Theory Fun

theory Fun
imports Set
(*  Title:      HOL/Fun.thy
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
    Author:     Andrei Popescu, TU Muenchen
    Copyright   1994, 2012
*)

section ‹Notions about functions›

theory Fun
  imports Set
  keywords "functor" :: thy_goal
begin

lemma apply_inverse: "f x = u ⟹ (⋀x. P x ⟹ g (f x) = x) ⟹ P x ⟹ x = g u"
  by auto

text ‹Uniqueness, so NOT the axiom of choice.›
lemma uniq_choice: "∀x. ∃!y. Q x y ⟹ ∃f. ∀x. Q x (f x)"
  by (force intro: theI')

lemma b_uniq_choice: "∀x∈S. ∃!y. Q x y ⟹ ∃f. ∀x∈S. Q x (f x)"
  by (force intro: theI')


subsection ‹The Identity Function ‹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)

lemma eq_id_iff: "(∀x. f x = x) ⟷ f = id"
  by auto

code_printing
  constant id  (Haskell) "id"


subsection ‹The Composition Operator ‹f ∘ g››

definition comp :: "('b ⇒ 'c) ⇒ ('a ⇒ 'b) ⇒ 'a ⇒ 'c"  (infixl "∘" 55)
  where "f ∘ g = (λx. f (g x))"

notation (ASCII)
  comp  (infixl "o" 55)

lemma comp_apply [simp]: "(f ∘ g) x = f (g x)"
  by (simp add: comp_def)

lemma comp_assoc: "(f ∘ g) ∘ h = f ∘ (g ∘ h)"
  by (simp add: fun_eq_iff)

lemma id_comp [simp]: "id ∘ g = g"
  by (simp add: fun_eq_iff)

lemma comp_id [simp]: "f ∘ id = f"
  by (simp add: fun_eq_iff)

lemma comp_eq_dest: "a ∘ b = c ∘ d ⟹ a (b v) = c (d v)"
  by (simp add: fun_eq_iff)

lemma comp_eq_elim: "a ∘ b = c ∘ d ⟹ ((⋀v. a (b v) = c (d v)) ⟹ R) ⟹ R"
  by (simp add: fun_eq_iff)

lemma comp_eq_dest_lhs: "a ∘ b = c ⟹ a (b v) = c v"
  by clarsimp

lemma comp_eq_id_dest: "a ∘ b = id ∘ c ⟹ a (b v) = c v"
  by clarsimp

lemma image_comp: "f ` (g ` r) = (f ∘ g) ` r"
  by auto

lemma vimage_comp: "f -` (g -` x) = (g ∘ f) -` x"
  by auto

lemma image_eq_imp_comp: "f ` A = g ` B ⟹ (h ∘ f) ` A = (h ∘ g) ` B"
  by (auto simp: comp_def elim!: equalityE)

lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f ∘ g)"
  by (auto simp add: Set.bind_def)

lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g ∘ f)"
  by (auto simp add: Set.bind_def)

lemma (in group_add) minus_comp_minus [simp]: "uminus ∘ uminus = id"
  by (simp add: fun_eq_iff)

lemma (in boolean_algebra) minus_comp_minus [simp]: "uminus ∘ uminus = id"
  by (simp add: fun_eq_iff)

code_printing
  constant comp  (SML) infixl 5 "o" and (Haskell) infixr 9 "."


subsection ‹The Forward Composition Operator ‹fcomp››

definition fcomp :: "('a ⇒ 'b) ⇒ ('b ⇒ 'c) ⇒ 'a ⇒ 'c"  (infixl "∘>" 60)
  where "f ∘> g = (λx. g (f x))"

lemma fcomp_apply [simp]:  "(f ∘> g) x = g (f x)"
  by (simp add: fcomp_def)

lemma fcomp_assoc: "(f ∘> g) ∘> h = f ∘> (g ∘> h)"
  by (simp add: fcomp_def)

lemma id_fcomp [simp]: "id ∘> g = g"
  by (simp add: fcomp_def)

lemma fcomp_id [simp]: "f ∘> id = f"
  by (simp add: fcomp_def)

lemma fcomp_comp: "fcomp f g = comp g f"
  by (simp add: ext)

