(*  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"

 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 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:

   "\<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