src/HOL/Fun.thy

more zproofs;

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

section \<open>Notions about functions\<close>

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

lemma apply_inverse: "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
  by auto

text \<open>Uniqueness, so NOT the axiom of choice.\<close>
lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
  by (force intro: theI')

lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
  by (force intro: theI')


subsection \<open>The Identity Function \<open>id\<close>\<close>

definition id :: "'a \<Rightarrow> 'a"
  where "id = (\<lambda>x. 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: "(\<forall>x. f x = x) \<longleftrightarrow> f = id"
  by auto

code_printing
  constant id \<rightharpoonup> (Haskell) "id"


subsection \<open>The Composition Operator \<open>f \<circ> g\<close>\<close>

definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c"  (infixl "\<circ>" 55)
  where "f \<circ> g = (\<lambda>x. f (g x))"

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

lemma comp_apply [simp]: "(f \<circ> g) x = f (g x)"
  by (simp add: comp_def)

lemma comp_assoc: "(f \<circ> g) \<circ> h = f \<circ> (g \<circ> h)"
  by (simp add: fun_eq_iff)

lemma id_comp [simp]: "id \<circ> g = g"
  by (simp add: fun_eq_iff)

lemma comp_id [simp]: "f \<circ> id = f"
  by (simp add: fun_eq_iff)

lemma comp_eq_dest: "a \<circ> b = c \<circ> d \<Longrightarrow> a (b v) = c (d v)"
  by (simp add: fun_eq_iff)

lemma comp_eq_elim: "a \<circ> b = c \<circ> d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
  by (simp add: fun_eq_iff)

lemma comp_eq_dest_lhs: "a \<circ> b = c \<Longrightarrow> a (b v) = c v"
  by clarsimp

lemma comp_eq_id_dest: "a \<circ> b = id \<circ> c \<Longrightarrow> a (b v) = c v"
  by clarsimp

lemma image_comp: "f ` (g ` r) = (f \<circ> g) ` r"
  by auto

lemma vimage_comp: "f -` (g -` x) = (g \<circ> f) -` x"
  by auto

lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h \<circ> f) ` A = (h \<circ> g) ` B"
  by (auto simp: comp_def elim!: equalityE)

lemma image_bind: "f ` (Set.bind A g) = Set.bind A ((`) f \<circ> g)"
  by (auto simp add: Set.bind_def)

lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"
  by (auto simp add: Set.bind_def)

lemma (in group_add) minus_comp_minus [simp]: "uminus \<circ> uminus = id"
  by (simp add: fun_eq_iff)

lemma (in boolean_algebra) minus_comp_minus [simp]: "uminus \<circ> uminus = id"
  by (simp add: fun_eq_iff)

code_printing
  constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "."


subsection \<open>The Forward Composition Operator \<open>fcomp\<close>\<close>

definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"  (infixl "\<circ>>" 60)
  where "f \<circ>> g = (\<lambda>x. g (f x))"

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

lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
  by (simp add: fcomp_def)

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

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

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

code_printing
  constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>"

no_notation fcomp (infixl "\<circ>>" 60)


subsection \<open>Mapping functions\<close>

definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd"
  where "map_fun f g h = g \<circ> h \<circ> f"

lemma map_fun_apply [simp]: "map_fun f g h x = g (h (f x))"
  by (simp add: map_fun_def)


subsection \<open>Injectivity and Bijectivity\<close>

definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool"  \<comment> \<open>injective\<close>
  where "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"

definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"  \<comment> \<open>bijective\<close>
  where "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"

text \<open>
  A common special case: functions injective, surjective or bijective over
  the entire domain type.
\<close>

abbreviation inj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "inj f \<equiv> inj_on f UNIV"

abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "surj f \<equiv> range f = UNIV"

translations \<comment> \<open>The negated case:\<close>
  "\<not> CONST surj f" \<leftharpoondown> "CONST range f \<noteq> CONST UNIV"

abbreviation bij :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "bij f \<equiv> bij_betw f UNIV UNIV"

lemma inj_def: "inj f \<longleftrightarrow> (\<forall>x y. f x = f y \<longrightarrow> x = y)"
  unfolding inj_on_def by blast

lemma injI: "(\<And>x y. f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj f"
  unfolding inj_def by blast

theorem range_ex1_eq: "inj f \<Longrightarrow> b \<in> range f \<longleftrightarrow> (\<exists>!x. b = f x)"
  unfolding inj_def by blast

lemma injD: "inj f \<Longrightarrow> f x = f y \<Longrightarrow> x = y"
  by (simp add: inj_def)

lemma inj_on_eq_iff: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
  by (auto simp: inj_on_def)

lemma inj_on_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A \<longleftrightarrow> inj_on g A"
  by (auto simp: inj_on_def)

lemma image_strict_mono: "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B"
  unfolding inj_on_def by blast

lemma inj_compose: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
  by (simp add: inj_def)

lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
  by (simp add: inj_def fun_eq_iff)

lemma inj_eq: "inj f \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"
  by (simp add: inj_on_eq_iff)

lemma inj_on_iff_Uniq: "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<exists>\<^sub>\<le>\<^sub>1y. y\<in>A \<and> 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 (\<lambda>x. x) A"
  by (simp add: inj_on_def)

lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> 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 \<Rightarrow> 'a::group_add)"
  unfolding bij_betw_def inj_on_def
  by (force intro: minus_minus [symmetric])

lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
  unfolding bij_betw_def by auto

lemma inj_onI [intro?]: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj_on f A"
  by (simp add: inj_on_def)

text \<open>For those frequent proofs by contradiction\<close>
lemma inj_onCI: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> x \<noteq> y \<Longrightarrow> False) \<Longrightarrow> inj_on f A"
  by (force simp: inj_on_def)

lemma inj_on_inverseI: "(\<And>x. x \<in> A \<Longrightarrow> g (f x) = x) \<Longrightarrow> inj_on f A"
  by (auto dest: arg_cong [of concl: g] simp add: inj_on_def)

lemma inj_onD: "inj_on f A \<Longrightarrow> f x = f y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y"
  unfolding inj_on_def by blast

lemma inj_on_subset:
  assumes "inj_on f A"
    and "B \<subseteq> A"
  shows "inj_on f B"
proof (rule inj_onI)
  fix a b
  assume "a \<in> B" and "b \<in> B"
  with assms have "a \<in> A" and "b \<in> 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 \<Longrightarrow> inj_on g (f ` A) \<Longrightarrow> inj_on (g \<circ> f) A"
  by (simp add: comp_def inj_on_def)

lemma inj_on_imageI: "inj_on (g \<circ> f) A \<Longrightarrow> inj_on g (f ` A)"
  by (auto simp add: inj_on_def)

lemma inj_on_image_iff:
  "\<forall>x\<in>A. \<forall>y\<in>A. g (f x) = g (f y) \<longleftrightarrow> g x = g y \<Longrightarrow> inj_on f A \<Longrightarrow> inj_on g (f ` A) \<longleftrightarrow> inj_on g A"
  unfolding inj_on_def by blast

lemma inj_on_contraD: "inj_on f A \<Longrightarrow> x \<noteq> y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x \<noteq> f y"
  unfolding inj_on_def by blast

lemma inj_singleton [simp]: "inj_on (\<lambda>x. {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 \<Longrightarrow> A \<subseteq> B \<Longrightarrow> inj_on f A"
  unfolding inj_on_def by blast

lemma inj_on_Un: "inj_on f (A \<union> B) \<longleftrightarrow> inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {}"
  unfolding inj_on_def by (blast intro: sym)

lemma inj_on_insert [iff]: "inj_on f (insert a A) \<longleftrightarrow> inj_on f A \<and> f a \<notin> f ` (A - {a})"
  unfolding inj_on_def by (blast intro: sym)

lemma inj_on_diff: "inj_on f A \<Longrightarrow> inj_on f (A - B)"
  unfolding inj_on_def by blast

lemma comp_inj_on_iff: "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' \<circ> f) A"
  by (auto simp: comp_inj_on inj_on_def)

lemma inj_on_imageI2: "inj_on (f' \<circ> f) A \<Longrightarrow> inj_on f A"
  by (auto simp: comp_inj_on inj_on_def)

lemma inj_img_insertE:
  assumes "inj_on f A"
  assumes "x \<notin> B"
    and "insert x B = f ` A"
  obtains x' A' where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'"
proof -
  from assms have "x \<in> f ` A" by auto
  then obtain x' where *: "x' \<in> 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' \<notin> A - {x'}" by simp
  from this A \<open>x = f x'\<close> B show ?thesis ..
qed

lemma linorder_inj_onI:
  fixes A :: "'a::order set"
  assumes ne: "\<And>x y. \<lbrakk>x < y; x\<in>A; y\<in>A\<rbrakk> \<Longrightarrow> f x \<noteq> f y" and lin: "\<And>x y. \<lbrakk>x\<in>A; y\<in>A\<rbrakk> \<Longrightarrow> x\<le>y \<or> y\<le>x"
  shows "inj_on f A"
proof (rule inj_onI)
  fix x y
  assume eq: "f x = f y" and "x\<in>A" "y\<in>A"
  then show "x = y"
    using lin [of x y] ne by (force simp: dual_order.order_iff_strict)
qed

lemma linorder_inj_onI':
  fixes A :: "'a :: linorder set"
  assumes "\<And>i j. i \<in> A \<Longrightarrow> j \<in> A \<Longrightarrow> i < j \<Longrightarrow> f i \<noteq> f j"
  shows   "inj_on f A"
  by (intro linorder_inj_onI) (auto simp add: assms)

lemma linorder_injI:
  assumes "\<And>x y::'a::linorder. x < y \<Longrightarrow> f x \<noteq> f y"
  shows "inj f"
    \<comment> \<open>Courtesy of Stephan Merz\<close>
using assms by (simp add: linorder_inj_onI')

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

lemma bij_betw_image_Pow: "bij_betw f A B \<Longrightarrow> 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 \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
  by auto