code_printing
  constant fcomp  (Eval) infixl 1 "#>"

no_notation fcomp (infixl "∘>" 60)


subsection ‹Mapping functions›

definition map_fun :: "('c ⇒ 'a) ⇒ ('b ⇒ 'd) ⇒ ('a ⇒ 'b) ⇒ 'c ⇒ 'd"
  where "map_fun f g h = g ∘ h ∘ 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"   ‹injective›
  where "inj_on f A ⟷ (∀x∈A. ∀y∈A. f x = f y ⟶ x = y)"

definition bij_betw :: "('a ⇒ 'b) ⇒ 'a set ⇒ 'b set ⇒ bool"   ‹bijective›
  where "bij_betw f A B ⟷ inj_on f A ∧ f ` A = B"

text ‹
  A common special case: functions injective, surjective or bijective over
  the entire domain type.
›

abbreviation inj :: "('a ⇒ 'b) ⇒ bool"
  where "inj f ≡ inj_on f UNIV"

abbreviation surj :: "('a ⇒ 'b) ⇒ bool"
  where "surj f ≡ range f = UNIV"

translations -- ‹The negated case:›
  "¬ CONST surj f"  "CONST range f ≠ CONST UNIV"

abbreviation bij :: "('a ⇒ 'b) ⇒ bool"
  where "bij f ≡ bij_betw f UNIV UNIV"

lemma inj_def: "inj f ⟷ (∀x y. f x = f y ⟶ x = y)"
  unfolding inj_on_def by blast

lemma injI: "(⋀x y. f x = f y ⟹ x = y) ⟹ inj f"
  unfolding inj_def by blast

theorem range_ex1_eq: "inj f ⟹ b ∈ range f ⟷ (∃!x. b = f x)"
  unfolding inj_def by blast

lemma injD: "inj f ⟹ f x = f y ⟹ x = y"
  by (simp add: inj_def)

lemma inj_on_eq_iff: "inj_on f A ⟹ x ∈ A ⟹ y ∈ A ⟹ f x = f y ⟷ x = y"
  by (auto simp: inj_on_def)

lemma inj_on_cong: "(⋀a. a ∈ A ⟹ f a = g a) ⟹ inj_on f A ⟷ inj_on g A"
  by (auto simp: inj_on_def)

lemma inj_on_strict_subset: "inj_on f B ⟹ A ⊂ B ⟹ f ` A ⊂ f ` B"
  unfolding inj_on_def by blast

lemma inj_comp: "inj f ⟹ inj g ⟹ inj (f ∘ g)"
  by (simp add: inj_def)

lemma inj_fun: "inj f ⟹ inj (λx y. f x)"
  by (simp add: inj_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 (λx. 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 blast

lemma surj_id: "surj id"
  by simp

lemma bij_id[simp]: "bij id"
  by (simp add: bij_betw_def)

lemma bij_uminus: "bij (uminus :: 'a ⇒ 'a::ab_group_add)"
  unfolding bij_betw_def inj_on_def
  by (force intro: minus_minus [symmetric])

lemma inj_onI [intro?]: "(⋀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"
  unfolding inj_on_def by blast

lemma inj_on_subset:
  assumes "inj_on f A"
    and "B ⊆ A"
  shows "inj_on f B"
proof (rule inj_onI)
  fix a b
  assume "a ∈ B" and "b ∈ B"
  with assms have "a ∈ A" and "b ∈ A"
    by auto
  moreover assume "f a = f b"
  ultimately show "a = b"
    using assms by (auto dest: inj_onD)
qed

lemma comp_inj_on: "inj_on f A ⟹ inj_on g (f ` A) ⟹ inj_on (g ∘ f) A"
  by (simp add: comp_def inj_on_def)

lemma inj_on_imageI: "inj_on (g ∘ f) A ⟹ inj_on g (f ` A)"
  by (auto simp add: inj_on_def)

lemma inj_on_image_iff:
  "∀x∈A. ∀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"
  unfolding inj_on_def by blast

lemma inj_on_contraD: "inj_on f A ⟹ x ≠ y ⟹ x ∈ A ⟹ y ∈ A ⟹ f x ≠ f y"
  unfolding inj_on_def by blast

