clasohm@1475: (*  Title:      HOL/Fun.thy
clasohm@1475:     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
blanchet@55019:     Author:     Andrei Popescu, TU Muenchen
blanchet@55019:     Copyright   1994, 2012
huffman@18154: *)
clasohm@923: 
wenzelm@60758: section \<open>Notions about functions\<close>
clasohm@923: 
paulson@15510: theory Fun
haftmann@56015: imports Set
blanchet@55467: keywords "functor" :: thy_goal
nipkow@15131: begin
nipkow@2912: 
haftmann@26147: lemma apply_inverse:
haftmann@26357:   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
haftmann@26147:   by auto
nipkow@2912: 
wenzelm@60758: text\<open>Uniqueness, so NOT the axiom of choice.\<close>
lp15@59504: lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
lp15@59504:   by (force intro: theI')
lp15@59504: 
lp15@59504: lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
lp15@59504:   by (force intro: theI')
wenzelm@12258: 
wenzelm@61799: subsection \<open>The Identity Function \<open>id\<close>\<close>
paulson@6171: 
haftmann@44277: definition id :: "'a \<Rightarrow> 'a" where
haftmann@22744:   "id = (\<lambda>x. x)"
nipkow@13910: 
haftmann@26147: lemma id_apply [simp]: "id x = x"
haftmann@26147:   by (simp add: id_def)
haftmann@26147: 
huffman@47579: lemma image_id [simp]: "image id = id"
huffman@47579:   by (simp add: id_def fun_eq_iff)
haftmann@26147: 
huffman@47579: lemma vimage_id [simp]: "vimage id = id"
huffman@47579:   by (simp add: id_def fun_eq_iff)
haftmann@26147: 
haftmann@52435: code_printing
haftmann@52435:   constant id \<rightharpoonup> (Haskell) "id"
haftmann@52435: 
haftmann@26147: 
wenzelm@61799: subsection \<open>The Composition Operator \<open>f \<circ> g\<close>\<close>
haftmann@26147: 
wenzelm@61955: definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c"  (infixl "\<circ>" 55)
wenzelm@61955:   where "f \<circ> g = (\<lambda>x. f (g x))"
oheimb@11123: 
wenzelm@61955: notation (ASCII)
wenzelm@61955:   comp  (infixl "o" 55)
wenzelm@19656: 
haftmann@49739: lemma comp_apply [simp]: "(f o g) x = f (g x)"
haftmann@49739:   by (simp add: comp_def)
paulson@13585: 
haftmann@49739: lemma comp_assoc: "(f o g) o h = f o (g o h)"
haftmann@49739:   by (simp add: fun_eq_iff)
paulson@13585: 
haftmann@49739: lemma id_comp [simp]: "id o g = g"
haftmann@49739:   by (simp add: fun_eq_iff)
paulson@13585: 
haftmann@49739: lemma comp_id [simp]: "f o id = f"
haftmann@49739:   by (simp add: fun_eq_iff)
haftmann@49739: 
haftmann@49739: lemma comp_eq_dest:
haftmann@34150:   "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
haftmann@49739:   by (simp add: fun_eq_iff)
haftmann@34150: 
haftmann@49739: lemma comp_eq_elim:
haftmann@34150:   "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
paulson@61204:   by (simp add: fun_eq_iff)
haftmann@34150: 
blanchet@55066: lemma comp_eq_dest_lhs: "a o b = c \<Longrightarrow> a (b v) = c v"
blanchet@55066:   by clarsimp
blanchet@55066: 
blanchet@55066: lemma comp_eq_id_dest: "a o b = id o c \<Longrightarrow> a (b v) = c v"
blanchet@55066:   by clarsimp
blanchet@55066: 
haftmann@49739: lemma image_comp:
haftmann@56154:   "f ` (g ` r) = (f o g) ` r"
paulson@33044:   by auto
paulson@33044: 
haftmann@49739: lemma vimage_comp:
haftmann@56154:   "f -` (g -` x) = (g \<circ> f) -` x"
haftmann@49739:   by auto
haftmann@49739: 
lp15@59504: lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B"
lp15@59504:   by (auto simp: comp_def elim!: equalityE)
lp15@59504: 
Andreas@59512: lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)"
Andreas@59512: by(auto simp add: Set.bind_def)
Andreas@59512: 
Andreas@59512: lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
Andreas@59512: by(auto simp add: Set.bind_def)
Andreas@59512: 
haftmann@60929: lemma (in group_add) minus_comp_minus [simp]:
haftmann@60929:   "uminus \<circ> uminus = id"
haftmann@60929:   by (simp add: fun_eq_iff)
haftmann@60929: 
haftmann@60929: lemma (in boolean_algebra) minus_comp_minus [simp]:
haftmann@60929:   "uminus \<circ> uminus = id"
haftmann@60929:   by (simp add: fun_eq_iff)
haftmann@60929: 
haftmann@52435: code_printing
haftmann@52435:   constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."
