src/HOL/Fun.thy
 author haftmann Sat, 21 May 2022 14:07:24 +0000 changeset 75456 160c9c18a707 parent 74123 7c5842b06114 permissions -rw-r--r--
»nil« seems to be a reserved constructor word in PolyML
```
(*  Title:      HOL/Fun.thy
Author:     Tobias Nipkow, Cambridge University Computer Laboratory
Author:     Andrei Popescu, TU Muenchen
*)

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"

lemma image_id [simp]: "image id = id"

lemma vimage_id [simp]: "vimage id = id"

lemma eq_id_iff: "(\<forall>x. f x = x) \<longleftrightarrow> f = id"
by auto

code_printing

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)"

lemma comp_assoc: "(f \<circ> g) \<circ> h = f \<circ> (g \<circ> h)"

lemma id_comp [simp]: "id \<circ> g = g"

lemma comp_id [simp]: "f \<circ> id = f"

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

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

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)"

lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)"

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

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

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)"

lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"

lemma id_fcomp [simp]: "id \<circ>> g = g"

lemma fcomp_id [simp]: "f \<circ>> id = f"

lemma fcomp_comp: "fcomp f g = comp g f"

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))"

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"

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 inj_on_strict_subset: "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)"

lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"

lemma inj_eq: "inj f \<Longrightarrow> f x = f y \<longleftrightarrow> x = y"

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"

lemma inj_on_id2[simp]: "inj_on (\<lambda>x. x) A"

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"

lemma bij_uminus: "bij (uminus :: 'a \<Rightarrow> '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 \<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"

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"

lemma inj_on_imageI: "inj_on (g \<circ> f) A \<Longrightarrow> inj_on g (f ` A)"

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"

lemma inj_on_empty[iff]: "inj_on f {}"

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_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 (auto intro: linorder_inj_onI linear)

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"

lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"

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_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"

lemma bij_is_surj: "bij f \<Longrightarrow> surj f"

lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"

lemma bij_betw_trans: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g \<circ> f) A C"

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''"
using assms
assume *: "bij_betw (f' \<circ> f) A A''"
then show "bij_betw f A A'"
using img
assume "inj_on (f' \<circ> f) A"
then show "inj_on f A"
using inj_on_imageI2 by blast
next
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
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 (auto simp: image_def)
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] show "\<exists>b\<in>B. a = ?g b" by blast
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"

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_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>
show ?Q
proof
fix y
from \<open>surj f\<close> obtain x where \<open>y = f x\<close>
with \<open>inj f\<close> show \<open>\<exists>!x. f x = y\<close>
qed
next
assume ?Q
then have \<open>inj f\<close>
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>
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>
then have \<open>inj_on f A\<close> and \<open>f ` (A - C) \<inter> f ` (C - A) = {}\<close>
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
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 (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 \<Longrightarrow> inj_on f A"
by (auto intro!: inj_onI dest: strict_mono_eq)

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 auto
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>
qed
qed

subsubsection \<open>Important examples\<close>

begin

"inj_on ((+) a) A"
by (rule inj_onI) simp

\<open>inj ((+) a)\<close>
by simp

"inj_on (\<lambda>b. b + a) A"
by (rule inj_onI) simp

"bij_betw ((+) a) A B \<longleftrightarrow> (+) a ` A = B"

end

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]:
\<open>inj (\<lambda>b. b - a)\<close>
proof -
have \<open>inj ((+) (- a))\<close>
also have \<open>(+) (- a) = (\<lambda>b. b - a)\<close>
finally show ?thesis .
qed

lemma surj_diff_right [simp]:
"surj (\<lambda>x. 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 \<in> (+) a ` (- t) \<longleftrightarrow> b \<in> - (+) a ` t"
by (auto simp: image_iff algebra_simps intro!: bexI [of _ "b - a"])
qed

lemma translation_subtract_Compl:
"(\<lambda>x. x - a) ` (- t) = - ((\<lambda>x. 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:
"(\<lambda>x. x - a) ` (s - t) = ((\<lambda>x. x - a) ` s) - ((\<lambda>x. x - a) ` t)"
using translation_diff [of "- a"] by (simp cong: image_cong_simp)

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)"
using translation_Int [of " -a"] by (simp cong: image_cong_simp)

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 (rule_tac [2] ext)
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)"

(* 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)"

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"

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

lemma override_on_apply_in[simp]: "a \<in> A \<Longrightarrow> (override_on f g A) a = g a"

lemma override_on_insert: "override_on f g (insert x X) = (override_on f g X)(x:=g x)"

lemma override_on_insert': "override_on f g (insert x X) = (override_on (f(x:=g x)) g X)"

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"

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)
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
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 \<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)

theorem Cantors_paradox: "\<nexists>f. f ` A = Pow A"
proof
assume "\<exists>f. f ` A = Pow A"
then obtain f where f: "f ` A = Pow A" ..
let ?X = "{a \<in> A. a \<notin> f a}"
have "?X \<in> Pow A" by blast
then have "?X \<in> f ` A" by (simp only: f)
then obtain x where "x \<in> A" and "f x = ?X" by blast
then show False by blast
qed

subsection \<open>Monotonic functions over a set\<close>

definition "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"

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 f A"
unfolding mono_on_def by simp

lemma mono_onD:
"\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
unfolding mono_on_def by simp

lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
unfolding mono_def mono_on_def by auto

lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
unfolding mono_on_def by auto

definition "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"

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 f A"
unfolding strict_mono_on_def by simp

lemma strict_mono_onD:
"\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
unfolding strict_mono_on_def by simp

lemma mono_on_greaterD:
assumes "mono_on g A" "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

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 (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A"
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 (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y"
shows "f x \<le> 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) \<Rightarrow> (_ :: preorder)"
assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> 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 (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A"
by (rule mono_onI, rule strict_mono_on_leD)

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>fn _ =>
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 proc end
\<close>

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

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

functor map_fun: map_fun

functor vimage

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
```