(*  Title:      HOL/Fun.thy

    Author:     Tobias Nipkow, Cambridge University Computer Laboratory

    Copyright   1994  University of Cambridge

*)

header {* Notions about functions *}

theory Fun

imports Complete_Lattice

uses ("Tools/enriched_type.ML")

begin

text{*As a simplification rule, it replaces all function equalities by

  first-order equalities.*}

lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"

apply (rule iffI)

apply (simp (no_asm_simp))

apply (rule ext)

apply (simp (no_asm_simp))

done

lemma apply_inverse:

  "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"

  by auto

subsection {* The Identity Function @{text id} *}

definition

  id :: "'a \<Rightarrow> 'a"

where

  "id = (\<lambda>x. x)"

lemma id_apply [simp]: "id x = x"

  by (simp add: id_def)

lemma image_ident [simp]: "(%x. x) ` Y = Y"

by blast

lemma image_id [simp]: "id ` Y = Y"

by (simp add: id_def)

lemma vimage_ident [simp]: "(%x. x) -` Y = Y"

by blast

lemma vimage_id [simp]: "id -` A = A"

by (simp add: id_def)

subsection {* The Composition Operator @{text "f \<circ> g"} *}

definition

  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)

where

  "f o g = (\<lambda>x. f (g x))"

notation (xsymbols)

  comp  (infixl "\<circ>" 55)

notation (HTML output)

  comp  (infixl "\<circ>" 55)

text{*compatibility*}

lemmas o_def = comp_def

lemma o_apply [simp]: "(f o g) x = f (g x)"

by (simp add: comp_def)

lemma o_assoc: "f o (g o h) = f o g o h"

by (simp add: comp_def)

lemma id_o [simp]: "id o g = g"

by (simp add: comp_def)

lemma o_id [simp]: "f o id = f"

by (simp add: comp_def)

lemma o_eq_dest:

  "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"

  by (simp only: o_def) (fact fun_cong)

lemma o_eq_elim:

  "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"

  by (erule meta_mp) (fact o_eq_dest) 

lemma image_compose: "(f o g) ` r = f`(g`r)"

by (simp add: comp_def, blast)

lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"

  by auto

lemma UN_o: "UNION A (g o f) = UNION (f`A) g"

by (unfold comp_def, blast)

subsection {* The Forward Composition Operator @{text fcomp} *}

definition

  fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60)

where

  "f \<circ>> g = (\<lambda>x. g (f x))"

lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"

  by (simp add: fcomp_def)

lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"

  by (simp add: fcomp_def)

lemma id_fcomp [simp]: "id \<circ>> g = g"

  by (simp add: fcomp_def)

lemma fcomp_id [simp]: "f \<circ>> id = f"

  by (simp add: fcomp_def)

code_const fcomp

  (Eval infixl 1 "#>")

no_notation fcomp (infixl "\<circ>>" 60)

subsection {* Mapping functions *}

definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where

  "map_fun f g h = g \<circ> h \<circ> f"

lemma map_fun_apply [simp]:

  "map_fun f g h x = g (h (f x))"

  by (simp add: map_fun_def)

subsection {* Injectivity and Bijectivity *}

definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective"

  "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"

definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"

  "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"

text{*A common special case: functions injective, surjective or bijective over

the entire domain type.*}

abbreviation

  "inj f \<equiv> inj_on f UNIV"

abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective"

  "surj f \<equiv> (range f = UNIV)"

abbreviation

  "bij f \<equiv> bij_betw f UNIV UNIV"

text{* The negated case: *}

translations

"\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"

lemma injI:

  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"

  shows "inj f"

  using assms unfolding inj_on_def by auto

theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"

  by (unfold inj_on_def, blast)

lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"

by (simp add: inj_on_def)

lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"

by (force simp add: inj_on_def)

lemma inj_on_cong:

  "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"

unfolding inj_on_def by auto

lemma inj_on_strict_subset:

  "\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B"

unfolding inj_on_def unfolding image_def by blast

lemma inj_comp:

