src/HOL/Fun.thy
author haftmann
Wed Apr 16 21:51:41 2014 +0200 (2014-04-16)
changeset 56608 8e3c848008fa
parent 56154 f0a927235162
child 57282 7da3e398804c
permissions -rw-r--r--
more simp rules for Fun.swap
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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_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"
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  unfolding bij_betw_def ..
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lemma bijI: "[| inj f; surj f |] ==> bij f"
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by (simp add: bij_def)
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lemma bij_is_inj: "bij f ==> inj f"
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by (simp add: bij_def)
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   324
lemma bij_is_surj: "bij f ==> surj f"
paulson@13585
   325
by (simp add: bij_def)
paulson@13585
   326
nipkow@26105
   327
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105
   328
by (simp add: bij_betw_def)
nipkow@26105
   329
nipkow@31438
   330
lemma bij_betw_trans:
nipkow@31438
   331
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31438
   332
by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31438
   333
hoelzl@40702
   334
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
hoelzl@40702
   335
  by (rule bij_betw_trans)
hoelzl@40702
   336
hoelzl@40703
   337
lemma bij_betw_comp_iff:
hoelzl@40703
   338
  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   339
by(auto simp add: bij_betw_def inj_on_def)
hoelzl@40703
   340
hoelzl@40703
   341
lemma bij_betw_comp_iff2:
hoelzl@40703
   342
  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
hoelzl@40703
   343
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703
   344
using assms
hoelzl@40703
   345
proof(auto simp add: bij_betw_comp_iff)
hoelzl@40703
   346
  assume *: "bij_betw (f' \<circ> f) A A''"
hoelzl@40703
   347
  thus "bij_betw f A A'"
hoelzl@40703
   348
  using IM
hoelzl@40703
   349
  proof(auto simp add: bij_betw_def)
hoelzl@40703
   350
    assume "inj_on (f' \<circ> f) A"
hoelzl@40703
   351
    thus "inj_on f A" using inj_on_imageI2 by blast
hoelzl@40703
   352
  next
hoelzl@40703
   353
    fix a' assume **: "a' \<in> A'"
hoelzl@40703
   354
    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
hoelzl@40703
   355
    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
hoelzl@40703
   356
    unfolding bij_betw_def by force
hoelzl@40703
   357
    hence "f a \<in> A'" using IM by auto
hoelzl@40703
   358
    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
hoelzl@40703
   359
    thus "a' \<in> f ` A" using 1 by auto
hoelzl@40703
   360
  qed
hoelzl@40703
   361
qed
hoelzl@40703
   362
nipkow@26105
   363
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105
   364
proof -
nipkow@26105
   365
  have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105
   366
    using assms by(auto simp:bij_betw_def)
nipkow@26105
   367
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105
   368
  { fix a b assume P: "?P b a"
haftmann@56077
   369
    hence ex1: "\<exists>a. ?P b a" using s by blast
nipkow@26105
   370
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105
   371
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105
   372
  } note g = this
nipkow@26105
   373
  have "inj_on ?g B"
nipkow@26105
   374
  proof(rule inj_onI)
nipkow@26105
   375
    fix x y assume "x:B" "y:B" "?g x = ?g y"
haftmann@56077
   376
    from s `x:B` obtain a1 where a1: "?P x a1" by blast
haftmann@56077
   377
    from s `y:B` obtain a2 where a2: "?P y a2" by blast
nipkow@26105
   378
    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
nipkow@26105
   379
  qed
nipkow@26105
   380
  moreover have "?g ` B = A"
haftmann@56077
   381
  proof(auto simp: image_def)
nipkow@26105
   382
    fix b assume "b:B"
haftmann@56077
   383
    with s obtain a where P: "?P b a" by blast
nipkow@26105
   384
    thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105
   385
  next
nipkow@26105
   386
    fix a assume "a:A"
haftmann@56077
   387
    then obtain b where P: "?P b a" using s by blast
haftmann@56077
   388
    then have "b:B" using s by blast
nipkow@26105
   389
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105
   390
  qed
nipkow@26105
   391
  ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105
   392
qed
nipkow@26105
   393
hoelzl@40703
   394
lemma bij_betw_cong:
hoelzl@40703
   395
  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
hoelzl@40703