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

lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
  by (simp add: surj_def)

lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
  by (simp add: surj_def) blast

lemma comp_surj: "surj f \<Longrightarrow> surj g \<Longrightarrow> surj (g \<circ> f)"
  using image_comp [of g f UNIV] by simp

lemma bij_betw_imageI: "inj_on f A \<Longrightarrow> f ` A = B \<Longrightarrow> bij_betw f A B"
  unfolding bij_betw_def by clarify

lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B"
  unfolding bij_betw_def by clarify

lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
  unfolding bij_betw_def by auto

lemma bij_betw_empty1: "bij_betw f {} A \<Longrightarrow> A = {}"
  unfolding bij_betw_def by blast

lemma bij_betw_empty2: "bij_betw f A {} \<Longrightarrow> A = {}"
  unfolding bij_betw_def by blast

lemma inj_on_imp_bij_betw: "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
  unfolding bij_betw_def by simp

lemma bij_betw_DiffI:
  assumes "bij_betw f A B" "bij_betw f C D" "C \<subseteq> A" "D \<subseteq> B"
  shows   "bij_betw f (A - C) (B - D)"
  using assms unfolding bij_betw_def inj_on_def by auto

lemma bij_betw_singleton_iff [simp]: "bij_betw f {x} {y} \<longleftrightarrow> f x = y"
  by (auto simp: bij_betw_def)

lemma bij_betw_singletonI [intro]: "f x = y \<Longrightarrow> bij_betw f {x} {y}"
  by auto

lemma bij_betw_apply: "\<lbrakk>bij_betw f A B; a \<in> A\<rbrakk> \<Longrightarrow> f a \<in> B"
  unfolding bij_betw_def by auto

lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
  by (rule bij_betw_def)

lemma bijI: "inj f \<Longrightarrow> surj f \<Longrightarrow> bij f"
  by (rule bij_betw_imageI)

lemma bij_is_inj: "bij f \<Longrightarrow> inj f"
  by (simp add: bij_def)

lemma bij_is_surj: "bij f \<Longrightarrow> surj f"
  by (simp add: bij_def)

lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
  by (simp add: bij_betw_def)

lemma bij_betw_trans: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g \<circ> f) A C"
  by (auto simp add:bij_betw_def comp_inj_on)

lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g \<circ> f)"
  by (rule bij_betw_trans)

lemma bij_betw_comp_iff: "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' \<circ> 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 \<le> A'"
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''" (is "?L \<longleftrightarrow> ?R")
proof
  assume "?L"
  then show "?R"
    using assms by (auto simp add: bij_betw_comp_iff)
  next
    assume *: "?R"
    have "inj_on (f' \<circ> f) A \<Longrightarrow> inj_on f A"
      using inj_on_imageI2 by blast
    moreover have "A' \<subseteq> f ` A"
    proof
      fix a'
      assume **: "a' \<in> A'"
      with bij have "f' a' \<in> A''"
        unfolding bij_betw_def by auto
      with * obtain a where 1: "a \<in> A \<and> f' (f a) = f' a'"
        unfolding bij_betw_def by force
      with img have "f a \<in> A'" by auto
      with bij ** 1 have "f a = a'"
        unfolding bij_betw_def inj_on_def by auto
      with 1 show "a' \<in> f ` A" by auto
    qed
    ultimately show "?L"
      using img * by (auto simp add: bij_betw_def)
qed

lemma bij_betw_inv:
  assumes "bij_betw f A B"
  shows "\<exists>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 = "\<lambda>b a. a \<in> A \<and> f a = b"
  let ?g = "\<lambda>b. The (?P b)"
  have g: "?g b = a" if P: "?P b a" for a b
  proof -
    from that s have ex1: "\<exists>a. ?P b a" by blast
    then have uex1: "\<exists>!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 \<in> B" "y \<in> B" "?g x = ?g y"
    from s \<open>x \<in> B\<close> obtain a1 where a1: "?P x a1" by blast
    from s \<open>y \<in> B\<close> obtain a2 where a2: "?P y a2" by blast
    from g [OF a1] a1 g [OF a2] a2 \<open>?g x = ?g y\<close> show "x = y" by simp
  qed
  moreover have "?g ` B = A"
  proof safe
    fix b
    assume "b \<in> B"
    with s obtain a where P: "?P b a" by blast
    with g[OF P] show "?g b \<in> A" by auto
  next
    fix a
    assume "a \<in> A"
    with s obtain b where P: "?P b a" by blast
    with s have "b \<in> B" by blast
    with g[OF P] have "\<exists>b\<in>B. a = ?g b" by blast
    then show "a \<in> ?g ` B"
      by auto
  qed
  ultimately show ?thesis
    by (auto simp: bij_betw_def)
qed

lemma bij_betw_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
  unfolding bij_betw_def inj_on_def by 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 \<longleftrightarrow> A = B"
  by (auto simp add: bij_betw_def)

lemma bij_betw_combine:
  "bij_betw f A B \<Longrightarrow> bij_betw f C D \<Longrightarrow> B \<inter> D = {} \<Longrightarrow> bij_betw f (A \<union> C) (B \<union> D)"
  unfolding bij_betw_def inj_on_Un image_Un by auto

lemma bij_betw_subset: "bij_betw f A A' \<Longrightarrow> B \<subseteq> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw f B B'"
  by (auto simp add: bij_betw_def inj_on_def)

lemma bij_betw_ball: "bij_betw f A B \<Longrightarrow> (\<forall>b \<in> B. phi b) = (\<forall>a \<in> 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 "\<And>x'. y = f x' \<Longrightarrow> 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 \<in> range f" by simp
  ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq)
  with that show thesis by blast
qed