  "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"

  by (simp add: inj_on_def)

lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"

  by (simp add: inj_on_def fun_eq_iff)

lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"

by (simp add: inj_on_eq_iff)

lemma inj_on_id[simp]: "inj_on id A"

  by (simp add: inj_on_def)

lemma inj_on_id2[simp]: "inj_on (%x. x) A"

by (simp add: inj_on_def)

lemma inj_on_Int: "\<lbrakk>inj_on f A; inj_on f B\<rbrakk> \<Longrightarrow> inj_on f (A \<inter> B)"

unfolding inj_on_def by blast

lemma inj_on_INTER:

  "\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)\<rbrakk> \<Longrightarrow> inj_on f (\<Inter> i \<in> I. A i)"

unfolding inj_on_def by blast

lemma inj_on_Inter:

  "\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)"

unfolding inj_on_def by blast

lemma inj_on_UNION_chain:

  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and

         INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"

  shows "inj_on f (\<Union> i \<in> I. A i)"

proof(unfold inj_on_def UNION_def, auto)

  fix i j x y

  assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"

         and ***: "f x = f y"

  show "x = y"

  proof-

    {assume "A i \<le> A j"

     with ** have "x \<in> A j" by auto

     with INJ * ** *** have ?thesis

     by(auto simp add: inj_on_def)

    }

    moreover

    {assume "A j \<le> A i"

     with ** have "y \<in> A i" by auto

     with INJ * ** *** have ?thesis

     by(auto simp add: inj_on_def)

    }

    ultimately show ?thesis using  CH * by blast

  qed

qed

lemma surj_id: "surj id"

by simp

lemma bij_id[simp]: "bij id"

by (simp add: bij_betw_def)

lemma inj_onI:

    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"

by (simp add: inj_on_def)

lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"

by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)

lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"

by (unfold inj_on_def, blast)

lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"

by (blast dest!: inj_onD)

lemma comp_inj_on:

     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"

by (simp add: comp_def inj_on_def)

lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"

apply(simp add:inj_on_def image_def)

apply blast

done

lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);

  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"

apply(unfold inj_on_def)

apply blast

done

lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"

by (unfold inj_on_def, blast)

lemma inj_singleton: "inj (%s. {s})"

by (simp add: inj_on_def)

lemma inj_on_empty[iff]: "inj_on f {}"

by(simp add: inj_on_def)

lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"

by (unfold inj_on_def, blast)

lemma inj_on_Un:

 "inj_on f (A Un B) =

  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"

apply(unfold inj_on_def)

apply (blast intro:sym)

done

lemma inj_on_insert[iff]:

  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"

apply(unfold inj_on_def)

apply (blast intro:sym)

done

lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"

apply(unfold inj_on_def)

apply (blast)

done

lemma comp_inj_on_iff:

  "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"

by(auto simp add: comp_inj_on inj_on_def)

lemma inj_on_imageI2:

  "inj_on (f' o f) A \<Longrightarrow> inj_on f A"

by(auto simp add: comp_inj_on inj_on_def)

lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"

  by auto

lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"

  using *[symmetric] by auto

lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"

  by (simp add: surj_def)

lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"

  by (simp add: surj_def, blast)

lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"

apply (simp add: comp_def surj_def, clarify)

apply (drule_tac x = y in spec, clarify)

apply (drule_tac x = x in spec, blast)

done

lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"

  unfolding bij_betw_def by auto

lemma bij_betw_empty1:

  assumes "bij_betw f {} A"

  shows "A = {}"

using assms unfolding bij_betw_def by blast

lemma bij_betw_empty2:

  assumes "bij_betw f A {}"

  shows "A = {}"

using assms unfolding bij_betw_def by blast

lemma inj_on_imp_bij_betw:

  "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"

unfolding bij_betw_def by simp

lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"

  unfolding bij_betw_def ..

lemma bijI: "[| inj f; surj f |] ==> bij f"

by (simp add: bij_def)

lemma bij_is_inj: "bij f ==> inj f"

by (simp add: bij_def)

lemma bij_is_surj: "bij f ==> surj f"

by (simp add: bij_def)

lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"

by (simp add: bij_betw_def)

lemma bij_betw_trans:

