# Theory Fun

Up to index of Isabelle/HOL-Proofs

theory Fun
imports Complete_Lattices
`(*  Title:      HOL/Fun.thy    Author:     Tobias Nipkow, Cambridge University Computer Laboratory    Copyright   1994  University of Cambridge*)header {* Notions about functions *}theory Funimports Complete_Latticeskeywords "enriched_type" :: thy_goalbeginlemma apply_inverse:  "f x = u ==> (!!x. P x ==> g (f x) = x) ==> P x ==> x = g u"  by autosubsection {* The Identity Function @{text id} *}definition id :: "'a => 'a" where  "id = (λx. x)"lemma id_apply [simp]: "id x = x"  by (simp add: id_def)lemma image_id [simp]: "image id = id"  by (simp add: id_def fun_eq_iff)lemma vimage_id [simp]: "vimage id = id"  by (simp add: id_def fun_eq_iff)subsection {* The Composition Operator @{text "f o g"} *}definition comp :: "('b => 'c) => ('a => 'b) => 'a => 'c" (infixl "o" 55) where  "f o g = (λx. f (g x))"notation (xsymbols)  comp  (infixl "o" 55)notation (HTML output)  comp  (infixl "o" 55)lemma comp_apply [simp]: "(f o g) x = f (g x)"  by (simp add: comp_def)lemma comp_assoc: "(f o g) o h = f o (g o h)"  by (simp add: fun_eq_iff)lemma id_comp [simp]: "id o g = g"  by (simp add: fun_eq_iff)lemma comp_id [simp]: "f o id = f"  by (simp add: fun_eq_iff)lemma comp_eq_dest:  "a o b = c o d ==> a (b v) = c (d v)"  by (simp add: fun_eq_iff)lemma comp_eq_elim:  "a o b = c o d ==> ((!!v. a (b v) = c (d v)) ==> R) ==> R"  by (simp add: fun_eq_iff) lemma image_comp:  "(f o g) ` r = f ` (g ` r)"  by autolemma vimage_comp:  "(g o f) -` x = f -` (g -` x)"  by autolemma INF_comp:  "INFI A (g o f) = INFI (f ` A) g"  by (simp add: INF_def image_comp)lemma SUP_comp:  "SUPR A (g o f) = SUPR (f ` A) g"  by (simp add: SUP_def image_comp)subsection {* The Forward Composition Operator @{text fcomp} *}definition fcomp :: "('a => 'b) => ('b => 'c) => 'a => 'c" (infixl "o>" 60) where  "f o> g = (λx. g (f x))"lemma fcomp_apply [simp]:  "(f o> g) x = g (f x)"  by (simp add: fcomp_def)lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"  by (simp add: fcomp_def)lemma id_fcomp [simp]: "id o> g = g"  by (simp add: fcomp_def)lemma fcomp_id [simp]: "f o> id = f"  by (simp add: fcomp_def)code_const fcomp  (Eval infixl 1 "#>")no_notation fcomp (infixl "o>" 60)subsection {* Mapping functions *}definition map_fun :: "('c => 'a) => ('b => 'd) => ('a => 'b) => 'c => 'd" where  "map_fun f g h = g o h o 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 => 'b) => 'a set => bool" where -- "injective"  "inj_on f A <-> (∀x∈A. ∀y∈A. f x = f y --> x = y)"definition bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"  "bij_betw f A B <-> inj_on f A ∧ f ` A = B"text{*A common special case: functions injective, surjective or bijective overthe entire domain type.*}abbreviation  "inj f ≡ inj_on f UNIV"abbreviation surj :: "('a => 'b) => bool" where -- "surjective"  "surj f ≡ (range f = UNIV)"abbreviation  "bij f ≡ bij_betw f UNIV UNIV"text{* The negated case: *}translations"¬ CONST surj f" <= "CONST range f ≠ CONST UNIV"lemma injI:  assumes "!!x y. f x = f y ==> x = y"  shows "inj f"  using assms unfolding inj_on_def by autotheorem range_ex1_eq: "inj f ==> 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:  "(!! a. a : A ==> f a = g a) ==> inj_on f A = inj_on g A"unfolding inj_on_def by autolemma inj_on_strict_subset:  "[| inj_on f B; A < B |] ==> f`A < f`B"unfolding inj_on_def unfolding image_def by blastlemma inj_comp:  "inj f ==> inj