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

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

lemma surj_image_vimage_eq: "surj f \<Longrightarrow> f ` (f -` A) = A"
  by simp

lemma surj_vimage_empty:
  assumes "surj f"
  shows "f -` A = {} \<longleftrightarrow> A = {}"
  using surj_image_vimage_eq [OF \<open>surj f\<close>, of A]
  by (intro iffI) fastforce+

lemma inj_vimage_image_eq: "inj f \<Longrightarrow> f -` (f ` A) = A"
  unfolding inj_def by blast

lemma vimage_subsetD: "surj f \<Longrightarrow> f -` B \<subseteq> A \<Longrightarrow> B \<subseteq> f ` A"
  by (blast intro: sym)

lemma vimage_subsetI: "inj f \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> f -` B \<subseteq> A"
  unfolding inj_def by blast

lemma vimage_subset_eq: "bij f \<Longrightarrow> f -` B \<subseteq> A \<longleftrightarrow> B \<subseteq> f ` A"
  unfolding bij_def by (blast del: subsetI intro: vimage_subsetI vimage_subsetD)

lemma inj_on_image_eq_iff: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
  by (fastforce simp: inj_on_def)

lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
  by (erule inj_on_image_eq_iff) simp_all

lemma inj_on_image_Int: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B"
  unfolding inj_on_def by blast

lemma inj_on_image_set_diff: "inj_on f C \<Longrightarrow> A - B \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A - B) = f ` A - f ` B"
  unfolding inj_on_def by blast

lemma image_Int: "inj f \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B"
  unfolding inj_def by blast

lemma image_set_diff: "inj f \<Longrightarrow> f ` (A - B) = f ` A - f ` B"
  unfolding inj_def by blast

lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> a \<in> B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A"
  by (auto simp: inj_on_def)

lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A"
  by (blast dest: injD)

lemma inj_image_subset_iff: "inj f \<Longrightarrow> f ` A \<subseteq> f ` B \<longleftrightarrow> A \<subseteq> B"
  by (blast dest: injD)

lemma inj_image_eq_iff: "inj f \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
  by (blast dest: injD)

lemma surj_Compl_image_subset: "surj f \<Longrightarrow> - (f ` A) \<subseteq> f ` (- A)"
  by auto

lemma inj_image_Compl_subset: "inj f \<Longrightarrow> f ` (- A) \<subseteq> - (f ` A)"
  by (auto simp: inj_def)

lemma bij_image_Compl_eq: "bij f \<Longrightarrow> f ` (- A) = - (f ` A)"
  by (simp add: bij_def inj_image_Compl_subset surj_Compl_image_subset equalityI)

lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
  \<comment> \<open>The inverse image of a singleton under an injective function is included in a singleton.\<close>
  by (simp add: inj_def) (blast intro: the_equality [symmetric])

lemma inj_on_vimage_singleton: "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
  by (auto simp add: inj_on_def intro: the_equality [symmetric])

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

corollary notIn_Un_bij_betw:
  assumes "b \<notin> A"
    and "f b \<notin> A'"
    and "bij_betw f A A'"
  shows "bij_betw f (A \<union> {b}) (A' \<union> {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 \<notin> A"
    and "f b \<notin> A'"
  shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
proof
  assume "bij_betw f A A'"
  then show "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
    using assms notIn_Un_bij_betw [of b A f A'] by blast
next
  assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
  have "f ` A = A'"
  proof safe
    fix a
    assume **: "a \<in> A"
    then have "f a \<in> A' \<union> {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 \<open>b \<notin> A\<close> ** show ?thesis by blast
    qed
    ultimately show "f a \<in> A'" by blast
  next
    fix a'
    assume **: "a' \<in> A'"
    then have "a' \<in> f ` (A \<union> {b})"
      using * by (auto simp add: bij_betw_def)
    then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast
    moreover
    have False if "a = b" using 1 ** \<open>f b \<notin> A'\<close> that by blast
    ultimately have "a \<in> A" by blast
    with 1 show "a' \<in> f ` A" by blast
  qed
  then show "bij_betw f A A'"
    using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast
qed

lemma inj_on_disjoint_Un:
  assumes "inj_on f A" and "inj_on g B" 
  and "f ` A \<inter> g ` B = {}"
  shows "inj_on (\<lambda>x. if x \<in> A then f x else g x) (A \<union> 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 \<inter> B = {}"
  and "C \<inter> D = {}"
  shows "bij_betw (\<lambda>x. if x \<in> A then f x else g x) (A \<union> B) (C \<union> D)"
  using assms by (auto simp: inj_on_disjoint_Un bij_betw_def)

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


subsubsection \<open>Inj/surj/bij of Algebraic Operations\<close>

context cancel_semigroup_add
begin

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

lemma inj_on_add' [simp]:
  "inj_on (\<lambda>b. b + a) A"
  by (rule inj_onI) simp

