Theory Fun

(*  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_defn
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: "xS. ∃!y. Q x y  f. xS. 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 ((`) 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  (xA. yA. 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_compose: "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_iff_Uniq: "inj_on f A  (xA. 1y. yA  f x = f y)"
  by (auto simp: Uniq_def inj_on_def)

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 bij_betwE: "bij_betw f A B  aA. f a  B"
  unfolding bij_betw_def by auto

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:
  "xA. yA. 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_inj_onI:
  fixes A :: "'a::order set"
  assumes ne: "x y. x < y; xA; yA  f x  f y" and lin: "x y. xA; yA  xy  yx"
  shows "inj_on f A"
proof (rule inj_onI)
  fix x y
  assume eq: "f x = f y" and "xA" "yA"
  then show "x = y"
    using lin [of x y] ne by (force simp: dual_order.order_iff_strict)
qed

lemma linorder_injI:
  assumes "x y::'a::linorder. x < y  f x  f y"
  shows "inj f"
    ― ‹Courtesy of Stephan Merz
using assms by (auto intro: linorder_inj_onI linear)

lemma inj_on_image_Pow: "inj_on f A inj_on (image f) (Pow A)"
  unfolding Pow_def inj_on_def by blast

lemma bij_betw_image_Pow: "bij_betw f A B  bij_betw (image f) (Pow A) (Pow B)"
  by (auto simp add: bij_betw_def inj_on_image_Pow image_Pow_surj)

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)"
  using image_comp [of g f UNIV] by simp

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_betw_apply: "bij_betw f A B; a  A  f a  B"
  unfolding bij_betw_def by auto

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 "bB. 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_betw_ball: "bij_betw f A B  (b  B. phi b) = (a  A. phi (f a))"
  unfolding bij_betw_def inj_on_def by blast

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 bij_iff: contributor Amine Chaieb
  bij f  (x. ∃!y. f y = x)  (is ?P  ?Q)
proof
  assume ?P
  then have inj f surj f
    by (simp_all add: bij_def)
  show ?Q
  proof
    fix y
    from surj f obtain x where y = f x
      by (auto simp add: surj_def)
    with inj f show ∃!x. f x = y
      by (auto simp add: inj_def)
  qed
next
  assume ?Q
  then have inj f
    by (auto simp add: inj_def)
  moreover have x. y = f x for y
  proof -
    from ?Q obtain x where f x = y
      by blast
    then have y = f x
      by simp
    then show ?thesis ..
  qed
  then have surj f
    by (auto simp add: surj_def)
  ultimately show ?P
    by (rule bijI)
qed

lemma bij_betw_partition:
  bij_betw f A B
  if bij_betw f (A  C) (B  D) bij_betw f C D A  C = {} B  D = {}
proof -
  from that have inj_on f (A  C) inj_on f C f ` (A  C) = B  D f ` C = D
    by (simp_all add: bij_betw_def)
  then have inj_on f A and f ` (A - C)  f ` (C - A) = {}
    by (simp_all add: inj_on_Un)
  with A  C = {} have f ` A  f ` C = {}
    by auto
  with f ` (A  C) = B  D f ` C = D  B  D = {}
  have f ` A = B
    by blast
  with inj_on f A show ?thesis
    by (simp add: bij_betw_def)
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)

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

lemma inj_on_disjoint_Un:
  assumes "inj_on f A" and "inj_on g B" 
  and "f ` A  g ` B = {}"
  shows "inj_on (λx. if x  A then f x else g x) (A  B)"
  using assms by (simp add: inj_on_def disjoint_iff) (blast)

lemma bij_betw_disjoint_Un:
  assumes "bij_betw f A C" and "bij_betw g B D" 
  and "A  B = {}"
  and "C  D = {}"
  shows "bij_betw (λx. if x  A then f x else g x) (A  B) (C  D)"
  using assms by (auto simp: inj_on_disjoint_Un bij_betw_def)

lemma involuntory_imp_bij:
  bij f if x. f (f x) = x
proof (rule bijI)
  from that show surj f
    by (rule surjI)
  show inj f
  proof (rule injI)
    fix x y
    assume f x = f y
    then have f (f x) = f (f y)
      by simp
    then show x = y
      by (simp add: that)
  qed
qed


subsubsection Important examples

context cancel_semigroup_add
begin

lemma inj_on_add [simp]:
  "inj_on ((+) a) A"
  by (rule inj_onI) simp

lemma inj_add_left:
  inj ((+) a)
  by simp

lemma inj_on_add' [simp]:
  "inj_on (λb. b + a) A"
  by (rule inj_onI) simp

lemma bij_betw_add [simp]:
  "bij_betw ((+) a) A B  (+) a ` A = B"
  by (simp add: bij_betw_def)