haftmann@52435: 
paulson@13585: 
wenzelm@61799: subsection \<open>The Forward Composition Operator \<open>fcomp\<close>\<close>
haftmann@26357: 
haftmann@44277: definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
haftmann@37751:   "f \<circ>> g = (\<lambda>x. g (f x))"
haftmann@26357: 
haftmann@37751: lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@37751: lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@37751: lemma id_fcomp [simp]: "id \<circ>> g = g"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
haftmann@37751: lemma fcomp_id [simp]: "f \<circ>> id = f"
haftmann@26357:   by (simp add: fcomp_def)
haftmann@26357: 
lp15@61699: lemma fcomp_comp: "fcomp f g = comp g f" 
lp15@61699:   by (simp add: ext)
lp15@61699: 
haftmann@52435: code_printing
haftmann@52435:   constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"
haftmann@31202: 
haftmann@37751: no_notation fcomp (infixl "\<circ>>" 60)
haftmann@26588: 
haftmann@26357: 
wenzelm@60758: subsection \<open>Mapping functions\<close>
haftmann@40602: 
haftmann@40602: definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
haftmann@40602:   "map_fun f g h = g \<circ> h \<circ> f"
haftmann@40602: 
haftmann@40602: lemma map_fun_apply [simp]:
haftmann@40602:   "map_fun f g h x = g (h (f x))"
haftmann@40602:   by (simp add: map_fun_def)
haftmann@40602: 
haftmann@40602: 
wenzelm@60758: subsection \<open>Injectivity and Bijectivity\<close>
hoelzl@39076: 
wenzelm@61799: definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where \<comment> "injective"
hoelzl@39076:   "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
haftmann@26147: 
wenzelm@61799: definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where \<comment> "bijective"
hoelzl@39076:   "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
haftmann@26147: 
wenzelm@60758: text\<open>A common special case: functions injective, surjective or bijective over
wenzelm@60758: the entire domain type.\<close>
haftmann@26147: 
haftmann@26147: abbreviation
hoelzl@39076:   "inj f \<equiv> inj_on f UNIV"
haftmann@26147: 
wenzelm@61799: abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where \<comment> "surjective"
hoelzl@40702:   "surj f \<equiv> (range f = UNIV)"
paulson@13585: 
hoelzl@39076: abbreviation
hoelzl@39076:   "bij f \<equiv> bij_betw f UNIV UNIV"
haftmann@26147: 
wenzelm@60758: text\<open>The negated case:\<close>
nipkow@43705: translations
nipkow@43705: "\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"
nipkow@43705: 
haftmann@26147: lemma injI:
haftmann@26147:   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
haftmann@26147:   shows "inj f"
haftmann@26147:   using assms unfolding inj_on_def by auto
paulson@13585: 
berghofe@13637: theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
berghofe@13637:   by (unfold inj_on_def, blast)
berghofe@13637: 
paulson@13585: lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
lp15@61520: lemma inj_on_eq_iff: "\<lbrakk>inj_on f A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y)"
paulson@13585: by (force simp add: inj_on_def)
paulson@13585: 
hoelzl@40703: lemma inj_on_cong:
hoelzl@40703:   "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
hoelzl@40703: unfolding inj_on_def by auto
hoelzl@40703: 
hoelzl@40703: lemma inj_on_strict_subset:
haftmann@56077:   "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
haftmann@56077:   unfolding inj_on_def by blast
hoelzl@40703: 
haftmann@38620: lemma inj_comp:
haftmann@38620:   "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
haftmann@38620:   by (simp add: inj_on_def)
haftmann@38620: 
haftmann@38620: lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
nipkow@39302:   by (simp add: inj_on_def fun_eq_iff)
haftmann@38620: 
nipkow@32988: lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
nipkow@32988: by (simp add: inj_on_eq_iff)
nipkow@32988: 
haftmann@26147: lemma inj_on_id[simp]: "inj_on id A"
hoelzl@39076:   by (simp add: inj_on_def)
paulson@13585: 
haftmann@26147: lemma inj_on_id2[simp]: "inj_on (%x. x) A"
hoelzl@39076: by (simp add: inj_on_def)
haftmann@26147: 
bulwahn@46586: lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"
hoelzl@40703: unfolding inj_on_def by blast
hoelzl@40703: 
hoelzl@40702: lemma surj_id: "surj id"
hoelzl@40702: by simp
haftmann@26147: 
hoelzl@39101: lemma bij_id[simp]: "bij id"
hoelzl@39076: by (simp add: bij_betw_def)
paulson@13585: 
paulson@13585: lemma inj_onI:
paulson@13585:     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
paulson@13585: lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
paulson@13585: by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
paulson@13585: 
paulson@13585: lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma comp_inj_on:
paulson@13585:      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
paulson@13585: by (simp add: comp_def inj_on_def)
paulson@13585: 
nipkow@15303: lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