   396
unfolding bij_betw_def inj_on_def by force
hoelzl@40703
   397
hoelzl@40703
   398
lemma bij_betw_id[intro, simp]:
hoelzl@40703
   399
  "bij_betw id A A"
hoelzl@40703
   400
unfolding bij_betw_def id_def by auto
hoelzl@40703
   401
hoelzl@40703
   402
lemma bij_betw_id_iff:
hoelzl@40703
   403
  "bij_betw id A B \<longleftrightarrow> A = B"
hoelzl@40703
   404
by(auto simp add: bij_betw_def)
hoelzl@40703
   405
hoelzl@39075
   406
lemma bij_betw_combine:
hoelzl@39075
   407
  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
hoelzl@39075
   408
  shows "bij_betw f (A \<union> C) (B \<union> D)"
hoelzl@39075
   409
  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
hoelzl@39075
   410
hoelzl@40703
   411
lemma bij_betw_subset:
hoelzl@40703
   412
  assumes BIJ: "bij_betw f A A'" and
hoelzl@40703
   413
          SUB: "B \<le> A" and IM: "f ` B = B'"
hoelzl@40703
   414
  shows "bij_betw f B B'"
hoelzl@40703
   415
using assms
hoelzl@40703
   416
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
hoelzl@40703
   417
paulson@13585
   418
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
hoelzl@40702
   419
by simp
paulson@13585
   420
hoelzl@42903
   421
lemma surj_vimage_empty:
hoelzl@42903
   422
  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
hoelzl@42903
   423
  using surj_image_vimage_eq[OF `surj f`, of A]
nipkow@44890
   424
  by (intro iffI) fastforce+
hoelzl@42903
   425
paulson@13585
   426
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585
   427
by (simp add: inj_on_def, blast)
paulson@13585
   428
paulson@13585
   429
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
hoelzl@40702
   430
by (blast intro: sym)
paulson@13585
   431
paulson@13585
   432
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585
   433
by (unfold inj_on_def, blast)
paulson@13585
   434
paulson@13585
   435
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585
   436
apply (unfold bij_def)
paulson@13585
   437
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   438
done
paulson@13585
   439
Andreas@53927
   440
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
   441
by(fastforce simp add: inj_on_def)
Andreas@53927
   442
nipkow@31438
   443
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
Andreas@53927
   444
by(erule inj_on_image_eq_iff) simp_all
nipkow@31438
   445
paulson@13585
   446
lemma inj_on_image_Int:
paulson@13585
   447
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   448
apply (simp add: inj_on_def, blast)
paulson@13585
   449
done
paulson@13585
   450
paulson@13585
   451
lemma inj_on_image_set_diff:
paulson@13585
   452
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
paulson@13585
   453
apply (simp add: inj_on_def, blast)
paulson@13585
   454
done
paulson@13585
   455
paulson@13585
   456
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   457
by (simp add: inj_on_def, blast)
paulson@13585
   458
paulson@13585
   459
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585
   460
by (simp add: inj_on_def, blast)
paulson@13585
   461
paulson@13585
   462
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
paulson@13585
   463
by (blast dest: injD)
paulson@13585
   464
paulson@13585
   465
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
paulson@13585
   466
by (simp add: inj_on_def, blast)
paulson@13585
   467
paulson@13585
   468
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
paulson@13585
   469
by (blast dest: injD)
paulson@13585
   470
paulson@13585
   471
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
hoelzl@40702
   472
by auto
paulson@13585
   473
paulson@13585
   474
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585
   475
by (auto simp add: inj_on_def)
paulson@5852
   476
paulson@13585
   477
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585
   478
apply (simp add: bij_def)
paulson@13585
   479
apply (rule equalityI)
paulson@13585
   480
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585
   481
done
paulson@13585
   482
haftmann@41657
   483
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
haftmann@41657
   484
  -- {* The inverse image of a singleton under an injective function
haftmann@41657
   485
         is included in a singleton. *}
haftmann@41657
   486
  apply (auto simp add: inj_on_def)
haftmann@41657
   487
  apply (blast intro: the_equality [symmetric])
haftmann@41657
   488
  done
haftmann@41657
   489
hoelzl@43991
   490
lemma inj_on_vimage_singleton:
hoelzl@43991
   491
  "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
hoelzl@43991
   492