lemma inj_singleton [simp]: "inj_on (λx. {x}) A"
  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"
  unfolding inj_on_def by blast

lemma inj_on_Un: "inj_on f (A ∪ B) ⟷ inj_on f A ∧ inj_on f B ∧ f ` (A - B) ∩ f ` (B - A) = {}"
  unfolding inj_on_def by (blast intro: sym)

lemma inj_on_insert [iff]: "inj_on f (insert a A) ⟷ inj_on f A ∧ f a ∉ f ` (A - {a})"
  unfolding inj_on_def by (blast intro: sym)

lemma inj_on_diff: "inj_on f A ⟹ inj_on f (A - B)"
  unfolding inj_on_def by blast

lemma comp_inj_on_iff: "inj_on f A ⟹ inj_on f' (f ` A) ⟷ inj_on (f' ∘ f) A"
  by (auto simp: comp_inj_on inj_on_def)

lemma inj_on_imageI2: "inj_on (f' ∘ f) A ⟹ inj_on f A"
  by (auto simp: comp_inj_on inj_on_def)

lemma inj_img_insertE:
  assumes "inj_on f A"
  assumes "x ∉ B"
    and "insert x B = f ` A"
  obtains x' A' where "x' ∉ A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
proof -
  from assms have "x ∈ f ` A" by auto
  then obtain x' where *: "x' ∈ A" "x = f x'" by auto
  then have A: "A = insert x' (A - {x'})" by auto
  with assms * have B: "B = f ` (A - {x'})" by (auto dest: inj_on_contraD)
  have "x' ∉ A - {x'}" by simp
  from this A ‹x = f x'› B show ?thesis ..
qed

lemma linorder_injI:
  assumes "⋀x y::'a::linorder. x < y ⟹ f x ≠ f y"
  shows "inj f"
   ‹Courtesy of Stephan Merz›
proof (rule inj_onI)
  show "x = y" if "f x = f y" for x y
   by (rule linorder_cases) (auto dest: assms simp: that)
qed

lemma surj_def: "surj f ⟷ (∀y. ∃x. y = f x)"
  by auto

lemma surjI:
  assumes "⋀x. g (f x) = x"
  shows "surj g"
  using assms [symmetric] by auto

lemma 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 ∘ f)"
  by (simp add: image_comp [symmetric])

lemma bij_betw_imageI: "inj_on f A ⟹ f ` A = B ⟹ bij_betw f A B"
  unfolding bij_betw_def by clarify

lemma bij_betw_imp_surj_on: "bij_betw f A B ⟹ f ` A = B"
  unfolding bij_betw_def by clarify

lemma bij_betw_imp_surj: "bij_betw f A UNIV ⟹ surj f"
  unfolding bij_betw_def by auto

lemma bij_betw_empty1: "bij_betw f {} A ⟹ A = {}"
  unfolding bij_betw_def by blast

lemma bij_betw_empty2: "bij_betw f A {} ⟹ A = {}"
  unfolding bij_betw_def by blast

lemma inj_on_imp_bij_betw: "inj_on f A ⟹ bij_betw f A (f ` A)"
  unfolding bij_betw_def by simp

lemma bij_def: "bij f ⟷ inj f ∧ surj f"
  by (rule bij_betw_def)

lemma bijI: "inj f ⟹ surj f ⟹ bij f"
  by (rule bij_betw_imageI)

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 ∘ f) A C"
  by (auto simp add:bij_betw_def comp_inj_on)

lemma bij_comp: "bij f ⟹ bij g ⟹ bij (g ∘ f)"
  by (rule bij_betw_trans)

lemma bij_betw_comp_iff: "bij_betw f A A' ⟹ bij_betw f' A' A'' ⟷ bij_betw (f' ∘ 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 img: "f ` A ≤ A'"
  shows "bij_betw f A A' ⟷ bij_betw (f' ∘ f) A A''"
  using assms
proof (auto simp add: bij_betw_comp_iff)
  assume *: "bij_betw (f' ∘ f) A A''"
  then show "bij_betw f A A'"
    using img
  proof (auto simp add: bij_betw_def)
    assume "inj_on (f' ∘ f) A"
    then show "inj_on f A"
      using inj_on_imageI2 by blast
  next
    fix a'
    assume **: "a' ∈ A'"
    with bij have "f' a' ∈ A''"
      unfolding bij_betw_def by auto
    with * obtain a where 1: "a ∈ A ∧ f' (f a) = f' a'"
      unfolding bij_betw_def by force
    with img have "f a ∈ A'" by auto
    with bij ** 1 have "f a = a'"
      unfolding bij_betw_def inj_on_def by auto
    with 1 show "a' ∈ f ` A" by auto
  qed
qed

