src/HOL/Fun.thy
author hoelzl
Thu Sep 02 10:45:51 2010 +0200 (2010-09-02)
changeset 39075 a18e5946d63c
parent 39074 211e4f6aad63
child 39076 b3a9b6734663
permissions -rw-r--r--
Permutation implies bij function
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(*  Title:      HOL/Fun.thy
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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header {* Notions about functions *}
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theory Fun
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imports Complete_Lattice
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begin
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text{*As a simplification rule, it replaces all function equalities by
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  first-order equalities.*}
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lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
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apply (rule iffI)
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apply (simp (no_asm_simp))
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apply (rule ext)
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apply (simp (no_asm_simp))
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done
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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
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  id :: "'a \<Rightarrow> 'a"
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where
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  "id = (\<lambda>x. x)"
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lemma id_apply [simp]: "id x = x"
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  by (simp add: id_def)
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lemma image_ident [simp]: "(%x. x) ` Y = Y"
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by blast
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lemma image_id [simp]: "id ` Y = Y"
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by (simp add: id_def)
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lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
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by blast
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lemma vimage_id [simp]: "id -` A = A"
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by (simp add: id_def)
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subsection {* The Composition Operator @{text "f \<circ> g"} *}
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definition
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  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
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where
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  "f o g = (\<lambda>x. f (g x))"
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notation (xsymbols)
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  comp  (infixl "\<circ>" 55)
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notation (HTML output)
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  comp  (infixl "\<circ>" 55)
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text{*compatibility*}
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lemmas o_def = comp_def
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lemma o_apply [simp]: "(f o g) x = f (g x)"
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by (simp add: comp_def)
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lemma o_assoc: "f o (g o h) = f o g o h"
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by (simp add: comp_def)
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lemma id_o [simp]: "id o g = g"
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by (simp add: comp_def)
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lemma o_id [simp]: "f o id = f"
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by (simp add: comp_def)
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lemma o_eq_dest:
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  "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
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  by (simp only: o_def) (fact fun_cong)
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lemma o_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 (erule meta_mp) (fact o_eq_dest) 
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lemma image_compose: "(f o g) ` r = f`(g`r)"
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by (simp add: comp_def, blast)
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lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
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  by auto
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lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
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by (unfold comp_def, blast)
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subsection {* The Forward Composition Operator @{text fcomp} *}
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definition
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  fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60)
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where
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  "f \<circ>> g = (\<lambda>x. g (f x))"
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lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
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  by (simp add: fcomp_def)
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lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
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  by (simp add: fcomp_def)
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lemma id_fcomp [simp]: "id \<circ>> g = g"
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  by (simp add: fcomp_def)
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lemma fcomp_id [simp]: "f \<circ>> id = f"
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  by (simp add: fcomp_def)
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code_const fcomp
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  (Eval infixl 1 "#>")
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no_notation fcomp (infixl "\<circ>>" 60)
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subsection {* Injectivity and Surjectivity *}
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definition
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  inj_on :: "['a => 'b, 'a set] => bool" where
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  -- "injective"
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  "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
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text{*A common special case: functions injective over the entire domain type.*}
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abbreviation
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  "inj f == inj_on f UNIV"
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definition