  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"

by(auto simp add:bij_betw_def comp_inj_on)

lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"

  by (rule bij_betw_trans)

lemma bij_betw_comp_iff:

  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"

by(auto simp add: bij_betw_def inj_on_def)

lemma bij_betw_comp_iff2:

  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"

  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"

using assms

proof(auto simp add: bij_betw_comp_iff)

  assume *: "bij_betw (f' \<circ> f) A A''"

  thus "bij_betw f A A'"

  using IM

  proof(auto simp add: bij_betw_def)

    assume "inj_on (f' \<circ> f) A"

    thus "inj_on f A" using inj_on_imageI2 by blast

  next

    fix a' assume **: "a' \<in> A'"

    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto

    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *

    unfolding bij_betw_def by force

    hence "f a \<in> A'" using IM by auto

    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto

    thus "a' \<in> f ` A" using 1 by auto

  qed

qed

lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"

proof -

  have i: "inj_on f A" and s: "f ` A = B"

    using assms by(auto simp:bij_betw_def)

  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"

  { fix a b assume P: "?P b a"

    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast

    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])

    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp

  } note g = this

  have "inj_on ?g B"

  proof(rule inj_onI)

    fix x y assume "x:B" "y:B" "?g x = ?g y"

    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast

    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast

    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp

  qed

  moreover have "?g ` B = A"

  proof(auto simp:image_def)

    fix b assume "b:B"

    with s obtain a where P: "?P b a" unfolding image_def by blast

    thus "?g b \<in> A" using g[OF P] by auto

  next

    fix a assume "a:A"

    then obtain b where P: "?P b a" using s unfolding image_def by blast

    then have "b:B" using s unfolding image_def by blast

    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast

  qed

  ultimately show ?thesis by(auto simp:bij_betw_def)

qed

lemma bij_betw_cong:

  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"

unfolding bij_betw_def inj_on_def by force

lemma bij_betw_id[intro, simp]:

  "bij_betw id A A"

unfolding bij_betw_def id_def by auto

lemma bij_betw_id_iff:

  "bij_betw id A B \<longleftrightarrow> A = B"

by(auto simp add: bij_betw_def)

lemma bij_betw_combine:

  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"

  shows "bij_betw f (A \<union> C) (B \<union> D)"

  using assms unfolding bij_betw_def inj_on_Un image_Un by auto

lemma bij_betw_UNION_chain:

  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and

         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"

  shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"

proof(unfold bij_betw_def, auto simp add: image_def)

  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"

  using BIJ bij_betw_def[of f] by auto

  thus "inj_on f (\<Union> i \<in> I. A i)"

  using CH inj_on_UNION_chain[of I A f] by auto

next

  fix i x

  assume *: "i \<in> I" "x \<in> A i"

  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto

  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast

next

  fix i x'

  assume *: "i \<in> I" "x' \<in> A' i"

  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast

  thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"

  using * by blast

qed

lemma bij_betw_Disj_Un:

  assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and

          B1: "bij_betw f A A'" and B2: "bij_betw f B B'"

  shows "bij_betw f (A \<union> B) (A' \<union> B')"

proof-

  have 1: "inj_on f A \<and> inj_on f B"

  using B1 B2 by (auto simp add: bij_betw_def)

  have 2: "f`A = A' \<and> f`B = B'"

  using B1 B2 by (auto simp add: bij_betw_def)

  hence "f`(A - B) \<inter> f`(B - A) = {}"

  using DISJ DISJ' by blast

  hence "inj_on f (A \<union> B)"

  using 1 by (auto simp add: inj_on_Un)

  (*  *)

  moreover

  have "f`(A \<union> B) = A' \<union> B'"

  using 2 by auto

  ultimately show ?thesis

  unfolding bij_betw_def by auto

qed

lemma bij_betw_subset:

  assumes BIJ: "bij_betw f A A'" and

          SUB: "B \<le> A" and IM: "f ` B = B'"

  shows "bij_betw f B B'"

using assms

by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)

lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"

by simp

lemma surj_vimage_empty:

  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"

  using surj_image_vimage_eq[OF `surj f`, of A]

  by (intro iffI) fastsimp+

lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"

by (simp add: inj_on_def, blast)

lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"

by (blast intro: sym)

lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"

by (unfold inj_on_def, blast)

lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"

apply (unfold bij_def)

apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)

done

lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"

by(blast dest: inj_onD)

lemma inj_on_image_Int:

   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"

apply (simp add: inj_on_def, blast)

done

lemma inj_on_image_set_diff:

   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"

apply (simp add: inj_on_def, blast)

done

lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"

by (simp add: inj_on_def, blast)

lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"

by (simp add: inj_on_def, blast)

lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"

by (blast dest: injD)

lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"

by (simp add: inj_on_def, blast)

lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"

by (blast dest: injD)

(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)

lemma image_INT:

   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]

    ==> f ` (INTER A B) = (INT x:A. f ` B x)"

apply (simp add: inj_on_def, blast)

done

(*Compare with image_INT: no use of inj_on, and if f is surjective then

  it doesn't matter whether A is empty*)

lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"

apply (simp add: bij_def)

apply (simp add: inj_on_def surj_def, blast)

done

lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"

by auto

lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"

by (auto simp add: inj_on_def)

lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"

apply (simp add: bij_def)

apply (rule equalityI)

apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)

done

lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"

  -- {* The inverse image of a singleton under an injective function

         is included in a singleton. *}

  apply (auto simp add: inj_on_def)

  apply (blast intro: the_equality [symmetric])

  done

lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"

  by (auto intro!: inj_onI)

lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"

  by (auto intro!: inj_onI dest: strict_mono_eq)

subsection{*Function Updating*}

definition

  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where

  "fun_upd f a b == % x. if x=a then b else f x"

nonterminal updbinds and updbind

syntax

  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")

  ""         :: "updbind => updbinds"             ("_")

  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")

  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)

translations

  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"

  "f(x:=y)" == "CONST fun_upd f x y"

(* Hint: to define the sum of two functions (or maps), use sum_case.

         A nice infix syntax could be defined (in Datatype.thy or below) by

notation

  sum_case  (infixr "'(+')"80)

*)

lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"

apply (simp add: fun_upd_def, safe)

apply (erule subst)

apply (rule_tac [2] ext, auto)

done

(* f x = y ==> f(x:=y) = f *)

lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]

(* f(x := f x) = f *)

lemmas fun_upd_triv = refl [THEN fun_upd_idem]

declare fun_upd_triv [iff]

lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"

by (simp add: fun_upd_def)

(* fun_upd_apply supersedes these two,   but they are useful

   if fun_upd_apply is intentionally removed from the simpset *)

lemma fun_upd_same: "(f(x:=y)) x = y"

by simp

lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"

by simp

lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"

by (simp add: fun_eq_iff)

lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"

by (rule ext, auto)

lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"

by (fastsimp simp:inj_on_def image_def)

lemma fun_upd_image:

     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"

by auto

lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"

by (auto intro: ext)

subsection {* @{text override_on} *}

definition

  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"

where

  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"

lemma override_on_emptyset[simp]: "override_on f g {} = f"

by(simp add:override_on_def)

lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"

by(simp add:override_on_def)

lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"

by(simp add:override_on_def)

subsection {* @{text swap} *}

definition

  swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"

where

  "swap a b f = f (a := f b, b:= f a)"

lemma swap_self [simp]: "swap a a f = f"

by (simp add: swap_def)

lemma swap_commute: "swap a b f = swap b a f"

by (rule ext, simp add: fun_upd_def swap_def)

lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"

by (rule ext, simp add: fun_upd_def swap_def)

lemma swap_triple:

  assumes "a \<noteq> c" and "b \<noteq> c"

  shows "swap a b (swap b c (swap a b f)) = swap a c f"

  using assms by (simp add: fun_eq_iff swap_def)

lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"

by (rule ext, simp add: fun_upd_def swap_def)

lemma swap_image_eq [simp]:

  assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"

proof -

  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"

    using assms by (auto simp: image_iff swap_def)

  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .

  with subset[of f] show ?thesis by auto

qed

lemma inj_on_imp_inj_on_swap:

  "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"

  by (simp add: inj_on_def swap_def, blast)

lemma inj_on_swap_iff [simp]:

  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"

proof

  assume "inj_on (swap a b f) A"

  with A have "inj_on (swap a b (swap a b f)) A"

    by (iprover intro: inj_on_imp_inj_on_swap)

  thus "inj_on f A" by simp

next

  assume "inj_on f A"

  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)

qed

lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"

  by simp

lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"

  by simp

lemma bij_betw_swap_iff [simp]:

  "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"

  by (auto simp: bij_betw_def)

lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"

  by simp

hide_const (open) swap

subsection {* Inversion of injective functions *}

definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where

"the_inv_into A f == %x. THE y. y : A & f y = x"

lemma the_inv_into_f_f:

  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"

apply (simp add: the_inv_into_def inj_on_def)

apply blast

done

lemma f_the_inv_into_f:

  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"

apply (simp add: the_inv_into_def)

apply (rule the1I2)

 apply(blast dest: inj_onD)

apply blast

done

lemma the_inv_into_into:

  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"

apply (simp add: the_inv_into_def)

apply (rule the1I2)

 apply(blast dest: inj_onD)

apply blast

done

lemma the_inv_into_onto[simp]:

  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"

by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])

lemma the_inv_into_f_eq:

  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"

  apply (erule subst)

  apply (erule the_inv_into_f_f, assumption)

  done

lemma the_inv_into_comp:

  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>

  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"

apply (rule the_inv_into_f_eq)

  apply (fast intro: comp_inj_on)

 apply (simp add: f_the_inv_into_f the_inv_into_into)

apply (simp add: the_inv_into_into)

done

lemma inj_on_the_inv_into:

  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"

by (auto intro: inj_onI simp: image_def the_inv_into_f_f)

lemma bij_betw_the_inv_into:

  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"

by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)

abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where

  "the_inv f \<equiv> the_inv_into UNIV f"

lemma the_inv_f_f:

  assumes "inj f"

  shows "the_inv f (f x) = x" using assms UNIV_I

  by (rule the_inv_into_f_f)

subsection {* Cantor's Paradox *}

lemma Cantors_paradox [no_atp]:

  "\<not>(\<exists>f. f ` A = Pow A)"

proof clarify

  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast

  let ?X = "{a \<in> A. a \<notin> f a}"

  have "?X \<in> Pow A" unfolding Pow_def by auto

  with * obtain x where "x \<in> A \<and> f x = ?X" by blast

  thus False by best

qed

subsection {* Setup *} 

subsubsection {* Proof tools *}

text {* simplifies terms of the form

  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}

simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>

let

  fun gen_fun_upd NONE T _ _ = NONE

    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)

  fun dest_fun_T1 (Type (_, T :: Ts)) = T

  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =

    let

      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =

            if v aconv x then SOME g else gen_fun_upd (find g) T v w

        | find t = NONE

    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end

  fun proc ss ct =

    let

      val ctxt = Simplifier.the_context ss

      val t = Thm.term_of ct

    in

      case find_double t of

        (T, NONE) => NONE

      | (T, SOME rhs) =>

          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))

            (fn _ =>

              rtac eq_reflection 1 THEN

              rtac ext 1 THEN

              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))

    end

in proc end

*}

subsubsection {* Code generator *}

types_code

  "fun"  ("(_ ->/ _)")

attach (term_of) {*

fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);

*}

attach (test) {*

fun gen_fun_type aF aT bG bT i =

  let

    val tab = Unsynchronized.ref [];

    fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",

      (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()

  in

    (fn x =>

       case AList.lookup op = (!tab) x of

         NONE =>

           let val p as (y, _) = bG i

           in (tab := (x, p) :: !tab; y) end

       | SOME (y, _) => y,

     fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))

  end;

*}

code_const "op \<circ>"

  (SML infixl 5 "o")

  (Haskell infixr 9 ".")

code_const "id"

  (Haskell "id")

subsubsection {* Functorial structure of types *}

use "Tools/enriched_type.ML"

end