haftmann@56077:   by (simp add: inj_on_def) blast
nipkow@15303: 
nipkow@15439: lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
nipkow@15439:   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
nipkow@15439: apply(unfold inj_on_def)
nipkow@15439: apply blast
nipkow@15439: done
nipkow@15439: 
paulson@13585: lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
paulson@13585: by (unfold inj_on_def, blast)
wenzelm@12258: 
paulson@13585: lemma inj_singleton: "inj (%s. {s})"
paulson@13585: by (simp add: inj_on_def)
paulson@13585: 
nipkow@15111: lemma inj_on_empty[iff]: "inj_on f {}"
nipkow@15111: by(simp add: inj_on_def)
nipkow@15111: 
nipkow@15303: lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
nipkow@15111: lemma inj_on_Un:
nipkow@15111:  "inj_on f (A Un B) =
nipkow@15111:   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
nipkow@15111: apply(unfold inj_on_def)
nipkow@15111: apply (blast intro:sym)
nipkow@15111: done
nipkow@15111: 
nipkow@15111: lemma inj_on_insert[iff]:
nipkow@15111:   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
nipkow@15111: apply(unfold inj_on_def)
nipkow@15111: apply (blast intro:sym)
nipkow@15111: done
nipkow@15111: 
nipkow@15111: lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
nipkow@15111: apply(unfold inj_on_def)
nipkow@15111: apply (blast)
nipkow@15111: done
nipkow@15111: 
hoelzl@40703: lemma comp_inj_on_iff:
hoelzl@40703:   "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
hoelzl@40703: by(auto simp add: comp_inj_on inj_on_def)
hoelzl@40703: 
hoelzl@40703: lemma inj_on_imageI2:
hoelzl@40703:   "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
hoelzl@40703: by(auto simp add: comp_inj_on inj_on_def)
hoelzl@40703: 
haftmann@51598: lemma inj_img_insertE:
haftmann@51598:   assumes "inj_on f A"
haftmann@51598:   assumes "x \<notin> B" and "insert x B = f ` A"
haftmann@51598:   obtains x' A' where "x' \<notin> A'" and "A = insert x' A'"
blanchet@55019:     and "x = f x'" and "B = f ` A'"
haftmann@51598: proof -
haftmann@51598:   from assms have "x \<in> f ` A" by auto
haftmann@51598:   then obtain x' where *: "x' \<in> A" "x = f x'" by auto
haftmann@51598:   then have "A = insert x' (A - {x'})" by auto
haftmann@51598:   with assms * have "B = f ` (A - {x'})"
haftmann@51598:     by (auto dest: inj_on_contraD)
haftmann@51598:   have "x' \<notin> A - {x'}" by simp
wenzelm@60758:   from \<open>x' \<notin> A - {x'}\<close> \<open>A = insert x' (A - {x'})\<close> \<open>x = f x'\<close> \<open>B = image f (A - {x'})\<close>
haftmann@51598:   show ?thesis ..
haftmann@51598: qed
haftmann@51598: 
traytel@54578: lemma linorder_injI:
traytel@54578:   assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y"
traytel@54578:   shows "inj f"
wenzelm@61799:   \<comment> \<open>Courtesy of Stephan Merz\<close>
traytel@54578: proof (rule inj_onI)
traytel@54578:   fix x y
traytel@54578:   assume f_eq: "f x = f y"
traytel@54578:   show "x = y" by (rule linorder_cases) (auto dest: hyp simp: f_eq)
traytel@54578: qed
traytel@54578: 
hoelzl@40702: lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
hoelzl@40702:   by auto
hoelzl@39076: 
hoelzl@40702: lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
hoelzl@40702:   using *[symmetric] by auto
paulson@13585: 
hoelzl@39076: lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
hoelzl@39076:   by (simp add: surj_def)
paulson@13585: 
hoelzl@39076: lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
hoelzl@39076:   by (simp add: surj_def, blast)
paulson@13585: 
paulson@13585: lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
paulson@13585: apply (simp add: comp_def surj_def, clarify)
paulson@13585: apply (drule_tac x = y in spec, clarify)
paulson@13585: apply (drule_tac x = x in spec, blast)
paulson@13585: done
paulson@13585: 
ballarin@57282: lemma bij_betw_imageI:
ballarin@57282:   "\<lbrakk> inj_on f A; f ` A = B \<rbrakk> \<Longrightarrow> bij_betw f A B"
ballarin@57282: unfolding bij_betw_def by clarify
ballarin@57282: 
ballarin@57282: lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
ballarin@57282:   unfolding bij_betw_def by clarify
ballarin@57282: 
hoelzl@39074: lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
hoelzl@40702:   unfolding bij_betw_def by auto
hoelzl@39074: 
hoelzl@40703: lemma bij_betw_empty1:
hoelzl@40703:   assumes "bij_betw f {} A"
hoelzl@40703:   shows "A = {}"
hoelzl@40703: using assms unfolding bij_betw_def by blast
hoelzl@40703: 
hoelzl@40703: lemma bij_betw_empty2:
hoelzl@40703:   assumes "bij_betw f A {}"
hoelzl@40703:   shows "A = {}"
hoelzl@40703: using assms unfolding bij_betw_def by blast
hoelzl@40703: 
hoelzl@40703: lemma inj_on_imp_bij_betw:
hoelzl@40703:   "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
hoelzl@40703: unfolding bij_betw_def by simp
hoelzl@40703: 
hoelzl@39076: lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
hoelzl@40702:   unfolding bij_betw_def ..