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  bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
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  "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
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definition
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  surj :: "('a => 'b) => bool" where
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  -- "surjective"
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  "surj f == ! y. ? x. y=f(x)"
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definition
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  bij :: "('a => 'b) => bool" where
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  -- "bijective"
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  "bij f == inj f & surj f"
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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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text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
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lemma datatype_injI:
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    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
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by (simp add: inj_on_def)
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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_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 expand_fun_eq)
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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 surj_id[simp]: "surj id"
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by (simp add: surj_def) 
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lemma bij_id[simp]: "bij id"
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by (simp add: bij_def)
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lemma inj_onI:
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    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
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by (simp add: inj_on_def)
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lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
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by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
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lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
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by (unfold inj_on_def, blast)
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lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
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by (blast dest!: inj_onD)
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lemma comp_inj_on:
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     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
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by (simp add: comp_def inj_on_def)
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lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
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apply(simp add:inj_on_def image_def)
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apply blast
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done
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lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
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  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
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apply(unfold inj_on_def)
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apply blast
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done
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lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
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by (unfold inj_on_def, blast)
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lemma inj_singleton: "inj (%s. {s})"
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by (simp add: inj_on_def)
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lemma inj_on_empty[iff]: "inj_on f {}"
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by(simp add: inj_on_def)
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lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
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by (unfold inj_on_def, blast)
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lemma inj_on_Un:
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 "inj_on f (A Un B) =
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  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
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apply(unfold inj_on_def)
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apply (blast intro:sym)
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done
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lemma inj_on_insert[iff]:
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  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
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apply(unfold inj_on_def)
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apply (blast intro:sym)
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done
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lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
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apply(unfold inj_on_def)
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apply (blast)
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done
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lemma surjI: "(!! x. g(f x) = x) ==> surj g"
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apply (simp add: surj_def)
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apply (blast intro: sym)
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done
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lemma surj_range: "surj f ==> range f = UNIV"
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by (auto simp add: surj_def)
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lemma surjD: "surj f ==> EX x. y = f x"
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by (simp add: surj_def)
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lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> 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 surj_range_iff: "surj f \<longleftrightarrow> range f = UNIV"
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  unfolding expand_set_eq image_iff surj_def by auto
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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 surj_range_iff by auto
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lemma bij_eq_bij_betw: "bij f \<longleftrightarrow> bij_betw f UNIV UNIV"
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  unfolding bij_def surj_range_iff 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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lemma bij_is_surj: "bij f ==> surj f"
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by (simp add: bij_def)