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

end

context group_add
begin

lemma diff_left_imp_eq: "a - b = a - c \<Longrightarrow> b = c"
unfolding add_uminus_conv_diff[symmetric]
by(drule local.add_left_imp_eq) simp

lemma inj_uminus[simp, intro]: "inj_on uminus A"
  by (auto intro!: inj_onI)

lemma surj_uminus[simp]: "surj uminus"
using surjI minus_minus by blast

lemma surj_plus [simp]:
  "surj ((+) a)"
proof (standard, simp, standard, simp)
  fix x
  have "x = a + (-a + x)" by (simp add: add.assoc)
  thus "x \<in> range ((+) a)" by blast
qed

lemma surj_plus_right [simp]:
  "surj (\<lambda>b. b+a)"
proof (standard, simp, standard, simp)
  fix b show "b \<in> range (\<lambda>b. b+a)"
    using diff_add_cancel[of b a, symmetric] by blast
qed

lemma inj_on_diff_left [simp]:
  \<open>inj_on ((-) a) A\<close>
by (auto intro: inj_onI dest!: diff_left_imp_eq)

lemma inj_on_diff_right [simp]:
  \<open>inj_on (\<lambda>b. b - a) A\<close>
by (auto intro: inj_onI simp add: algebra_simps)

lemma surj_diff [simp]:
  "surj ((-) a)"
proof (standard, simp, standard, simp)
  fix x
  have "x = a - (- x + a)" by (simp add: algebra_simps)
  thus "x \<in> range ((-) a)" by blast
qed

lemma surj_diff_right [simp]:
  "surj (\<lambda>x. x - a)"
proof (standard, simp, standard, simp)
  fix x
  have "x = x + a - a" by simp
  thus "x \<in> range (\<lambda>x. x - a)" by fast
qed

lemma shows bij_plus: "bij ((+) a)" and bij_plus_right: "bij (\<lambda>x. x + a)"
  and bij_uminus: "bij uminus"
  and bij_diff: "bij ((-) a)" and bij_diff_right: "bij (\<lambda>x. x - a)"
by(simp_all add: bij_def)

lemma translation_subtract_Compl:
  "(\<lambda>x. x - a) ` (- t) = - ((\<lambda>x. x - a) ` t)"
by(rule bij_image_Compl_eq)
  (auto simp add: bij_def surj_def inj_def diff_eq_eq intro!: add_diff_cancel[symmetric])

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

lemma translation_subtract_diff:
  "(\<lambda>x. x - a) ` (s - t) = ((\<lambda>x. x - a) ` s) - ((\<lambda>x. x - a) ` t)"
by(rule image_set_diff)(simp add: inj_on_def diff_eq_eq)

lemma translation_Int:
  "(+) a ` (s \<inter> t) = ((+) a ` s) \<inter> ((+) a ` t)"
  by auto

lemma translation_subtract_Int:
  "(\<lambda>x. x - a) ` (s \<inter> t) = ((\<lambda>x. x - a) ` s) \<inter> ((\<lambda>x. x - a) ` t)"
by(rule image_Int)(simp add: inj_on_def diff_eq_eq)

end

(* TODO: prove in group_add *)
context ab_group_add
begin

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

end


subsection \<open>Function Updating\<close>

definition fun_upd :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)"
  where "fun_upd f a b = (\<lambda>x. if x = a then b else f x)"

nonterminal updbinds and updbind

syntax
  "_updbind" :: "'a \<Rightarrow> 'a \<Rightarrow> updbind"             ("(2_ :=/ _)")
  ""         :: "updbind \<Rightarrow> updbinds"             ("_")
  "_updbinds":: "updbind \<Rightarrow> updbinds \<Rightarrow> updbinds" ("_,/ _")
  "_Update"  :: "'a \<Rightarrow> updbinds \<Rightarrow> 'a"            ("_/'((_)')" [1000, 0] 900)

translations
  "_Update f (_updbinds b bs)" \<rightleftharpoons> "_Update (_Update f b) bs"
  "f(x:=y)" \<rightleftharpoons> "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 \<longleftrightarrow> f x = y"
  unfolding fun_upd_def
  apply safe
   apply (erule subst)
   apply auto
  done

lemma fun_upd_idem: "f x = y \<Longrightarrow> 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 \<noteq> x \<Longrightarrow> (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 \<noteq> c \<Longrightarrow> (m(a := b))(c := d) = (m(c := d))(a := b)"
  by auto

lemma inj_on_fun_updI: "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A"
  by (auto simp: inj_on_def)

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

lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
  by auto

lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z"
  by (simp add: fun_eq_iff split: if_split_asm)


subsection \<open>\<open>override_on\<close>\<close>

definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
  where "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"

lemma override_on_emptyset[simp]: "override_on f g {} = f"
  by (simp add: override_on_def)

lemma override_on_apply_notin[simp]: "a \<notin> A \<Longrightarrow> (override_on f g A) a = f a"
  by (simp add: override_on_def)

lemma override_on_apply_in[simp]: "a \<in> A \<Longrightarrow> (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 \<open>Inversion of injective functions\<close>

definition the_inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
  where "the_inv_into A f = (\<lambda>x. THE y. y \<in> A \<and> f y = x)"

lemma the_inv_into_f_f: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> 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 \<Longrightarrow> y \<in> f ` A  \<Longrightarrow> 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 \<Longrightarrow> (bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> 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 \<Longrightarrow> x \<in> f ` A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> the_inv_into A f x \<in> B"
  unfolding the_inv_into_def
  by (rule the1I2; blast dest: inj_onD)