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
hoelzl@43991
   493
hoelzl@35584
   494
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   495
  by (auto intro!: inj_onI)
paulson@13585
   496
hoelzl@35584
   497
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   498
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   499
blanchet@55019
   500
lemma bij_betw_byWitness:
blanchet@55019
   501
assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
blanchet@55019
   502
        RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
blanchet@55019
   503
        IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
blanchet@55019
   504
shows "bij_betw f A A'"
blanchet@55019
   505
using assms
blanchet@55019
   506
proof(unfold bij_betw_def inj_on_def, safe)
blanchet@55019
   507
  fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
blanchet@55019
   508
  have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
blanchet@55019
   509
  with ** show "a = b" by simp
blanchet@55019
   510
next
blanchet@55019
   511
  fix a' assume *: "a' \<in> A'"
blanchet@55019
   512
  hence "f' a' \<in> A" using IM2 by blast
blanchet@55019
   513
  moreover
blanchet@55019
   514
  have "a' = f(f' a')" using * RIGHT by simp
blanchet@55019
   515
  ultimately show "a' \<in> f ` A" by blast
blanchet@55019
   516
qed
blanchet@55019
   517
blanchet@55019
   518
corollary notIn_Un_bij_betw:
blanchet@55019
   519
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
blanchet@55019
   520
       BIJ: "bij_betw f A A'"
blanchet@55019
   521
shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   522
proof-
blanchet@55019
   523
  have "bij_betw f {b} {f b}"
blanchet@55019
   524
  unfolding bij_betw_def inj_on_def by simp
blanchet@55019
   525
  with assms show ?thesis
blanchet@55019
   526
  using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
blanchet@55019
   527
qed
blanchet@55019
   528
blanchet@55019
   529
lemma notIn_Un_bij_betw3:
blanchet@55019
   530
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
blanchet@55019
   531
shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   532
proof
blanchet@55019
   533
  assume "bij_betw f A A'"
blanchet@55019
   534
  thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   535
  using assms notIn_Un_bij_betw[of b A f A'] by blast
blanchet@55019
   536
next
blanchet@55019
   537
  assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019
   538
  have "f ` A = A'"
blanchet@55019
   539
  proof(auto)
blanchet@55019
   540
    fix a assume **: "a \<in> A"
blanchet@55019
   541
    hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
blanchet@55019
   542
    moreover
blanchet@55019
   543
    {assume "f a = f b"
blanchet@55019
   544
     hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
blanchet@55019
   545
     with NIN ** have False by blast
blanchet@55019
   546
    }
blanchet@55019
   547
    ultimately show "f a \<in> A'" by blast
blanchet@55019
   548
  next
blanchet@55019
   549
    fix a' assume **: "a' \<in> A'"
blanchet@55019
   550
    hence "a' \<in> f`(A \<union> {b})"
blanchet@55019
   551
    using * by (auto simp add: bij_betw_def)
blanchet@55019
   552
    then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
blanchet@55019
   553
    moreover
blanchet@55019
   554
    {assume "a = b" with 1 ** NIN' have False by blast
blanchet@55019
   555
    }
blanchet@55019
   556
    ultimately have "a \<in> A" by blast
blanchet@55019
   557
    with 1 show "a' \<in> f ` A" by blast
blanchet@55019
   558
  qed
blanchet@55019
   559
  thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
blanchet@55019
   560
qed
blanchet@55019
   561
haftmann@41657
   562
paulson@13585
   563
subsection{*Function Updating*}
paulson@13585
   564
haftmann@44277
   565
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147
   566
  "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147
   567
wenzelm@41229
   568
nonterminal updbinds and updbind
wenzelm@41229
   569
haftmann@26147
   570
syntax
haftmann@26147
   571
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147
   572
  ""         :: "updbind => updbinds"             ("_")
haftmann@26147
   573
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115
   574
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   575
haftmann@26147
   576
translations
wenzelm@35115
   577
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115
   578
  "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147
   579
blanchet@55414
   580
(* Hint: to define the sum of two functions (or maps), use case_sum.
haftmann@26147
   581
         A nice infix syntax could be defined (in Datatype.thy or below) by
wenzelm@35115
   582
notation
blanchet@55414
   583
  case_sum  (infixr "'(+')"80)
haftmann@26147
   584
*)
haftmann@26147
   585
paulson@13585
   586
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585
   587
apply (simp add: fun_upd_def, safe)
paulson@13585
   588
apply (erule subst)
paulson@13585
   589
apply (rule_tac [2] ext, auto)
paulson@13585
   590
done
paulson@13585
   591
wenzelm@45603
   592
lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
wenzelm@45603
   593
  by (simp only: fun_upd_idem_iff)
paulson@13585
   594
wenzelm@45603
   595
lemma fun_upd_triv [iff]: "f(x := f x) = f"
wenzelm@45603
   596
  by (simp only: fun_upd_idem)
paulson@13585
   597
paulson@13585
   598
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084
   599
by (simp add: fun_upd_def)
paulson@13585
   600
paulson@13585
   601
(* fun_upd_apply supersedes these two,   but they are useful
paulson@13585
   602
   if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585
   603
lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585
   604
by simp
paulson@13585
   605
paulson@13585
   606
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585
   607
by simp
paulson@13585
   608
paulson@13585
   609
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
nipkow@39302
   610
by (simp add: fun_eq_iff)
paulson@13585
   611
paulson@13585
   612
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585
   613
by (rule ext, auto)
paulson@13585
   614
haftmann@56077
   615
lemma inj_on_fun_updI:
haftmann@56077
   616
  "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
haftmann@56077
   617
  by (fastforce simp: inj_on_def)
nipkow@15303
   618
paulson@15510
   619
lemma fun_upd_image:
paulson@15510
   620
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510
   621
by auto
paulson@15510
   622
nipkow@31080
   623
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
huffman@44921
   624
  by auto
nipkow@31080
   625
haftmann@26147
   626
haftmann@26147
   627
subsection {* @{text override_on} *}
haftmann@26147
   628
haftmann@44277
   629
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
haftmann@26147
   630
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   631
nipkow@15691
   632
lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691
   633
by(simp add:override_on_def)
nipkow@13910
   634
nipkow@15691
   635
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691
   636
by(simp add:override_on_def)
nipkow@13910
   637
nipkow@15691
   638
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691
   639
by(simp add:override_on_def)
nipkow@13910
   640
haftmann@26147
   641
haftmann@26147
   642
subsection {* @{text swap} *}
paulson@15510
   643
haftmann@56608
   644
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@56608
   645
where
haftmann@22744
   646
  "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   647
haftmann@56608
   648
lemma swap_apply [simp]:
haftmann@56608
   649
  "swap a b f a = f b"
haftmann@56608
   650
  "swap a b f b = f a"
haftmann@56608
   651
  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
haftmann@56608
   652
  by (simp_all add: swap_def)
haftmann@56608
   653
haftmann@56608
   654
lemma swap_self [simp]:
haftmann@56608
   655
  "swap a a f = f"
haftmann@56608
   656
  by (simp add: swap_def)
paulson@15510
   657
haftmann@56608
   658
lemma swap_commute:
haftmann@56608
   659
  "swap a b f = swap b a f"
haftmann@56608
   660
  by (simp add: fun_upd_def swap_def fun_eq_iff)
paulson@15510
   661
haftmann@56608
   662
lemma swap_nilpotent [simp]:
haftmann@56608
   663
  "swap a b (swap a b f) = f"
haftmann@56608
   664
  by (rule ext, simp add: fun_upd_def swap_def)
haftmann@56608
   665
haftmann@56608
   666
lemma swap_comp_involutory [simp]:
haftmann@56608
   667
  "swap a b \<circ> swap a b = id"
haftmann@56608
   668
  by (rule ext) simp
paulson@15510
   669
huffman@34145
   670
lemma swap_triple:
huffman@34145
   671
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   672
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
nipkow@39302
   673
  using assms by (simp add: fun_eq_iff swap_def)
huffman@34145
   674
huffman@34101
   675
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
haftmann@56608
   676
  by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101
   677
hoelzl@39076
   678
lemma swap_image_eq [simp]:
hoelzl@39076
   679
  assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
hoelzl@39076
   680
proof -
hoelzl@39076
   681
  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
hoelzl@39076
   682
    using assms by (auto simp: image_iff swap_def)
hoelzl@39076
   683
  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
hoelzl@39076
   684
  with subset[of f] show ?thesis by auto
hoelzl@39076
   685
qed
hoelzl@39076
   686
paulson@15510
   687
lemma inj_on_imp_inj_on_swap:
hoelzl@39076
   688
  "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
hoelzl@39076
   689
  by (simp add: inj_on_def swap_def, blast)
paulson@15510
   690
paulson@15510
   691
lemma inj_on_swap_iff [simp]:
hoelzl@39076
   692
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
hoelzl@39075
   693
proof
paulson@15510
   694
  assume "inj_on (swap a b f) A"
hoelzl@39075
   695
  with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075
   696
    by (iprover intro: inj_on_imp_inj_on_swap)
hoelzl@39075
   697
  thus "inj_on f A" by simp
paulson@15510
   698
next
paulson@15510
   699
  assume "inj_on f A"
krauss@34209
   700
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   701
qed
paulson@15510
   702
hoelzl@39076
   703
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
hoelzl@40702
   704
  by simp
paulson@15510
   705
hoelzl@39076
   706
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
hoelzl@40702
   707
  by simp
haftmann@21547
   708
hoelzl@39076
   709
lemma bij_betw_swap_iff [simp]:
hoelzl@39076
   710
  "\<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
   711
  by (auto simp: bij_betw_def)
hoelzl@39076
   712
hoelzl@39076
   713
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
hoelzl@39076
   714
  by simp
hoelzl@39075
   715
wenzelm@36176
   716
hide_const (open) swap
haftmann@21547
   717
haftmann@56608
   718
haftmann@31949
   719
subsection {* Inversion of injective functions *}
haftmann@31949
   720
nipkow@33057
   721
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
haftmann@44277
   722
  "the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961
   723
nipkow@33057
   724
lemma the_inv_into_f_f:
nipkow@33057
   725
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057
   726
apply (simp add: the_inv_into_def inj_on_def)
krauss@34209
   727
apply blast
nipkow@32961
   728
done
nipkow@32961
   729
nipkow@33057
   730
lemma f_the_inv_into_f:
nipkow@33057
   731
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057
   732
apply (simp add: the_inv_into_def)
nipkow@32961
   733
apply (rule the1I2)
nipkow@32961
   734
 apply(blast dest: inj_onD)
nipkow@32961
   735
apply blast
nipkow@32961
   736
done
nipkow@32961
   737
nipkow@33057
   738
lemma the_inv_into_into:
nipkow@33057
   739
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057
   740
apply (simp add: the_inv_into_def)
nipkow@32961
   741
apply (rule the1I2)
nipkow@32961
   742
 apply(blast dest: inj_onD)
nipkow@32961
   743
apply blast
nipkow@32961
   744
done
nipkow@32961
   745
nipkow@33057
   746
lemma the_inv_into_onto[simp]:
nipkow@33057
   747
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057
   748
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961
   749
nipkow@33057
   750
lemma the_inv_into_f_eq:
nipkow@33057
   751
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961
   752
  apply (erule subst)
nipkow@33057
   753
  apply (erule the_inv_into_f_f, assumption)
nipkow@32961
   754
  done
nipkow@32961
   755
nipkow@33057
   756
lemma the_inv_into_comp:
nipkow@32961
   757
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   758
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057
   759
apply (rule the_inv_into_f_eq)
nipkow@32961
   760
  apply (fast intro: comp_inj_on)
nipkow@33057
   761
 apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057
   762
apply (simp add: the_inv_into_into)
nipkow@32961
   763
done
nipkow@32961
   764
nipkow@33057
   765
lemma inj_on_the_inv_into:
nipkow@33057
   766
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
haftmann@56077
   767
by (auto intro: inj_onI simp: the_inv_into_f_f)
nipkow@32961
   768
nipkow@33057
   769
lemma bij_betw_the_inv_into:
nipkow@33057
   770
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057
   771
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   772
berghofe@32998
   773
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057
   774
  "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   775
berghofe@32998
   776
lemma the_inv_f_f:
berghofe@32998
   777
  assumes "inj f"
berghofe@32998
   778
  shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057
   779
  by (rule the_inv_into_f_f)
berghofe@32998
   780
haftmann@44277
   781
hoelzl@40703
   782
subsection {* Cantor's Paradox *}
hoelzl@40703
   783
blanchet@54147
   784
lemma Cantors_paradox:
hoelzl@40703
   785
  "\<not>(\<exists>f. f ` A = Pow A)"
hoelzl@40703
   786
proof clarify
hoelzl@40703
   787
  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
hoelzl@40703
   788
  let ?X = "{a \<in> A. a \<notin> f a}"
hoelzl@40703
   789
  have "?X \<in> Pow A" unfolding Pow_def by auto
hoelzl@40703
   790
  with * obtain x where "x \<in> A \<and> f x = ?X" by blast
hoelzl@40703
   791
  thus False by best
hoelzl@40703
   792
qed
haftmann@31949
   793
haftmann@40969
   794
subsection {* Setup *} 
haftmann@40969
   795
haftmann@40969
   796
subsubsection {* Proof tools *}
haftmann@22845
   797
haftmann@22845
   798
text {* simplifies terms of the form
haftmann@22845
   799
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
haftmann@22845
   800
wenzelm@24017
   801
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
haftmann@22845
   802
let
haftmann@22845
   803
  fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017
   804
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845
   805
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845
   806
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845
   807
    let
haftmann@22845
   808
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845
   809
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845
   810
        | find t = NONE
haftmann@22845
   811
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   812
wenzelm@51717
   813
  val ss = simpset_of @{context}
wenzelm@51717
   814
wenzelm@51717
   815
  fun proc ctxt ct =
wenzelm@24017
   816
    let
wenzelm@24017
   817
      val t = Thm.term_of ct
wenzelm@24017
   818
    in
wenzelm@24017
   819
      case find_double t of
wenzelm@24017
   820
        (T, NONE) => NONE
wenzelm@24017
   821
      | (T, SOME rhs) =>
wenzelm@27330
   822
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017
   823
            (fn _ =>
wenzelm@24017
   824
              rtac eq_reflection 1 THEN
wenzelm@55990
   825
              rtac @{thm ext} 1 THEN
wenzelm@51717
   826
              simp_tac (put_simpset ss ctxt) 1))
wenzelm@24017
   827
    end
wenzelm@24017
   828
in proc end
haftmann@22845
   829
*}
haftmann@22845
   830
haftmann@22845
   831
haftmann@40969
   832
subsubsection {* Functorial structure of types *}
haftmann@40969
   833
blanchet@55467
   834
ML_file "Tools/functor.ML"
haftmann@40969
   835
blanchet@55467
   836
functor map_fun: map_fun
haftmann@47488
   837
  by (simp_all add: fun_eq_iff)
haftmann@47488
   838
blanchet@55467
   839
functor vimage
haftmann@49739
   840
  by (simp_all add: fun_eq_iff vimage_comp)
haftmann@49739
   841
haftmann@49739
   842
text {* Legacy theorem names *}
haftmann@49739
   843
haftmann@49739
   844
lemmas o_def = comp_def
haftmann@49739
   845
lemmas o_apply = comp_apply
haftmann@49739
   846
lemmas o_assoc = comp_assoc [symmetric]
haftmann@49739
   847
lemmas id_o = id_comp
haftmann@49739
   848
lemmas o_id = comp_id
haftmann@49739
   849
lemmas o_eq_dest = comp_eq_dest
haftmann@49739
   850
lemmas o_eq_elim = comp_eq_elim
blanchet@55066
   851
lemmas o_eq_dest_lhs = comp_eq_dest_lhs
blanchet@55066
   852
lemmas o_eq_id_dest = comp_eq_id_dest
haftmann@47488
   853
nipkow@2912
   854
end
haftmann@56015
   855