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lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
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by (simp add: bij_betw_def)
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lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
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by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
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lemma bij_betw_trans:
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  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
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by(auto simp add:bij_betw_def comp_inj_on)
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lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
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proof -
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  have i: "inj_on f A" and s: "f ` A = B"
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    using assms by(auto simp:bij_betw_def)
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  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
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  { fix a b assume P: "?P b a"
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    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
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    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
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    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
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  } note g = this
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  have "inj_on ?g B"
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  proof(rule inj_onI)
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    fix x y assume "x:B" "y:B" "?g x = ?g y"
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    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
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    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
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    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
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  qed
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  moreover have "?g ` B = A"
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  proof(auto simp:image_def)
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    fix b assume "b:B"
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    with s obtain a where P: "?P b a" unfolding image_def by blast
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    thus "?g b \<in> A" using g[OF P] by auto
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  next
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    fix a assume "a:A"
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    then obtain b where P: "?P b a" using s unfolding image_def by blast
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    then have "b:B" using s unfolding image_def by blast
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    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
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  qed
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  ultimately show ?thesis by(auto simp:bij_betw_def)
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qed
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lemma bij_betw_combine:
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  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
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  shows "bij_betw f (A \<union> C) (B \<union> D)"
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  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
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lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
paulson@13585
   330
by (simp add: surj_range)
paulson@13585
   331
paulson@13585
   332
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585
   333
by (simp add: inj_on_def, blast)
paulson@13585
   334
paulson@13585
   335
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
paulson@13585
   336
apply (unfold surj_def)
paulson@13585
   337
apply (blast intro: sym)
paulson@13585
   338
done
paulson@13585
   339
paulson@13585
   340
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585
   341
by (unfold inj_on_def, blast)
paulson@13585
   342
paulson@13585
   343
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585
   344
apply (unfold bij_def)
paulson@13585
   345
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   346
done
paulson@13585
   347
nipkow@31438
   348
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
nipkow@31438
   349
by(blast dest: inj_onD)
nipkow@31438
   350
paulson@13585
   351
lemma inj_on_image_Int:
paulson@13585
   352
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   353
apply (simp add: inj_on_def, blast)
paulson@13585
   354
done
paulson@13585
   355
paulson@13585
   356
lemma inj_on_image_set_diff:
paulson@13585
   357
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
paulson@13585
   358
apply (simp add: inj_on_def, blast)
paulson@13585
   359
done
paulson@13585
   360
paulson@13585
   361
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   362
by (simp add: inj_on_def, blast)
paulson@13585
   363
paulson@13585
   364
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585
   365
by (simp add: inj_on_def, blast)
paulson@13585
   366
paulson@13585
   367
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
paulson@13585
   368
by (blast dest: injD)
paulson@13585
   369
paulson@13585
   370
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
paulson@13585
   371
by (simp add: inj_on_def, blast)
paulson@13585
   372
paulson@13585
   373
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
paulson@13585
   374
by (blast dest: injD)
paulson@13585
   375
paulson@13585
   376
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
paulson@13585
   377
lemma image_INT:
paulson@13585
   378
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
paulson@13585
   379
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585
   380
apply (simp add: inj_on_def, blast)
paulson@13585
   381
done
paulson@13585
   382
paulson@13585
   383
(*Compare with image_INT: no use of inj_on, and if f is surjective then
paulson@13585
   384
  it doesn't matter whether A is empty*)
paulson@13585
   385
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585
   386
apply (simp add: bij_def)
paulson@13585
   387
apply (simp add: inj_on_def surj_def, blast)
paulson@13585
   388
done
paulson@13585
   389
paulson@13585
   390
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
paulson@13585
   391
by (auto simp add: surj_def)
paulson@13585
   392
paulson@13585
   393
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585
   394
by (auto simp add: inj_on_def)
paulson@5852
   395
paulson@13585
   396
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585
   397
apply (simp add: bij_def)
paulson@13585
   398