lemma the_inv_into_onto [simp]: "inj_on f A \<Longrightarrow> 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 \<Longrightarrow> f x = y \<Longrightarrow> x \<in> A \<Longrightarrow> 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) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow>
    the_inv_into A (f \<circ> g) x = (the_inv_into A g \<circ> the_inv_into (g ` A) f) x"
  apply (rule the_inv_into_f_eq)
    apply (fast intro: comp_inj_on)
   apply (simp add: f_the_inv_into_f the_inv_into_into)
  apply (simp add: the_inv_into_into)
  done

lemma inj_on_the_inv_into: "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
  by (auto intro: inj_onI simp: the_inv_into_f_f)

lemma bij_betw_the_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
  by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)

lemma bij_betw_iff_bijections:
  "bij_betw f A B \<longleftrightarrow> (\<exists>g. (\<forall>x \<in> A. f x \<in> B \<and> g(f x) = x) \<and> (\<forall>y \<in> B. g y \<in> A \<and> f(g y) = y))"
  (is "?lhs = ?rhs")
proof
  show "?lhs \<Longrightarrow> ?rhs"
    by (auto simp: bij_betw_def f_the_inv_into_f the_inv_into_f_f the_inv_into_into
        exI[where ?x="the_inv_into A f"])
next
  show "?rhs \<Longrightarrow> ?lhs"
    by (force intro: bij_betw_byWitness)
qed

abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
  where "the_inv f \<equiv> the_inv_into UNIV f"

lemma the_inv_f_f: "the_inv f (f x) = x" if "inj f"
  using that UNIV_I by (rule the_inv_into_f_f)


subsection \<open>Monotonicity\<close>

definition monotone_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "monotone_on A orda ordb f \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. orda x y \<longrightarrow> ordb (f x) (f y))"

abbreviation monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "monotone \<equiv> monotone_on UNIV"

lemma monotone_def[no_atp]: "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
  by (simp add: monotone_on_def)

text \<open>Lemma @{thm [source] monotone_def} is provided for backward compatibility.\<close>

lemma monotone_onI:
  "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)) \<Longrightarrow> monotone_on A orda ordb f"
  by (simp add: monotone_on_def)

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

lemma monotone_onD:
  "monotone_on A orda ordb f \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
  by (simp add: monotone_on_def)

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

lemma monotone_on_subset: "monotone_on A orda ordb f \<Longrightarrow> B \<subseteq> A \<Longrightarrow> 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 \<subseteq> A"
  shows "monotone_on B ordc ordb (f \<circ> g)"
proof (rule monotone_onI)
  fix x y assume "x \<in> B" and "y \<in> B" and "ordc x y"
  hence "orda (g x) (g y)"
    by (rule mono_g[THEN monotone_onD])
  moreover from \<open>g ` B \<subseteq> A\<close> \<open>x \<in> B\<close> \<open>y \<in> B\<close> have "g x \<in> A" and "g y \<in> A"
    unfolding image_subset_iff by simp_all
  ultimately show "ordb ((f \<circ> g) x) ((f \<circ> g) y)"
    using mono_f[THEN monotone_onD] by simp
qed


subsubsection \<open>Specializations For @{class ord} Type Class And More\<close>

context ord begin

abbreviation mono_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b :: ord) \<Rightarrow> bool"
  where "mono_on A \<equiv> monotone_on A (\<le>) (\<le>)"

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

abbreviation antimono_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b :: ord) \<Rightarrow> bool"
  where "antimono_on A \<equiv> monotone_on A (\<le>) (\<ge>)"

abbreviation strict_antimono_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b :: ord) \<Rightarrow> bool"
  where "strict_antimono_on A \<equiv> monotone_on A (<) (>)"

lemma mono_on_def[no_atp]: "mono_on A f \<longleftrightarrow> (\<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s)"
  by (auto simp add: monotone_on_def)

lemma strict_mono_on_def[no_atp]:
  "strict_mono_on A f \<longleftrightarrow> (\<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s)"
  by (auto simp add: monotone_on_def)

text \<open>Lemmas @{thm [source] mono_on_def} and @{thm [source] strict_mono_on_def} are provided for
backward compatibility.\<close>

lemma mono_onI:
  "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on A f"
  by (rule monotone_onI)

lemma strict_mono_onI:
  "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on A f"
  by (rule monotone_onI)

lemma mono_onD: "\<lbrakk>mono_on A f; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
  by (rule monotone_onD)

lemma strict_mono_onD: "\<lbrakk>strict_mono_on A f; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
  by (rule monotone_onD)

lemma mono_on_subset: "mono_on A f \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on B f"
  by (rule monotone_on_subset)

end

lemma mono_on_greaterD:
  assumes "mono_on A g" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)"
  shows "x > y"
proof (rule ccontr)
  assume "\<not>x > y"
  hence "x \<le> y" by (simp add: not_less)
  from assms(1-3) and this have "g x \<le> g y" by (rule mono_onD)
  with assms(4) show False by simp
qed

context order begin

abbreviation mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool"
  where "mono \<equiv> mono_on UNIV"