g ==> inj (f o g)"  by (simp add: inj_on_def)lemma inj_fun: "inj f ==> inj (λx 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: "inj_on f A ∨ inj_on f B ==> inj_on f (A ∩ B)"unfolding inj_on_def by blastlemma inj_on_INTER:  "[|I ≠ {}; !! i. i ∈ I ==> inj_on f (A i)|] ==> inj_on f (\<Inter> i ∈ I. A i)"unfolding inj_on_def by blastlemma inj_on_Inter:  "[|S ≠ {}; !! A. A ∈ S ==> inj_on f A|] ==> inj_on f (Inter S)"unfolding inj_on_def by blastlemma inj_on_UNION_chain:  assumes CH: "!! i j. [|i ∈ I; j ∈ I|] ==> A i ≤ A j ∨ A j ≤ A i" and         INJ: "!! i. i ∈ I ==> inj_on f (A i)"  shows "inj_on f (\<Union> i ∈ I. A i)"proof -  {    fix i j x y    assume *: "i ∈ I" "j ∈ I" and **: "x ∈ A i" "y ∈ A j"      and ***: "f x = f y"    have "x = y"    proof -      {        assume "A i ≤ A j"        with ** have "x ∈ A j" by auto        with INJ * ** *** have ?thesis        by(auto simp add: inj_on_def)      }      moreover      {        assume "A j ≤ A i"        with ** have "y ∈ A i" by auto        with INJ * ** *** have ?thesis        by(auto simp add: inj_on_def)      }      ultimately show ?thesis using CH * by blast    qed  }  then show ?thesis by (unfold inj_on_def UNION_eq) autoqedlemma surj_id: "surj id"by simplemma 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 ==> inj_on g (f ` A)"apply(simp add:inj_on_def image_def)apply blastdonelemma inj_on_image_iff: "[| ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);  inj_on f A |] ==> inj_on g (f ` A) = inj_on g A"apply(unfold inj_on_def)apply blastdonelemma 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)donelemma 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)donelemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"apply(unfold inj_on_def)apply (blast)donelemma comp_inj_on_iff:  "inj_on f A ==> inj_on f' (f ` A) <-> 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 ==> inj_on f A"by(auto simp add: comp_inj_on inj_on_def)lemma surj_def: "surj f <-> (∀y. ∃x. y = f x)"  by autolemma surjI: assumes *: "!! x. g (f x) = x" shows "surj g"  using *[symmetric] by autolemma surjD: "surj f ==> ∃x. y = f x"  by (simp add: surj_def)lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> 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)donelemma bij_betw_imp_surj: "bij_betw f A UNIV ==> surj f"  unfolding bij_betw_def by autolemma bij_betw_empty1:  assumes "bij_betw f {} A"  shows "A = {}"using assms unfolding bij_betw_def by blastlemma bij_betw_empty2:  assumes "bij_betw f A {}"  shows "A = {}"using assms unfolding bij_betw_def by blastlemma inj_on_imp_bij_betw:  "inj_on f A ==> bij_betw f A (f ` A)"unfolding bij_betw_def by simplemma bij_def: "bij f <-> inj f ∧ 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 ==> inj_on f A"by (simp add: bij_betw_def)lemma bij_betw_trans:  "bij_betw f A B ==> bij_betw g B C ==> bij_betw (g o f) A C"by(auto simp add:bij_betw_def comp_inj_on)lemma bij_comp: "bij f ==> bij g ==> bij (g o f)"  by (rule bij_betw_trans)lemma bij_betw_comp_iff:  "bij_betw f A A' ==> bij_betw f' A' A'' <-> 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 ≤ A'"  shows "bij_betw f A A' <-> bij_betw (f' o f) A A''"using assmsproof(auto simp add: bij_betw_comp_iff)  assume *: "bij_betw (f' o f) A A''"  thus "bij_betw f A A'"  using IM  proof(auto simp add: bij_betw_def)    assume "inj_on (f' o f) A"    thus "inj_on f A" using inj_on_imageI2 by blast  next    fix a' assume **: "a' ∈ A'"    hence "f' a' ∈ A''" using BIJ unfolding bij_betw_def by auto    then obtain a where 1: "a ∈ A ∧ f'(f a) = f' a'" using *    unfolding bij_betw_def by force    hence "f a ∈ A'" using IM by auto    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto    thus "a' ∈ f ` A" using 1 by auto  qedqedlemma 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 ∧ f a = b" let ?g = "%b. The (?P b)"  { fix a b assume P: "?P b a"    hence ex1: "∃a. ?P b a" using s unfolding image_def by blast    hence uex1: "∃!