apply (rule equalityI)
paulson@13585
   399
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585
   400
done
paulson@13585
   401
hoelzl@35584
   402
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   403
  by (auto intro!: inj_onI)
paulson@13585
   404
hoelzl@35584
   405
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   406
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   407
paulson@13585
   408
subsection{*Function Updating*}
paulson@13585
   409
haftmann@35416
   410
definition
haftmann@35416
   411
  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147
   412
  "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147
   413
haftmann@26147
   414
nonterminals
haftmann@26147
   415
  updbinds updbind
haftmann@26147
   416
syntax
haftmann@26147
   417
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147
   418
  ""         :: "updbind => updbinds"             ("_")
haftmann@26147
   419
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115
   420
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   421
haftmann@26147
   422
translations
wenzelm@35115
   423
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115
   424
  "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147
   425
haftmann@26147
   426
(* Hint: to define the sum of two functions (or maps), use sum_case.
haftmann@26147
   427
         A nice infix syntax could be defined (in Datatype.thy or below) by
wenzelm@35115
   428
notation
wenzelm@35115
   429
  sum_case  (infixr "'(+')"80)
haftmann@26147
   430
*)
haftmann@26147
   431
paulson@13585
   432
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585
   433
apply (simp add: fun_upd_def, safe)
paulson@13585
   434
apply (erule subst)
paulson@13585
   435
apply (rule_tac [2] ext, auto)
paulson@13585
   436
done
paulson@13585
   437
paulson@13585
   438
(* f x = y ==> f(x:=y) = f *)
paulson@13585
   439
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
paulson@13585
   440
paulson@13585
   441
(* f(x := f x) = f *)
paulson@17084
   442
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
paulson@17084
   443
declare fun_upd_triv [iff]
paulson@13585
   444
paulson@13585
   445
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084
   446
by (simp add: fun_upd_def)
paulson@13585
   447
paulson@13585
   448
(* fun_upd_apply supersedes these two,   but they are useful
paulson@13585
   449
   if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585
   450
lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585
   451
by simp
paulson@13585
   452
paulson@13585
   453
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585
   454
by simp
paulson@13585
   455
paulson@13585
   456
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
paulson@13585
   457
by (simp add: expand_fun_eq)
paulson@13585
   458
paulson@13585
   459
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585
   460
by (rule ext, auto)
paulson@13585
   461
nipkow@15303
   462
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
krauss@34209
   463
by (fastsimp simp:inj_on_def image_def)
nipkow@15303
   464
paulson@15510
   465
lemma fun_upd_image:
paulson@15510
   466
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510
   467
by auto
paulson@15510
   468
nipkow@31080
   469
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
krauss@34209
   470
by (auto intro: ext)
nipkow@31080
   471
haftmann@26147
   472
haftmann@26147
   473
subsection {* @{text override_on} *}
haftmann@26147
   474
haftmann@26147
   475
definition
haftmann@26147
   476
  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
haftmann@26147
   477
where
haftmann@26147
   478
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   479
nipkow@15691
   480
lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691
   481
by(simp add:override_on_def)
nipkow@13910
   482
nipkow@15691
   483
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691
   484
by(simp add:override_on_def)
nipkow@13910
   485
nipkow@15691
   486
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691
   487
by(simp add:override_on_def)
nipkow@13910
   488
haftmann@26147
   489
haftmann@26147
   490
subsection {* @{text swap} *}
paulson@15510
   491
haftmann@22744
   492
definition
haftmann@22744
   493
  swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@22744
   494
where
haftmann@22744
   495
  "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   496
huffman@34101
   497
lemma swap_self [simp]: "swap a a f = f"
nipkow@15691
   498
by (simp add: swap_def)
paulson@15510
   499
paulson@15510
   500
lemma swap_commute: "swap a b f = swap b a f"
paulson@15510
   501
by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510
   502
paulson@15510
   503
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
paulson@15510
   504
by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510
   505
huffman@34145
   506
lemma swap_triple:
huffman@34145
   507
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   508
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
huffman@34145
   509
  using assms by (simp add: expand_fun_eq swap_def)
huffman@34145
   510
huffman@34101
   511
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
huffman@34101
   512
by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101
   513
paulson@15510
   514
lemma inj_on_imp_inj_on_swap:
haftmann@22744
   515
  "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
paulson@15510
   516
by (simp add: inj_on_def swap_def, blast)
paulson@15510
   517
paulson@15510
   518
lemma inj_on_swap_iff [simp]:
paulson@15510
   519
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
hoelzl@39075
   520
proof
paulson@15510
   521
  assume "inj_on (swap a b f) A"
hoelzl@39075
   522
  with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075
   523
    by (iprover intro: inj_on_imp_inj_on_swap)
hoelzl@39075
   524
  thus "inj_on f A" by simp
paulson@15510
   525
next
paulson@15510
   526
  assume "inj_on f A"
krauss@34209
   527
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   528
qed
paulson@15510
   529
paulson@15510
   530
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
paulson@15510
   531
apply (simp add: surj_def swap_def, clarify)
wenzelm@27125
   532
apply (case_tac "y = f b", blast)
wenzelm@27125
   533
apply (case_tac "y = f a", auto)
paulson@15510
   534
done
paulson@15510
   535
paulson@15510
   536
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
hoelzl@39075
   537
proof
paulson@15510
   538
  assume "surj (swap a b f)"
hoelzl@39075
   539
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap)
hoelzl@39075
   540
  thus "surj f" by simp
paulson@15510
   541
next
paulson@15510
   542
  assume "surj f"
hoelzl@39075
   543
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap)
paulson@15510
   544
qed
paulson@15510
   545
paulson@15510
   546
lemma bij_swap_iff: "bij (swap a b f) = bij f"
paulson@15510
   547
by (simp add: bij_def)
haftmann@21547
   548
hoelzl@39075
   549
lemma bij_betw_swap:
hoelzl@39075
   550
  assumes "bij_betw f A B" "x \<in> A" "y \<in> A"
hoelzl@39075
   551
  shows "bij_betw (Fun.swap x y f) A B"
hoelzl@39075
   552
proof (unfold bij_betw_def, intro conjI)