end

context ab_group_add
begin

lemma surj_plus [simp]:
  "surj ((+) a)"
  by (auto intro!: range_eqI [of b "(+) a" "b - a" for b]) (simp add: algebra_simps)

lemma inj_diff_right [simp]:
  inj (λb. b - a)
proof -
  have inj ((+) (- a))
    by (fact inj_add_left)
  also have (+) (- a) = (λb. b - a)
    by (simp add: fun_eq_iff)
  finally show ?thesis .
qed

lemma surj_diff_right [simp]:
  "surj (λx. x - a)"
  using surj_plus [of "- a"] by (simp cong: image_cong_simp)

lemma translation_Compl:
  "(+) a ` (- t) = - ((+) a ` t)"
proof (rule set_eqI)
  fix b
  show "b  (+) a ` (- t)  b  - (+) a ` t"
    by (auto simp: image_iff algebra_simps intro!: bexI [of _ "b - a"])
qed

lemma translation_subtract_Compl:
  "(λx. x - a) ` (- t) = - ((λx. x - a) ` t)"
  using translation_Compl [of "- a" t] by (simp cong: image_cong_simp)

lemma translation_diff:
  "(+) a ` (s - t) = ((+) a ` s) - ((+) a ` t)"
  by auto

lemma translation_subtract_diff:
  "(λx. x - a) ` (s - t) = ((λx. x - a) ` s) - ((λx. x - a) ` t)"
  using translation_diff [of "- a"] by (simp cong: image_cong_simp)

lemma translation_Int:
  "(+) a ` (s  t) = ((+) a ` s)  ((+) a ` t)"
  by auto

lemma translation_subtract_Int:
  "(λx. x - a) ` (s  t) = ((λx. x - a) ` s)  ((λx. x - a) ` t)"
  using translation_Int [of " -a"] by (simp cong: image_cong_simp)

end


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 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 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"
  unfolding the_inv_into_def
  by (rule the1I2; blast dest: inj_onD)

lemma f_the_inv_into_f_bij_betw:
  "bij_betw f A B  (bij_betw f A B  x  B)  f (the_inv_into A f x) = x"
  unfolding bij_betw_def by (blast intro: f_the_inv_into_f)

lemma the_inv_into_into: "inj_on f A  x  f ` A  A  B  the_inv_into A f x  B"
  unfolding the_inv_into_def
  by (rule the1I2; blast dest: inj_onD)

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"
  by (force simp add: the_inv_into_f_f)

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)

lemma bij_betw_iff_bijections:
  "bij_betw f A B  (g. (x  A. f x  B  g(f x) = x)  (y  B. g y  A  f(g y) = y))"
  (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then show ?rhs
    apply (rule_tac x="the_inv_into A f" in exI)
    apply (auto simp: bij_betw_def f_the_inv_into_f the_inv_into_f_f the_inv_into_into)
    done
qed (force intro: bij_betw_byWitness)

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 Monotonic functions over a set

definition monotone_on :: "'a set  ('a  'a  bool)  ('b  'b  bool)  ('a  'b)  bool"
  where "monotone_on A orda ordb f  (xA. yA. orda x y  ordb (f x) (f y))"

abbreviation monotone :: "('a  'a  bool)  ('b  'b  bool)  ('a  'b)  bool"
  where "monotone  monotone_on UNIV"

lemma monotone_def[no_atp]: "monotone orda ordb f  (x y. orda x y  ordb (f x) (f y))"
  by (simp add: monotone_on_def)

text Lemma @{thm [source] monotone_def} is provided for backward compatibility.

lemma monotone_onI:
  "(x y. x  A  y  A  orda x y  ordb (f x) (f y))  monotone_on A orda ordb f"
  by (simp add: monotone_on_def)

lemma monotoneI[intro?]: "(x y. orda x y  ordb (f x) (f y))  monotone orda ordb f"
  by (rule monotone_onI)

lemma monotone_onD:
  "monotone_on A orda ordb f  x  A  y  A  orda x y  ordb (f x) (f y)"
  by (simp add: monotone_on_def)

lemma monotoneD[dest?]: "monotone orda ordb f  orda x y  ordb (f x) (f y)"
  by (rule monotone_onD[of UNIV, simplified])

lemma monotone_on_subset: "monotone_on A orda ordb f  B  A  monotone_on B orda ordb f"
  by (auto intro: monotone_onI dest: monotone_onD)

lemma monotone_on_empty[simp]: "monotone_on {} orda ordb f"
  by (auto intro: monotone_onI dest: monotone_onD)

lemma monotone_on_o:
  assumes
    mono_f: "monotone_on A orda ordb f" and
    mono_g: "monotone_on B ordc orda g" and
    "g ` B  A"
  shows "monotone_on B ordc ordb (f  g)"
proof (rule monotone_onI)
  fix x y assume "x  B" and "y  B" and "ordc x y"
  hence "orda (g x) (g y)"
    by (rule mono_g[THEN monotone_onD])
  moreover from g ` B  A x  B y  B have "g x  A" and "g y  A"
    unfolding image_subset_iff by simp_all
  ultimately show "ordb ((f  g) x) ((f  g) y)"
    using mono_f[THEN monotone_onD] by simp
qed

abbreviation mono_on :: "('a :: ord) set  ('a  'b :: ord)  bool"
  where "mono_on A  monotone_on A (≤) (≤)"

lemma mono_on_def: "mono_on A f  (r s. r  A  s  A  r  s  f r  f s)"
  by (auto simp add: monotone_on_def)

lemma mono_onI:
  "(r s. r  A  s  A  r  s  f r  f s)  mono_on A f"
  by (rule monotone_onI)

lemma mono_onD:
  "mono_on A f; r  A; s  A; r  s  f r  f s"
  by (rule monotone_onD)

lemma mono_imp_mono_on: "mono f  mono_on A f"
  unfolding mono_def mono_on_def by auto

lemma mono_on_subset: "mono_on A f  B  A  mono_on B f"
  by (rule monotone_on_subset)

abbreviation strict_mono_on :: "('a :: ord) set  ('a  'b :: ord)  bool"
  where "strict_mono_on A  monotone_on A (<) (<)"

lemma strict_mono_on_def: "strict_mono_on A f  (r s. r  A  s  A  r < s  f r < f s)"
  by (auto simp add: monotone_on_def)

lemma strict_mono_onI:
  "(r s. r  A  s  A  r < s  f r < f s)  strict_mono_on A f"
  by (rule monotone_onI)

lemma strict_mono_onD:
  "strict_mono_on A f; r  A; s  A; r < s  f r < f s"
  by (rule monotone_onD)

lemma mono_on_greaterD:
  assumes "mono_on A g" "x  A" "y  A" "g x > (g (y::_::linorder) :: _ :: linorder)"
  shows "x > y"
proof (rule ccontr)
  assume "¬x > y"
  hence "x  y" by (simp add: not_less)
  from assms(1-3) and this have "g x  g y" by (rule mono_onD)
  with assms(4) show False by simp
qed

lemma strict_mono_inv:
  fixes f :: "('a::linorder)  ('b::linorder)"
  assumes "strict_mono f" and "surj f" and inv: "x. g (f x) = x"
  shows "strict_mono g"
proof
  fix x y :: 'b assume "x < y"
  from surj f obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
  with x < y and strict_mono f have "x' < y'" by (simp add: strict_mono_less)
  with inv show "g x < g y" by simp
qed

lemma strict_mono_on_imp_inj_on:
  assumes "strict_mono_on A (f :: (_ :: linorder)  (_ :: preorder))"
  shows "inj_on f A"
proof (rule inj_onI)
  fix x y assume "x  A" "y  A" "f x = f y"
  thus "x = y"
    by (cases x y rule: linorder_cases)
       (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
qed

lemma strict_mono_on_leD:
  assumes "strict_mono_on A (f :: (_ :: linorder)  _ :: preorder)" "x  A" "y  A" "x  y"
  shows "f x  f y"
proof (insert le_less_linear[of y x], elim disjE)
  assume "x < y"
  with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all
  thus ?thesis by (rule less_imp_le)
qed (insert assms, simp)

lemma strict_mono_on_eqD:
  fixes f :: "(_ :: linorder)  (_ :: preorder)"
  assumes "strict_mono_on A f" "f x = f y" "x  A" "y  A"
  shows "y = x"
  using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)

lemma strict_mono_on_imp_mono_on:
  "strict_mono_on A (f :: (_ :: linorder)  _ :: preorder)  mono_on A f"
  by (rule mono_onI, rule strict_mono_on_leD)


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_namefun_upd, T) $ f $ x $ y)
    fun dest_fun_T1 (Type (_, T :: Ts)) = T
    fun find_double (t as Const (const_namefun_upd,T) $ f $ x $ y) =
      let
        fun find (Const (const_namefun_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