hoelzl@39074: 
paulson@13585: lemma bijI: "[| inj f; surj f |] ==> bij f"
paulson@13585: by (simp add: bij_def)
paulson@13585: 
paulson@13585: lemma bij_is_inj: "bij f ==> inj f"
paulson@13585: by (simp add: bij_def)
paulson@13585: 
paulson@13585: lemma bij_is_surj: "bij f ==> surj f"
paulson@13585: by (simp add: bij_def)
paulson@13585: 
nipkow@26105: lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105: by (simp add: bij_betw_def)
nipkow@26105: 
nipkow@31438: lemma bij_betw_trans:
nipkow@31438:   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31438: by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31438: 
hoelzl@40702: lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
hoelzl@40702:   by (rule bij_betw_trans)
hoelzl@40702: 
hoelzl@40703: lemma bij_betw_comp_iff:
hoelzl@40703:   "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703: by(auto simp add: bij_betw_def inj_on_def)
hoelzl@40703: 
hoelzl@40703: lemma bij_betw_comp_iff2:
hoelzl@40703:   assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
hoelzl@40703:   shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
hoelzl@40703: using assms
hoelzl@40703: proof(auto simp add: bij_betw_comp_iff)
hoelzl@40703:   assume *: "bij_betw (f' \<circ> f) A A''"
hoelzl@40703:   thus "bij_betw f A A'"
hoelzl@40703:   using IM
hoelzl@40703:   proof(auto simp add: bij_betw_def)
hoelzl@40703:     assume "inj_on (f' \<circ> f) A"
hoelzl@40703:     thus "inj_on f A" using inj_on_imageI2 by blast
hoelzl@40703:   next
hoelzl@40703:     fix a' assume **: "a' \<in> A'"
hoelzl@40703:     hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
hoelzl@40703:     then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
hoelzl@40703:     unfolding bij_betw_def by force
hoelzl@40703:     hence "f a \<in> A'" using IM by auto
hoelzl@40703:     hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
hoelzl@40703:     thus "a' \<in> f ` A" using 1 by auto
hoelzl@40703:   qed
hoelzl@40703: qed
hoelzl@40703: 
nipkow@26105: lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105: proof -
nipkow@26105:   have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105:     using assms by(auto simp:bij_betw_def)
nipkow@26105:   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105:   { fix a b assume P: "?P b a"
haftmann@56077:     hence ex1: "\<exists>a. ?P b a" using s by blast
nipkow@26105:     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105:     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105:   } note g = this
nipkow@26105:   have "inj_on ?g B"
nipkow@26105:   proof(rule inj_onI)
nipkow@26105:     fix x y assume "x:B" "y:B" "?g x = ?g y"
wenzelm@60758:     from s \<open>x:B\<close> obtain a1 where a1: "?P x a1" by blast
wenzelm@60758:     from s \<open>y:B\<close> obtain a2 where a2: "?P y a2" by blast
wenzelm@60758:     from g[OF a1] a1 g[OF a2] a2 \<open>?g x = ?g y\<close> show "x=y" by simp
nipkow@26105:   qed
nipkow@26105:   moreover have "?g ` B = A"
haftmann@56077:   proof(auto simp: image_def)
nipkow@26105:     fix b assume "b:B"
haftmann@56077:     with s obtain a where P: "?P b a" by blast
nipkow@26105:     thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105:   next
nipkow@26105:     fix a assume "a:A"
haftmann@56077:     then obtain b where P: "?P b a" using s by blast
haftmann@56077:     then have "b:B" using s by blast
nipkow@26105:     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105:   qed
nipkow@26105:   ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105: qed
nipkow@26105: 
hoelzl@40703: lemma bij_betw_cong:
hoelzl@40703:   "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
hoelzl@40703: unfolding bij_betw_def inj_on_def by force
hoelzl@40703: 
hoelzl@40703: lemma bij_betw_id[intro, simp]:
hoelzl@40703:   "bij_betw id A A"
hoelzl@40703: unfolding bij_betw_def id_def by auto
hoelzl@40703: 
hoelzl@40703: lemma bij_betw_id_iff:
hoelzl@40703:   "bij_betw id A B \<longleftrightarrow> A = B"
hoelzl@40703: by(auto simp add: bij_betw_def)
hoelzl@40703: 
hoelzl@39075: lemma bij_betw_combine:
hoelzl@39075:   assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
hoelzl@39075:   shows "bij_betw f (A \<union> C) (B \<union> D)"
hoelzl@39075:   using assms unfolding bij_betw_def inj_on_Un image_Un by auto
hoelzl@39075: 
hoelzl@40703: lemma bij_betw_subset:
hoelzl@40703:   assumes BIJ: "bij_betw f A A'" and
hoelzl@40703:           SUB: "B \<le> A" and IM: "f ` B = B'"
hoelzl@40703:   shows "bij_betw f B B'"
hoelzl@40703: using assms
hoelzl@40703: by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
hoelzl@40703: 
haftmann@58195: lemma bij_pointE:
haftmann@58195:   assumes "bij f"
haftmann@58195:   obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
haftmann@58195: proof -
haftmann@58195:   from assms have "inj f" by (rule bij_is_inj)