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 ∈ 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 "∃b∈B. a = ?g b" by blast  qed  ultimately show ?thesis by(auto simp:bij_betw_def)qedlemma bij_betw_cong:  "(!! a. a ∈ A ==> f a = g a) ==> bij_betw f A A' = bij_betw g A A'"unfolding bij_betw_def inj_on_def by forcelemma bij_betw_id[intro, simp]:  "bij_betw id A A"unfolding bij_betw_def id_def by autolemma bij_betw_id_iff:  "bij_betw id A B <-> 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 ∩ D = {}"  shows "bij_betw f (A ∪ C) (B ∪ D)"  using assms unfolding bij_betw_def inj_on_Un image_Un by autolemma bij_betw_UNION_chain:  assumes CH: "!! i j. [|i ∈ I; j ∈ I|] ==> A i ≤ A j ∨ A j ≤ A i" and         BIJ: "!! i. i ∈ I ==> bij_betw f (A i) (A' i)"  shows "bij_betw f (\<Union> i ∈ I. A i) (\<Union> i ∈ I. A' i)"proof (unfold bij_betw_def, auto)  have "!! i. i ∈ I ==> inj_on f (A i)"  using BIJ bij_betw_def[of f] by auto  thus "inj_on f (\<Union> i ∈ I. A i)"  using CH inj_on_UNION_chain[of I A f] by autonext  fix i x  assume *: "i ∈ I" "x ∈ A i"  hence "f x ∈ A' i" using BIJ bij_betw_def[of f] by auto  thus "∃j ∈ I. f x ∈ A' j" using * by blastnext  fix i x'  assume *: "i ∈ I" "x' ∈ A' i"  hence "∃x ∈ A i. x' = f x" using BIJ bij_betw_def[of f] by blast  then have "∃j ∈ I. ∃x ∈ A j. x' = f x"    using * by blast  then show "x' ∈ f ` (\<Union>x∈I. A x)" by (simp add: image_def)qedlemma bij_betw_subset:  assumes BIJ: "bij_betw f A A'" and          SUB: "B ≤ A" and IM: "f ` B = B'"  shows "bij_betw f B B'"using assmsby(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 simplemma surj_vimage_empty:  assumes "surj f" shows "f -` A = {} <-> A = {}"  using surj_image_vimage_eq[OF `surj f`, of A]  by (intro iffI) fastforce+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)donelemma inj_on_Un_image_eq_iff: "inj_on f (A ∪ B) ==> f ` A = f ` B <-> 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)donelemma 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)donelemma 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)donelemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"by autolemma 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)donelemma inj_vimage_singleton: "inj f ==> f -` {a} ⊆ {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])  donelemma inj_on_vimage_singleton:  "inj_on f A ==> f -` {a} ∩ A ⊆ {THE x. x ∈ A ∧ f x = a}"  by (auto simp add: inj_on_def intro: the_equality [symmetric])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 ==> 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 updbindsyntax  "_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) bynotation  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)donelemma fun_upd_idem: "f x = y ==> f(x:=y) = f"  by (simp only: fun_upd_idem_iff)lemma fun_upd_triv [iff]: "f(x := f x) = f"  by (simp only: fun_upd_idem)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 simplemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"by simplemma 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: "[| inj_on f A; y ∉ f`A |] ==> inj_on (f(x:=y)) A"by (fastforce simp:inj_on_def image_def)lemma fun_upd_image:     "f(x:=y) ` A = (if x ∈ A then insert y (f ` (A-{x})) else f ` A)"by autolemma fun_upd_comp: "f o (g(x := y)) = (f o g)(x := f y)"  by autolemma UNION_fun_upd:  "UNION J (A(i:=B)) = (UNION (J-{i}) A ∪ (if i∈J then B else {}))"by (auto split: if_splits)subsection {* @{text override_on} *}definition override_on :: "('a => 'b) => ('a => 'b) => 'a set => 'a => 'b" where  "override_on f g A = (λa. if a ∈ 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 => 'a => ('a => 'b) => ('a => '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 ≠ c" and "b ≠ 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 o swap a b g = swap a b (f o g)"by (rule ext, simp add: fun_upd_def swap_def)lemma swap_image_eq [simp]:  assumes "a ∈ A" "b ∈ A" shows "swap a b f ` A = f ` A"proof -  have subset: "!!f. swap a b f ` A ⊆ f ` A"    using assms by (auto simp: image_iff swap_def)  then have "swap a b (swap a b f) ` A ⊆ (swap a b f) ` A" .  with subset[of f] show ?thesis by autoqedlemma inj_on_imp_inj_on_swap:  "[|inj_on f A; a ∈ A; b ∈ A|] ==> 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 ∈ A" "b ∈ A" shows "inj_on (swap a b f) A <-> 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 simpnext  assume "inj_on f A"  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)qedlemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"  by simplemma surj_swap_iff [simp]: "surj (swap a b f) <-> surj f"  by simplemma bij_betw_swap_iff [simp]:  "[| x ∈ A; y ∈ A |] ==> bij_betw (swap x y f) A B <-> bij_betw f A B"  by (auto simp: bij_betw_def)lemma bij_swap_iff [simp]: "bij (swap a b f) <-> bij f"  by simphide_const (open) swapsubsection {* 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 blastdonelemma 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 blastdonelemma 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 blastdonelemma 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)  donelemma 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)donelemma inj_on_the_inv_into:  "inj_on f A ==> 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 ==> 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 => 'b) => ('b => 'a)" where  "the_inv f ≡ 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]:  "¬(∃f. f ` A = Pow A)"proof clarify  fix f assume "f ` A = Pow A" hence *: "Pow A ≤ f ` A" by blast  let ?X = "{a ∈ A. a ∉ f a}"  have "?X ∈ Pow A" unfolding Pow_def by auto  with * obtain x where "x ∈ A ∧ f x = ?X" by blast  thus False by bestqedsubsection {* 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))    endin proc end*}subsubsection {* Code generator *}code_const "op o"  (SML infixl 5 "o")  (Haskell infixr 9 ".")code_const "id"  (Haskell "id")subsubsection {* Functorial structure of types *}ML_file "Tools/enriched_type.ML"enriched_type map_fun: map_fun  by (simp_all add: fun_eq_iff)enriched_type vimage  by (simp_all add: fun_eq_iff vimage_comp)text {* Legacy theorem names *}lemmas o_def = comp_deflemmas o_apply = comp_applylemmas o_assoc = comp_assoc [symmetric]lemmas id_o = id_complemmas o_id = comp_idlemmas o_eq_dest = comp_eq_destlemmas o_eq_elim = comp_eq_elimlemmas image_compose = image_complemmas vimage_compose = vimage_compend`