lemma bij_betw_inv:
  assumes "bij_betw f A B"
  shows "∃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 = "λb a. a ∈ A ∧ f a = b"
  let ?g = "λb. The (?P b)"
  have g: "?g b = a" if P: "?P b a" for a b
  proof -
    from that s have ex1: "∃a. ?P b a" by blast
    then have uex1: "∃!a. ?P b a" by (blast dest:inj_onD[OF i])
    then show ?thesis
      using the1_equality[OF uex1, OF P] P by simp
  qed
  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" by blast
    from s ‹y ∈ B› obtain a2 where a2: "?P y a2" 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" by blast
    with g[OF P] show "?g b ∈ A" by auto
  next
    fix a
    assume "a ∈ A"
    with s obtain b where P: "?P b a" by blast
    with s have "b ∈ B" by blast
    with g[OF P] show "∃b∈B. a = ?g b" by blast
  qed
  ultimately show ?thesis
    by (auto simp: bij_betw_def)
qed

lemma 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 safe force+  (* somewhat slow *)

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 ⟷ A = B"
  by (auto simp add: bij_betw_def)

lemma bij_betw_combine:
  "bij_betw f A B ⟹ bij_betw f C D ⟹ B ∩ D = {} ⟹ bij_betw f (A ∪ C) (B ∪ D)"
  unfolding bij_betw_def inj_on_Un image_Un by auto

lemma bij_betw_subset: "bij_betw f A A' ⟹ B ⊆ A ⟹ f ` B = B' ⟹ bij_betw f B B'"
  by (auto simp add: bij_betw_def inj_on_def)

lemma bij_pointE:
  assumes "bij f"
  obtains x where "y = f x" and "⋀x'. y = f x' ⟹ x' = x"
proof -
  from assms have "inj f" by (rule bij_is_inj)
  moreover from assms have "surj f" by (rule bij_is_surj)
  then have "y ∈ range f" by simp
  ultimately have "∃!x. y = f x" by (simp add: range_ex1_eq)
  with that show thesis by blast
qed

lemma surj_image_vimage_eq: "surj f ⟹ f ` (f -` A) = A"
  by simp

lemma 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"
  unfolding inj_def by 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"
  unfolding inj_def by blast

lemma vimage_subset_eq: "bij f ⟹ f -` B ⊆ A ⟷ B ⊆ f ` A"
  unfolding bij_def by (blast del: subsetI intro: vimage_subsetI vimage_subsetD)

lemma inj_on_image_eq_iff: "inj_on f C ⟹ A ⊆ C ⟹ B ⊆ C ⟹ f ` A = f ` B ⟷ A = B"
  by (fastforce simp: inj_on_def)

lemma inj_on_Un_image_eq_iff: "inj_on f (A ∪ B) ⟹ f ` A = f ` B ⟷ A = B"
  by (erule inj_on_image_eq_iff) simp_all

lemma inj_on_image_Int: "inj_on f C ⟹ A ⊆ C ⟹ B ⊆ C ⟹ f ` (A ∩ B) = f ` A ∩ f ` B"
  unfolding inj_on_def by blast

lemma inj_on_image_set_diff: "inj_on f C ⟹ A - B ⊆ C ⟹ B ⊆ C ⟹ f ` (A - B) = f ` A - f ` B"
  unfolding inj_on_def by blast

lemma image_Int: "inj f ⟹ f ` (A ∩ B) = f ` A ∩ f ` B"
  unfolding inj_def by blast

lemma image_set_diff: "inj f ⟹ f ` (A - B) = f ` A - f ` B"
  unfolding inj_def by blast

lemma inj_on_image_mem_iff: "inj_on f B ⟹ a ∈ B ⟹ A ⊆ B ⟹ f a ∈ f ` A ⟷ a ∈ A"
  by (auto simp: inj_on_def)

(*FIXME DELETE*)
lemma inj_on_image_mem_iff_alt: "inj_on f B ⟹ A ⊆ B ⟹ f a ∈ f ` A ⟹ a ∈ B ⟹ a ∈ A"
  by (blast dest: inj_onD)

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 (blast dest: injD)

lemma inj_image_eq_iff: "inj f ⟹ f ` A = f ` B ⟷ A = B"
  by (blast dest: injD)

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: inj_def)

lemma bij_image_Compl_eq: "bij f ⟹ f ` (- A) = - (f ` A)"
  by (simp add: bij_def inj_image_Compl_subset surj_Compl_image_subset equalityI)

lemma 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.›
  by (simp add: inj_def) (blast intro: the_equality [symmetric])

lemma 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)

lemma bij_betw_byWitness:
  assumes left: "∀a ∈ A. f' (f a) = a"
    and right: "∀a' ∈ A'. f (f' a') = a'"
    and "f ` A ⊆ A'"
    and img2: "f' ` A' ⊆ A"
  shows "bij_betw f A A'"
  using assms
  unfolding bij_betw_def inj_on_def
proof safe
  fix a b
  assume "a ∈ A" "b ∈ A"
  with left have "a = f' (f a) ∧ b = f' (f b)" by simp
  moreover assume "f a = f b"
  ultimately show "a = b" by simp
next
  fix a' assume *: "a' ∈ A'"
  with img2 have "f' a' ∈ A" by blast
  moreover from * right have "a' = f (f' a')" by simp
  ultimately show "a' ∈ f ` A" by blast
qed

corollary notIn_Un_bij_betw:
  assumes "b ∉ A"
    and "f b ∉ A'"
    and "bij_betw f A A'"
  shows "bij_betw f (A ∪ {b}) (A' ∪ {f b})"
proof -
  have "bij_betw f {b} {f b}"
    unfolding bij_betw_def inj_on_def by simp
  with assms show ?thesis
    using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast
qed

lemma notIn_Un_bij_betw3:
  assumes "b ∉ A"
    and "f b ∉ A'"
  shows "bij_betw f A A' = bij_betw f (A ∪ {b}) (A' ∪ {f b})"
proof
  assume "bij_betw f A A'"
  then show "bij_betw f (A ∪ {b}) (A' ∪ {f b})"
    using assms notIn_Un_bij_betw [of b A f A'] by blast
next
  assume *: "bij_betw f (A ∪ {b}) (A' ∪ {f b})"
  have "f ` A = A'"
  proof auto
    fix a
    assume **: "a ∈ A"
    then have "f a ∈ A' ∪ {f b}"
      using * unfolding bij_betw_def by blast
    moreover
    have False if "f a = f b"
    proof -
      have "a = b"
        using * ** that unfolding bij_betw_def inj_on_def by blast
      with ‹b ∉ A› ** show ?thesis by blast
    qed
    ultimately show "f a ∈ A'" by blast
  next
    fix a'
    assume **: "a' ∈ A'"
    then have "a' ∈ f ` (A ∪ {b})"
      using * by (auto simp add: bij_betw_def)
    then obtain a where 1: "a ∈ A ∪ {b} ∧ f a = a'" by blast
    moreover
    have False if "a = b" using 1 ** ‹f b ∉ A'› that by blast
    ultimately have "a ∈ A" by blast
    with 1 show "a' ∈ f ` A" by blast
  qed
  then show "bij_betw f A A'"
    using * bij_betw_subset[of f "A ∪ {b}" _ A] by blast
qed


subsection ‹Function Updating›

definition fun_upd :: "('a ⇒ 'b) ⇒ 'a ⇒ 'b ⇒ ('a ⇒ 'b)"
  where "fun_upd f a b = (λx. 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 case_sum.
         A nice infix syntax could be defined by
notation
  case_sum  (infixr "'(+')"80)
*)

lemma fun_upd_idem_iff: "f(x:=y) = f ⟷ f x = y"
  unfolding fun_upd_def
  apply safe
   apply (erule subst)
   apply (rule_tac [2] ext)
   apply auto
  done

lemma 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 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: "inj_on f A ⟹ y ∉ f ` A ⟹ inj_on (f(x := y)) A"
  by (auto simp: inj_on_def)

lemma fun_upd_image: "f(x := y) ` A = (if x ∈ A then insert y (f ` (A - {x})) else f ` A)"
  by auto

lemma fun_upd_comp: "f ∘ (g(x := y)) = (f ∘ g)(x := f y)"
  by auto

lemma fun_upd_eqD: "f(x := y) = g(x := z) ⟹ y = z"
  by (simp add: fun_eq_iff split: if_split_asm)


subsection ‹‹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)

lemma override_on_insert: "override_on f g (insert x X) = (override_on f g X)(x:=g x)"
  by (simp add: override_on_def fun_eq_iff)

lemma override_on_insert': "override_on f g (insert x X) = (override_on (f(x:=g x)) g X)"
  by (simp add: override_on_def fun_eq_iff)


subsection ‹‹swap››

definition swap :: "'a ⇒ 'a ⇒ ('a ⇒ 'b) ⇒ ('a ⇒ 'b)"
  where "swap a b f = f (a := f b, b:= f a)"

lemma swap_apply [simp]:
  "swap a b f a = f b"
  "swap a b f b = f a"
  "c ≠ a ⟹ c ≠ b ⟹ swap a b f c = f c"
  by (simp_all add: swap_def)

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 (simp add: fun_upd_def swap_def fun_eq_iff)

lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
  by (rule ext) (simp add: fun_upd_def swap_def)

lemma swap_comp_involutory [simp]: "swap a b ∘ swap a b = id"
  by (rule ext) simp

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 ∘ swap a b g = swap a b (f ∘ 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 auto
qed

lemma inj_on_imp_inj_on_swap: "inj_on f A ⟹ a ∈ A ⟹ b ∈ A ⟹ inj_on (swap a b f) A"
  by (auto simp add: inj_on_def swap_def)

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)
  then show "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 ⟹ surj (swap a b f)"
  by simp

lemma surj_swap_iff [simp]: "surj (swap a b f) ⟷ surj f"
  by simp

lemma 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 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 = (λx. 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"
  unfolding the_inv_into_def inj_on_def by blast

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)
  apply 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 ∘ g) x = (the_inv_into A g ∘ 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 ⟹ inj_on (the_inv_into A f) (f ` A)"
  by (auto intro: inj_onI simp: 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: "the_inv f (f x) = x" if "inj f"
  using that UNIV_I by (rule the_inv_into_f_f)


subsection ‹Cantor's Paradox›

theorem Cantors_paradox: "∄f. f ` A = Pow A"
proof
  assume "∃f. f ` A = Pow A"
  then obtain f where f: "f ` A = Pow A" ..
  let ?X = "{a ∈ A. a ∉ f a}"
  have "?X ∈ Pow A" by blast
  then have "?X ∈ f ` A" by (simp only: f)
  then obtain x where "x ∈ A" and "f x = ?X" by blast
  then show False by blast
qed


subsection ‹Setup›

subsubsection ‹Proof tools›

text ‹Simplify 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

    val ss = simpset_of @{context}

    fun proc ctxt ct =
      let
        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 _ =>
                resolve_tac ctxt [eq_reflection] 1 THEN
                resolve_tac ctxt @{thms ext} 1 THEN
                simp_tac (put_simpset ss ctxt) 1)))
      end
  in proc end
›


subsubsection ‹Functorial structure of types›

ML_file "Tools/functor.ML"

functor map_fun: map_fun
  by (simp_all add: fun_eq_iff)

functor vimage
  by (simp_all add: fun_eq_iff vimage_comp)


text ‹Legacy theorem names›

lemmas o_def = comp_def
lemmas o_apply = comp_apply
lemmas o_assoc = comp_assoc [symmetric]
lemmas id_o = id_comp
lemmas o_id = comp_id
lemmas o_eq_dest = comp_eq_dest
lemmas o_eq_elim = comp_eq_elim
lemmas o_eq_dest_lhs = comp_eq_dest_lhs
lemmas o_eq_id_dest = comp_eq_id_dest

end