haftmann@58195:   moreover from assms have "surj f" by (rule bij_is_surj)
haftmann@58195:   then have "y \<in> range f" by simp
haftmann@58195:   ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq)
haftmann@58195:   with that show thesis by blast
haftmann@58195: qed
haftmann@58195: 
paulson@13585: lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
hoelzl@40702: by simp
paulson@13585: 
hoelzl@42903: lemma surj_vimage_empty:
hoelzl@42903:   assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
wenzelm@60758:   using surj_image_vimage_eq[OF \<open>surj f\<close>, of A]
nipkow@44890:   by (intro iffI) fastforce+
hoelzl@42903: 
paulson@13585: lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585: by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
hoelzl@40702: by (blast intro: sym)
paulson@13585: 
paulson@13585: lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585: by (unfold inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585: apply (unfold bij_def)
paulson@13585: apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585: done
paulson@13585: 
Andreas@53927: lemma inj_on_image_eq_iff: "\<lbrakk> inj_on f C; A \<subseteq> C; B \<subseteq> C \<rbrakk> \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
Andreas@53927: by(fastforce simp add: inj_on_def)
Andreas@53927: 
nipkow@31438: lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
Andreas@53927: by(erule inj_on_image_eq_iff) simp_all
nipkow@31438: 
paulson@13585: lemma inj_on_image_Int:
paulson@13585:    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@60303:   by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma inj_on_image_set_diff:
paulson@60303:    "[| inj_on f C;  A-B \<subseteq> C;  B \<subseteq> C |] ==> f`(A-B) = f`A - f`B"
paulson@60303:   by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@60303:   by (simp add: inj_on_def, blast)
paulson@13585: 
paulson@13585: lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585: by (simp add: inj_on_def, blast)
paulson@13585: 
lp15@59504: lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
lp15@59504:   by (auto simp: inj_on_def)
lp15@59504: 
lp15@61520: (*FIXME DELETE*)
lp15@61520: lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f`A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A"
lp15@61520:   by (blast dest: inj_onD)
lp15@61520: 
lp15@59504: lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A"
lp15@59504:   by (blast dest: injD)
paulson@13585: 
paulson@13585: lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
lp15@59504:   by (blast dest: injD)
paulson@13585: 
paulson@13585: lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
lp15@59504:   by (blast dest: injD)
paulson@13585: 
paulson@13585: lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
hoelzl@40702: by auto
paulson@13585: 
paulson@13585: lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585: by (auto simp add: inj_on_def)
paulson@5852: 
paulson@13585: lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585: apply (simp add: bij_def)
paulson@13585: apply (rule equalityI)
paulson@13585: apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585: done
paulson@13585: 
haftmann@41657: lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
wenzelm@61799:   \<comment> \<open>The inverse image of a singleton under an injective function
wenzelm@60758:          is included in a singleton.\<close>
haftmann@41657:   apply (auto simp add: inj_on_def)
haftmann@41657:   apply (blast intro: the_equality [symmetric])
haftmann@41657:   done
haftmann@41657: 
hoelzl@43991: lemma inj_on_vimage_singleton:
hoelzl@43991:   "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
hoelzl@43991:   by (auto simp add: inj_on_def intro: the_equality [symmetric])
hoelzl@43991: 
hoelzl@35584: lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580:   by (auto intro!: inj_onI)
paulson@13585: 
hoelzl@35584: lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584:   by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584: 
blanchet@55019: lemma bij_betw_byWitness:
blanchet@55019: assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and
blanchet@55019:         RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and
blanchet@55019:         IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A"
blanchet@55019: shows "bij_betw f A A'"
blanchet@55019: using assms
blanchet@55019: proof(unfold bij_betw_def inj_on_def, safe)
blanchet@55019:   fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b"
blanchet@55019:   have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp
blanchet@55019:   with ** show "a = b" by simp
blanchet@55019: next
blanchet@55019:   fix a' assume *: "a' \<in> A'"
blanchet@55019:   hence "f' a' \<in> A" using IM2 by blast
blanchet@55019:   moreover
