fun
haftmann@47488
   844
  by (simp_all add: fun_eq_iff)
haftmann@47488
   845
blanchet@55467
   846
functor vimage
haftmann@49739
   847
  by (simp_all add: fun_eq_iff vimage_comp)
haftmann@49739
   848
haftmann@49739
   849
text {* Legacy theorem names *}
haftmann@49739
   850
haftmann@49739
   851
lemmas o_def = comp_def
haftmann@49739
   852
lemmas o_apply = comp_apply
haftmann@49739
   853
lemmas o_assoc = comp_assoc [symmetric]
haftmann@49739
   854
lemmas id_o = id_comp
haftmann@49739
   855
lemmas o_id = comp_id
haftmann@49739
   856
lemmas o_eq_dest = comp_eq_dest
haftmann@49739
   857
lemmas o_eq_elim = comp_eq_elim
blanchet@55066
   858
lemmas o_eq_dest_lhs = comp_eq_dest_lhs
blanchet@55066
   859
lemmas o_eq_id_dest = comp_eq_id_dest
haftmann@47488
   860
nipkow@2912
   861
end
haftmann@56015
   862