hoelzl@39075
   553
  show "inj_on (Fun.swap x y f) A" using assms
hoelzl@39075
   554
    by (intro inj_on_imp_inj_on_swap) (auto simp: bij_betw_def)
hoelzl@39075
   555
  show "Fun.swap x y f ` A = B"
hoelzl@39075
   556
  proof safe
hoelzl@39075
   557
    fix z assume "z \<in> A"
hoelzl@39075
   558
    then show "Fun.swap x y f z \<in> B"
hoelzl@39075
   559
      using assms unfolding bij_betw_def
hoelzl@39075
   560
      by (auto simp: image_iff Fun.swap_def
hoelzl@39075
   561
               intro!: bexI[of _ "if z = x then y else if z = y then x else z"])
hoelzl@39075
   562
  next
hoelzl@39075
   563
    fix z assume "z \<in> B"
hoelzl@39075
   564
    then obtain v where "v \<in> A" "z = f v" using assms unfolding bij_betw_def by auto
hoelzl@39075
   565
    then show "z \<in> Fun.swap x y f ` A" unfolding image_iff
hoelzl@39075
   566
      using assms
hoelzl@39075
   567
      by (auto simp add: Fun.swap_def split: split_if_asm
hoelzl@39075
   568
               intro!: bexI[of _ "if v = x then y else if v = y then x else v"])
hoelzl@39075
   569
  qed
hoelzl@39075
   570
qed
hoelzl@39075
   571
wenzelm@36176
   572
hide_const (open) swap
haftmann@21547
   573
haftmann@31949
   574
haftmann@31949
   575
subsection {* Inversion of injective functions *}
haftmann@31949
   576
nipkow@33057
   577
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
nipkow@33057
   578
"the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961
   579
nipkow@33057
   580
lemma the_inv_into_f_f:
nipkow@33057
   581
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057
   582
apply (simp add: the_inv_into_def inj_on_def)
krauss@34209
   583
apply blast
nipkow@32961
   584
done
nipkow@32961
   585
nipkow@33057
   586
lemma f_the_inv_into_f:
nipkow@33057
   587
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057
   588
apply (simp add: the_inv_into_def)
nipkow@32961
   589
apply (rule the1I2)
nipkow@32961
   590
 apply(blast dest: inj_onD)
nipkow@32961
   591
apply blast
nipkow@32961
   592
done
nipkow@32961
   593
nipkow@33057
   594
lemma the_inv_into_into:
nipkow@33057
   595
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057
   596
apply (simp add: the_inv_into_def)
nipkow@32961
   597
apply (rule the1I2)
nipkow@32961
   598
 apply(blast dest: inj_onD)
nipkow@32961
   599
apply blast
nipkow@32961
   600
done
nipkow@32961
   601
nipkow@33057
   602
lemma the_inv_into_onto[simp]:
nipkow@33057
   603
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057
   604
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961
   605
nipkow@33057
   606
lemma the_inv_into_f_eq:
nipkow@33057
   607
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961
   608
  apply (erule subst)
nipkow@33057
   609
  apply (erule the_inv_into_f_f, assumption)
nipkow@32961
   610
  done
nipkow@32961
   611
nipkow@33057
   612
lemma the_inv_into_comp:
nipkow@32961
   613
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   614
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057
   615
apply (rule the_inv_into_f_eq)
nipkow@32961
   616
  apply (fast intro: comp_inj_on)
nipkow@33057
   617
 apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057
   618
apply (simp add: the_inv_into_into)
nipkow@32961
   619
done
nipkow@32961
   620
nipkow@33057
   621
lemma inj_on_the_inv_into:
nipkow@33057
   622
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
nipkow@33057
   623
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
nipkow@32961
   624
nipkow@33057
   625
lemma bij_betw_the_inv_into:
nipkow@33057
   626
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057
   627
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   628
berghofe@32998
   629
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057
   630
  "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   631
berghofe@32998
   632
lemma the_inv_f_f:
berghofe@32998
   633
  assumes "inj f"
berghofe@32998
   634
  shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057
   635
  by (rule the_inv_into_f_f)
berghofe@32998
   636
haftmann@31949
   637
haftmann@22845
   638
subsection {* Proof tool setup *} 
haftmann@22845
   639
haftmann@22845
   640
text {* simplifies terms of the form
haftmann@22845
   641
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
haftmann@22845
   642
wenzelm@24017
   643
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
haftmann@22845
   644
let
haftmann@22845
   645
  fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017
   646
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845
   647
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845
   648
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845
   649
    let
haftmann@22845
   650
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845
   651
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845
   652
        | find t = NONE
haftmann@22845
   653
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   654
wenzelm@24017
   655
  fun proc ss ct =
wenzelm@24017
   656
    let
wenzelm@24017
   657
      val ctxt = Simplifier.the_context ss
wenzelm@24017
   658
      val t = Thm.term_of ct
wenzelm@24017
   659
    in
wenzelm@24017
   660
      case find_double t of
wenzelm@24017
   661
        (T, NONE) => NONE
wenzelm@24017
   662
      | (T, SOME rhs) =>
wenzelm@27330
   663
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017
   664
            (fn _ =>
wenzelm@24017
   665
              rtac eq_reflection 1 THEN
wenzelm@24017
   666
              rtac ext 1 THEN
wenzelm@24017
   667
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
wenzelm@24017
   668
    end
wenzelm@24017
   669
in proc end
haftmann@22845
   670
*}
haftmann@22845
   671
haftmann@22845
   672
haftmann@21870
   673
subsection {* Code generator setup *}
haftmann@21870
   674
berghofe@25886
   675
types_code
berghofe@25886
   676
  "fun"  ("(_ ->/ _)")
berghofe@25886
   677
attach (term_of) {*
berghofe@25886
   678
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
berghofe@25886
   679
*}
berghofe@25886
   680
attach (test) {*
berghofe@25886
   681
fun gen_fun_type aF aT bG bT i =
berghofe@25886
   682
  let
wenzelm@32740
   683
    val tab = Unsynchronized.ref [];
berghofe@25886
   684
    fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
berghofe@25886
   685
      (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
berghofe@25886
   686
  in
berghofe@25886
   687
    (fn x =>
berghofe@25886
   688
       case AList.lookup op = (!tab) x of
berghofe@25886
   689
         NONE =>
berghofe@25886
   690
           let val p as (y, _) = bG i
berghofe@25886
   691
           in (tab := (x, p) :: !tab; y) end
berghofe@25886
   692
       | SOME (y, _) => y,
berghofe@28711
   693
     fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
berghofe@25886
   694
  end;
berghofe@25886
   695
*}
berghofe@25886
   696
haftmann@21870
   697
code_const "op \<circ>"
haftmann@21870
   698
  (SML infixl 5 "o")
haftmann@21870
   699
  (Haskell infixr 9 ".")
haftmann@21870
   700
haftmann@21906
   701
code_const "id"
haftmann@21906
   702
  (Haskell "id")
haftmann@21906
   703
nipkow@2912
   704
end