blanchet@55019:   have "a' = f(f' a')" using * RIGHT by simp
blanchet@55019:   ultimately show "a' \<in> f ` A" by blast
blanchet@55019: qed
blanchet@55019: 
blanchet@55019: corollary notIn_Un_bij_betw:
blanchet@55019: assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and
blanchet@55019:        BIJ: "bij_betw f A A'"
blanchet@55019: shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019: proof-
blanchet@55019:   have "bij_betw f {b} {f b}"
blanchet@55019:   unfolding bij_betw_def inj_on_def by simp
blanchet@55019:   with assms show ?thesis
blanchet@55019:   using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
blanchet@55019: qed
blanchet@55019: 
blanchet@55019: lemma notIn_Un_bij_betw3:
blanchet@55019: assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'"
blanchet@55019: shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019: proof
blanchet@55019:   assume "bij_betw f A A'"
blanchet@55019:   thus "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019:   using assms notIn_Un_bij_betw[of b A f A'] by blast
blanchet@55019: next
blanchet@55019:   assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
blanchet@55019:   have "f ` A = A'"
blanchet@55019:   proof(auto)
blanchet@55019:     fix a assume **: "a \<in> A"
blanchet@55019:     hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast
blanchet@55019:     moreover
blanchet@55019:     {assume "f a = f b"
blanchet@55019:      hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast
blanchet@55019:      with NIN ** have False by blast
blanchet@55019:     }
blanchet@55019:     ultimately show "f a \<in> A'" by blast
blanchet@55019:   next
blanchet@55019:     fix a' assume **: "a' \<in> A'"
blanchet@55019:     hence "a' \<in> f`(A \<union> {b})"
blanchet@55019:     using * by (auto simp add: bij_betw_def)
blanchet@55019:     then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
blanchet@55019:     moreover
blanchet@55019:     {assume "a = b" with 1 ** NIN' have False by blast
blanchet@55019:     }
blanchet@55019:     ultimately have "a \<in> A" by blast
blanchet@55019:     with 1 show "a' \<in> f ` A" by blast
blanchet@55019:   qed
blanchet@55019:   thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
blanchet@55019: qed
blanchet@55019: 
haftmann@41657: 
wenzelm@60758: subsection\<open>Function Updating\<close>
paulson@13585: 
haftmann@44277: definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147:   "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147: 
wenzelm@41229: nonterminal updbinds and updbind
wenzelm@41229: 
haftmann@26147: syntax
haftmann@26147:   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147:   ""         :: "updbind => updbinds"             ("_")
haftmann@26147:   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115:   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147: 
haftmann@26147: translations
wenzelm@35115:   "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115:   "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147: 
blanchet@55414: (* Hint: to define the sum of two functions (or maps), use case_sum.
blanchet@58111:          A nice infix syntax could be defined by
wenzelm@35115: notation
blanchet@55414:   case_sum  (infixr "'(+')"80)
haftmann@26147: *)
haftmann@26147: 
paulson@13585: lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585: apply (simp add: fun_upd_def, safe)
paulson@13585: apply (erule subst)
paulson@13585: apply (rule_tac [2] ext, auto)
paulson@13585: done
paulson@13585: 
wenzelm@45603: lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
wenzelm@45603:   by (simp only: fun_upd_idem_iff)
paulson@13585: 
wenzelm@45603: lemma fun_upd_triv [iff]: "f(x := f x) = f"
wenzelm@45603:   by (simp only: fun_upd_idem)
paulson@13585: 
paulson@13585: lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084: by (simp add: fun_upd_def)
paulson@13585: 
paulson@13585: (* fun_upd_apply supersedes these two,   but they are useful
paulson@13585:    if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585: lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585: by simp
paulson@13585: 
paulson@13585: lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585: by simp
paulson@13585: 
paulson@13585: lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
nipkow@39302: by (simp add: fun_eq_iff)
paulson@13585: 
paulson@13585: lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585: by (rule ext, auto)
paulson@13585: 
haftmann@56077: lemma inj_on_fun_updI:
haftmann@56077:   "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
haftmann@56077:   by (fastforce simp: inj_on_def)
nipkow@15303: 
paulson@15510: lemma fun_upd_image:
paulson@15510:      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510: by auto
paulson@15510: 
nipkow@31080: lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