abbreviation strict_mono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool"
  where "strict_mono \<equiv> strict_mono_on UNIV"

abbreviation antimono :: "('a \<Rightarrow> 'b::order) \<Rightarrow> bool"
  where "antimono \<equiv> monotone (\<le>) (\<lambda>x y. y \<le> x)"

lemma mono_def[no_atp]: "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
  by (simp add: monotone_on_def)

lemma strict_mono_def[no_atp]: "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
  by (simp add: monotone_on_def)

lemma antimono_def[no_atp]: "antimono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<ge> f y)"
  by (simp add: monotone_on_def)

text \<open>Lemmas @{thm [source] mono_def}, @{thm [source] strict_mono_def}, and
@{thm [source] antimono_def} are provided for backward compatibility.\<close>

lemma monoI [intro?]: "(\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y) \<Longrightarrow> mono f"
  by (rule monotoneI)

lemma strict_monoI [intro?]: "(\<And>x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> strict_mono f"
  by (rule monotoneI)

lemma antimonoI [intro?]: "(\<And>x y. x \<le> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> antimono f"
  by (rule monotoneI)

lemma monoD [dest?]: "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
  by (rule monotoneD)

lemma strict_monoD [dest?]: "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
  by (rule monotoneD)

lemma antimonoD [dest?]: "antimono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
  by (rule monotoneD)

lemma monoE:
  assumes "mono f"
  assumes "x \<le> y"
  obtains "f x \<le> f y"
proof
  from assms show "f x \<le> f y" by (simp add: mono_def)
qed

lemma antimonoE:
  fixes f :: "'a \<Rightarrow> 'b::order"
  assumes "antimono f"
  assumes "x \<le> y"
  obtains "f x \<ge> f y"
proof
  from assms show "f x \<ge> f y" by (simp add: antimono_def)
qed

lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on A f"
  by (rule monotone_on_subset[OF _ subset_UNIV])

lemma strict_mono_mono [dest?]:
  assumes "strict_mono f"
  shows "mono f"
proof (rule monoI)
  fix x y
  assume "x \<le> y"
  show "f x \<le> f y"
  proof (cases "x = y")
    case True then show ?thesis by simp
  next
    case False with \<open>x \<le> y\<close> have "x < y" by simp
    with assms strict_monoD have "f x < f y" by auto
    then show ?thesis by simp

  qed
qed

end

context linorder begin

lemma mono_invE:
  fixes f :: "'a \<Rightarrow> 'b::order"
  assumes "mono f"
  assumes "f x < f y"
  obtains "x \<le> y"
proof
  show "x \<le> y"
  proof (rule ccontr)
    assume "\<not> x \<le> y"
    then have "y \<le> x" by simp
    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
    with \<open>f x < f y\<close> show False by simp
  qed
qed

lemma mono_strict_invE:
  fixes f :: "'a \<Rightarrow> 'b::order"
  assumes "mono f"
  assumes "f x < f y"
  obtains "x < y"
proof
  show "x < y"
  proof (rule ccontr)
    assume "\<not> x < y"
    then have "y \<le> x" by simp
    with \<open>mono f\<close> obtain "f y \<le> f x" by (rule monoE)
    with \<open>f x < f y\<close> show False by simp
  qed
qed

lemma strict_mono_eq:
  assumes "strict_mono f"
  shows "f x = f y \<longleftrightarrow> x = y"
proof
  assume "f x = f y"
  show "x = y" proof (cases x y rule: linorder_cases)
    case less with assms strict_monoD have "f x < f y" by auto
    with \<open>f x = f y\<close> show ?thesis by simp
  next
    case equal then show ?thesis .
  next
    case greater with assms strict_monoD have "f y < f x" by auto
    with \<open>f x = f y\<close> show ?thesis by simp
  qed
qed simp

lemma strict_mono_less_eq:
  assumes "strict_mono f"
  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
proof
  assume "x \<le> y"
  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
next
  assume "f x \<le> f y"
  show "x \<le> y" proof (rule ccontr)
    assume "\<not> x \<le> y" then have "y < x" by simp
    with assms strict_monoD have "f y < f x" by auto
    with \<open>f x \<le> f y\<close> show False by simp
  qed
qed

lemma strict_mono_less:
  assumes "strict_mono f"
  shows "f x < f y \<longleftrightarrow> x < y"
  using assms
    by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)

end

lemma strict_mono_inv:
  fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)"
  assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x"
  shows "strict_mono g"
proof
  fix x y :: 'b assume "x < y"
  from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
  with \<open>x < y\<close> and \<open>strict_mono f\<close> 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) \<Rightarrow> (_ :: preorder))"
  shows "inj_on f A"
proof (rule inj_onI)
  fix x y assume "x \<in> A" "y \<in> 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) \<Rightarrow> _ :: preorder)" "x \<in> A" "y \<in> A" "x \<le> y"
  shows "f x \<le> f y"
proof (cases "x = y")
  case True
  then show ?thesis by simp
next
  case False
  with assms have "f x < f y"
    using strict_mono_onD[OF assms(1)] by simp
  then show ?thesis by (rule less_imp_le)
qed

lemma strict_mono_on_eqD:
  fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)"
  assumes "strict_mono_on A f" "f x = f y" "x \<in> A" "y \<in> A"
  shows "y = x"
  using assms by (cases rule: linorder_cases) (auto dest: strict_mono_onD)

lemma strict_mono_on_imp_mono_on:
  "strict_mono_on A (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) \<Longrightarrow> mono_on A f"
  by (rule mono_onI, rule strict_mono_on_leD)

lemma mono_imp_strict_mono:
  fixes f :: "'a::order \<Rightarrow> 'b::order"
  shows "\<lbrakk>mono_on S f; inj_on f S\<rbrakk> \<Longrightarrow> strict_mono_on S f"
  by (auto simp add: monotone_on_def order_less_le inj_on_eq_iff)

lemma strict_mono_iff_mono:
  fixes f :: "'a::linorder \<Rightarrow> 'b::order"
  shows "strict_mono_on S f \<longleftrightarrow> mono_on S f \<and> inj_on f S"
proof
  show "strict_mono_on S f \<Longrightarrow> mono_on S f \<and> inj_on f S"
    by (simp add: strict_mono_on_imp_inj_on strict_mono_on_imp_mono_on)
qed (auto intro: mono_imp_strict_mono)

lemma antimono_imp_strict_antimono:
  fixes f :: "'a::order \<Rightarrow> 'b::order"
  shows "\<lbrakk>antimono_on S f; inj_on f S\<rbrakk> \<Longrightarrow> strict_antimono_on S f"
  by (auto simp add: monotone_on_def order_less_le inj_on_eq_iff)

lemma strict_antimono_iff_antimono:
  fixes f :: "'a::linorder \<Rightarrow> 'b::order"
  shows "strict_antimono_on S f \<longleftrightarrow> antimono_on S f \<and> inj_on f S"
proof
  show "strict_antimono_on S f \<Longrightarrow> antimono_on S f \<and> inj_on f S"
    by (force simp add: monotone_on_def intro: linorder_inj_onI)
qed (auto intro: antimono_imp_strict_antimono)

lemma mono_compose: "mono Q \<Longrightarrow> mono (\<lambda>i x. Q i (f x))"
  unfolding mono_def le_fun_def by auto

lemma mono_add:
  fixes a :: "'a::ordered_ab_semigroup_add" 
  shows "mono ((+) a)"
  by (simp add: add_left_mono monoI)

lemma (in semilattice_inf) mono_inf: "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
  for f :: "'a \<Rightarrow> 'b::semilattice_inf"
  by (auto simp add: mono_def intro: Lattices.inf_greatest)

lemma (in semilattice_sup) mono_sup: "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
  for f :: "'a \<Rightarrow> 'b::semilattice_sup"
  by (auto simp add: mono_def intro: Lattices.sup_least)

lemma (in linorder) min_of_mono: "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"
  by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym)

lemma (in linorder) max_of_mono: "mono f \<Longrightarrow> max (f m) (f n) = f (max m n)"
  by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym)

lemma (in linorder)
  max_of_antimono: "antimono f \<Longrightarrow> max (f x) (f y) = f (min x y)" and
  min_of_antimono: "antimono f \<Longrightarrow> min (f x) (f y) = f (max x y)"
  by (auto simp: antimono_def Orderings.max_def max_def Orderings.min_def min_def intro!: antisym)

lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
  by (auto intro!: inj_onI dest: strict_mono_eq)

lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
  by (fact mono_inf)

lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
  by (fact mono_sup)


subsubsection \<open>Least value operator\<close>

lemma Least_mono: "mono f \<Longrightarrow> \<exists>x\<in>S. \<forall>y\<in>S. x \<le> y \<Longrightarrow> (LEAST y. y \<in> f ` S) = f (LEAST x. x \<in> S)"
  for f :: "'a::order \<Rightarrow> 'b::order"
  \<comment> \<open>Courtesy of Stephan Merz\<close>
  apply clarify
  apply (erule_tac P = "\<lambda>x. x \<in> S" in LeastI2_order)
   apply fast
  apply (rule LeastI2_order)
    apply (auto elim: monoD intro!: order_antisym)
  done


subsection \<open>Setup\<close>

subsubsection \<open>Proof tools\<close>

text \<open>Simplify terms of the form \<open>f(\<dots>,x:=y,\<dots>,x:=z,\<dots>)\<close> to \<open>f(\<dots>,x:=z,\<dots>)\<close>\<close>

simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>
  let
    fun gen_fun_upd NONE T _ _ = NONE
      | gen_fun_upd (SOME f) T x y = SOME (Const (\<^const_name>\<open>fun_upd\<close>, T) $ f $ x $ y)
    fun dest_fun_T1 (Type (_, T :: Ts)) = T
    fun find_double (t as Const (\<^const_name>\<open>fun_upd\<close>,T) $ f $ x $ y) =
      let
        fun find (Const (\<^const_name>\<open>fun_upd\<close>,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 K proc end
\<close>


subsubsection \<open>Functorial structure of types\<close>

ML_file \<open>Tools/functor.ML\<close>

functor map_fun: map_fun
  by (simp_all add: fun_eq_iff)

functor vimage
  by (simp_all add: fun_eq_iff vimage_comp)


text \<open>Legacy theorem names\<close>

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