huffman@44921:   by auto
nipkow@31080: 
Andreas@61630: lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z"
Andreas@61630: by(simp add: fun_eq_iff split: split_if_asm)
haftmann@26147: 
wenzelm@61799: subsection \<open>\<open>override_on\<close>\<close>
haftmann@26147: 
haftmann@44277: definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
haftmann@26147:   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910: 
nipkow@15691: lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691: by(simp add:override_on_def)
nipkow@13910: 
nipkow@15691: lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691: by(simp add:override_on_def)
nipkow@13910: 
nipkow@15691: lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691: by(simp add:override_on_def)
nipkow@13910: 
haftmann@26147: 
wenzelm@61799: subsection \<open>\<open>swap\<close>\<close>
paulson@15510: 
haftmann@56608: definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@56608: where
haftmann@22744:   "swap a b f = f (a := f b, b:= f a)"
paulson@15510: 
haftmann@56608: lemma swap_apply [simp]:
haftmann@56608:   "swap a b f a = f b"
haftmann@56608:   "swap a b f b = f a"
haftmann@56608:   "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c"
haftmann@56608:   by (simp_all add: swap_def)
haftmann@56608: 
haftmann@56608: lemma swap_self [simp]:
haftmann@56608:   "swap a a f = f"
haftmann@56608:   by (simp add: swap_def)
paulson@15510: 
haftmann@56608: lemma swap_commute:
haftmann@56608:   "swap a b f = swap b a f"
haftmann@56608:   by (simp add: fun_upd_def swap_def fun_eq_iff)
paulson@15510: 
haftmann@56608: lemma swap_nilpotent [simp]:
haftmann@56608:   "swap a b (swap a b f) = f"
haftmann@56608:   by (rule ext, simp add: fun_upd_def swap_def)
haftmann@56608: 
haftmann@56608: lemma swap_comp_involutory [simp]:
haftmann@56608:   "swap a b \<circ> swap a b = id"
haftmann@56608:   by (rule ext) simp
paulson@15510: 
huffman@34145: lemma swap_triple:
huffman@34145:   assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145:   shows "swap a b (swap b c (swap a b f)) = swap a c f"
nipkow@39302:   using assms by (simp add: fun_eq_iff swap_def)
huffman@34145: 
huffman@34101: lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
haftmann@56608:   by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101: 
hoelzl@39076: lemma swap_image_eq [simp]:
hoelzl@39076:   assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
hoelzl@39076: proof -
hoelzl@39076:   have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
hoelzl@39076:     using assms by (auto simp: image_iff swap_def)
hoelzl@39076:   then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
hoelzl@39076:   with subset[of f] show ?thesis by auto
hoelzl@39076: qed
hoelzl@39076: 
paulson@15510: lemma inj_on_imp_inj_on_swap:
hoelzl@39076:   "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
hoelzl@39076:   by (simp add: inj_on_def swap_def, blast)
paulson@15510: 
paulson@15510: lemma inj_on_swap_iff [simp]:
hoelzl@39076:   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
hoelzl@39075: proof
paulson@15510:   assume "inj_on (swap a b f) A"
hoelzl@39075:   with A have "inj_on (swap a b (swap a b f)) A"
hoelzl@39075:     by (iprover intro: inj_on_imp_inj_on_swap)
hoelzl@39075:   thus "inj_on f A" by simp
paulson@15510: next
paulson@15510:   assume "inj_on f A"
krauss@34209:   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510: qed
paulson@15510: 
hoelzl@39076: lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
hoelzl@40702:   by simp
paulson@15510: 
hoelzl@39076: lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
hoelzl@40702:   by simp
haftmann@21547: 
hoelzl@39076: lemma bij_betw_swap_iff [simp]:
hoelzl@39076:   "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
hoelzl@39076:   by (auto simp: bij_betw_def)
hoelzl@39076: 
hoelzl@39076: lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
hoelzl@39076:   by simp
hoelzl@39075: 
wenzelm@36176: hide_const (open) swap
haftmann@21547: 
haftmann@56608: 
wenzelm@60758: subsection \<open>Inversion of injective functions\<close>
haftmann@31949: 
nipkow@33057: definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
haftmann@44277:   "the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961: 
nipkow@33057: lemma the_inv_into_f_f:
nipkow@33057:   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057: apply (simp add: the_inv_into_def inj_on_def)
krauss@34209: apply blast
nipkow@32961: done
nipkow@32961: 
nipkow@33057: lemma f_the_inv_into_f:
nipkow@33057:   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057: apply (simp add: the_inv_into_def)
nipkow@32961: apply (rule the1I2)
nipkow@32961:  apply(blast dest: inj_onD)
nipkow@32961: apply blast
nipkow@32961: done
nipkow@32961: 
nipkow@33057: lemma the_inv_into_into:
nipkow@33057:   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057: apply (simp add: the_inv_into_def)
nipkow@32961: apply (rule the1I2)
nipkow@32961:  apply(blast dest: inj_onD)
nipkow@32961: apply blast
nipkow@32961: done
nipkow@32961: 
nipkow@33057: lemma the_inv_into_onto[simp]:
nipkow@33057:   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057: by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961: 
nipkow@33057: lemma the_inv_into_f_eq:
nipkow@33057:   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961:   apply (erule subst)
nipkow@33057:   apply (erule the_inv_into_f_f, assumption)
nipkow@32961:   done
nipkow@32961: 
nipkow@33057: lemma the_inv_into_comp:
nipkow@32961:   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057:   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057: apply (rule the_inv_into_f_eq)
nipkow@32961:   apply (fast intro: comp_inj_on)
nipkow@33057:  apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057: apply (simp add: the_inv_into_into)
nipkow@32961: done
nipkow@32961: 
nipkow@33057: lemma inj_on_the_inv_into:
nipkow@33057:   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
haftmann@56077: by (auto intro: inj_onI simp: the_inv_into_f_f)
nipkow@32961: 
nipkow@33057: lemma bij_betw_the_inv_into:
nipkow@33057:   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057: by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961: 
berghofe@32998: abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057:   "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998: 
berghofe@32998: lemma the_inv_f_f:
berghofe@32998:   assumes "inj f"
berghofe@32998:   shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057:   by (rule the_inv_into_f_f)
berghofe@32998: 
haftmann@44277: 
wenzelm@60758: subsection \<open>Cantor's Paradox\<close>
hoelzl@40703: 
blanchet@54147: lemma Cantors_paradox:
hoelzl@40703:   "\<not>(\<exists>f. f ` A = Pow A)"
hoelzl@40703: proof clarify
hoelzl@40703:   fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
hoelzl@40703:   let ?X = "{a \<in> A. a \<notin> f a}"
hoelzl@40703:   have "?X \<in> Pow A" unfolding Pow_def by auto
hoelzl@40703:   with * obtain x where "x \<in> A \<and> f x = ?X" by blast
hoelzl@40703:   thus False by best
hoelzl@40703: qed
haftmann@31949: 
paulson@61204: subsection \<open>Setup\<close>
haftmann@40969: 
wenzelm@60758: subsubsection \<open>Proof tools\<close>
haftmann@22845: 
wenzelm@60758: text \<open>simplifies terms of the form
wenzelm@60758:   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close>
haftmann@22845: 
wenzelm@60758: simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ =>
haftmann@22845: let
haftmann@22845:   fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017:     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845:   fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845:   fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845:     let
haftmann@22845:       fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845:             if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845:         | find t = NONE
haftmann@22845:     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017: 
wenzelm@51717:   val ss = simpset_of @{context}
wenzelm@51717: 
wenzelm@51717:   fun proc ctxt ct =
wenzelm@24017:     let
wenzelm@24017:       val t = Thm.term_of ct
wenzelm@24017:     in
wenzelm@24017:       case find_double t of
wenzelm@24017:         (T, NONE) => NONE
wenzelm@24017:       | (T, SOME rhs) =>
wenzelm@27330:           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017:             (fn _ =>
wenzelm@59498:               resolve_tac ctxt [eq_reflection] 1 THEN
wenzelm@59498:               resolve_tac ctxt @{thms ext} 1 THEN
wenzelm@51717:               simp_tac (put_simpset ss ctxt) 1))
wenzelm@24017:     end
wenzelm@24017: in proc end
wenzelm@60758: \<close>
haftmann@22845: 
haftmann@22845: 
wenzelm@60758: subsubsection \<open>Functorial structure of types\<close>
haftmann@40969: 
blanchet@55467: ML_file "Tools/functor.ML"
haftmann@40969: 
blanchet@55467: functor map_fun: map_fun
haftmann@47488:   by (simp_all add: fun_eq_iff)
haftmann@47488: 
blanchet@55467: functor vimage
haftmann@49739:   by (simp_all add: fun_eq_iff vimage_comp)
haftmann@49739: 
wenzelm@60758: text \<open>Legacy theorem names\<close>
haftmann@49739: 
haftmann@49739: lemmas o_def = comp_def
haftmann@49739: lemmas o_apply = comp_apply
haftmann@49739: lemmas o_assoc = comp_assoc [symmetric]
haftmann@49739: lemmas id_o = id_comp
haftmann@49739: lemmas o_id = comp_id
haftmann@49739: lemmas o_eq_dest = comp_eq_dest
haftmann@49739: lemmas o_eq_elim = comp_eq_elim
blanchet@55066: lemmas o_eq_dest_lhs = comp_eq_dest_lhs
blanchet@55066: lemmas o_eq_id_dest = comp_eq_id_dest
haftmann@47488: 
nipkow@2912: end
haftmann@56015: