Theory Fun

```(*  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 ⟹ (⋀x. P x ⟹ g (f x) = x) ⟹ P x ⟹ x = g u"
by auto

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

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

subsection ‹The Identity Function ‹id››

definition id :: "'a ⇒ 'a"
where "id = (λx. x)"

lemma id_apply [simp]: "id x = x"

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

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

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

code_printing

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)"

lemma comp_assoc: "(f ∘ g) ∘ h = f ∘ (g ∘ h)"

lemma id_comp [simp]: "id ∘ g = g"

lemma comp_id [simp]: "f ∘ id = f"

lemma comp_eq_dest: "a ∘ b = c ∘ d ⟹ a (b v) = c (d v)"

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

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)"

lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g ∘ f)"

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

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

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)"

lemma fcomp_assoc: "(f ∘> g) ∘> h = f ∘> (g ∘> h)"

lemma id_fcomp [simp]: "id ∘> g = g"

lemma fcomp_id [simp]: "f ∘> id = f"

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

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))"

subsection ‹Injectivity and Bijectivity›

definition inj_on :: "('a ⇒ 'b) ⇒ 'a set ⇒ bool"  ― ‹injective›
where "inj_on f A ⟷ (∀x∈A. ∀y∈A. f x = f y ⟶ x = y)"

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

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

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

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

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

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

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

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

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

lemma injD: "inj f ⟹ f x = f y ⟹ x = y"

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)"

lemma inj_fun: "inj f ⟹ inj (λx y. f x)"

lemma inj_eq: "inj f ⟹ f x = f y ⟷ x = y"

lemma inj_on_iff_Uniq: "inj_on f A ⟷ (∀x∈A. ∃⇩≤⇩1y. y∈A ∧ 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 (λx. x) A"

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"

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 ⟹ ∀a∈A. 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"

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"

lemma inj_on_imageI: "inj_on (g ∘ f) A ⟹ inj_on g (f ` A)"

lemma inj_on_image_iff:
"∀x∈A. ∀y∈A. g (f x) = g (f y) ⟷ g x = g y ⟹ inj_on f A ⟹ inj_on g (f ` A) ⟷ inj_on g A"
unfolding inj_on_def by blast

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

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

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

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; x∈A; y∈A⟧ ⟹ f x ≠ f y" and lin: "⋀x y. ⟦x∈A; y∈A⟧ ⟹ x≤y ∨ y≤x"
shows "inj_on f A"
proof (rule inj_onI)
fix x y
assume eq: "f x = f y" and "x∈A" "y∈A"
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"

lemma surjE: "surj f ⟹ (⋀x. y = f x ⟹ C) ⟹ C"

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"

lemma bij_is_surj: "bij f ⟹ surj f"

lemma bij_betw_imp_inj_on: "bij_betw f A B ⟹ inj_on f A"

lemma bij_betw_trans: "bij_betw f A B ⟹ bij_betw g B C ⟹ bij_betw (g ∘ f) A C"

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''" (is "?L ⟷ ?R")
proof
assume "?L"
then show "?R"
using assms by (auto simp add: bij_betw_comp_iff)
next
assume *: "?R"
have "inj_on (f' ∘ f) A ⟹ inj_on f A"
using inj_on_imageI2 by blast
moreover have "A' ⊆ f ` A"
proof
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
ultimately show "?L"
using img * by (auto simp add: bij_betw_def)
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 safe
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] have "∃b∈B. a = ?g b" by blast
then show "a ∈ ?g ` B"
by auto
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"

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›
show ?Q
proof
fix y
from ‹surj f› obtain x where ‹y = f x›
with ‹inj f› show ‹∃!x. f x = y›
qed
next
assume ?Q
then have ‹inj f›
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›
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›
then have ‹inj_on f A› and ‹f ` (A - C) ∩ f ` (C - A) = {}›
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
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 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 safe
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›
qed
qed

subsubsection ‹Important examples›

begin

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

‹inj ((+) a)›
by simp

"inj_on (λb. b + a) A"
by (rule inj_onI) simp

"bij_betw ((+) a) A B ⟷ (+) 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]:
‹inj (λb. b - a)›
proof -
have ‹inj ((+) (- a))›
also have ‹(+) (- a) = (λb. b - a)›
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 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)"

(* 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)"

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"

lemma override_on_apply_notin[simp]: "a ∉ A ⟹ (override_on f g A) a = f a"

lemma override_on_apply_in[simp]: "a ∈ A ⟹ (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 ‹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"

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)
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
show "?lhs ⟹ ?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 ⟹ ?lhs"
by (force intro: bij_betw_byWitness)
qed

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)

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 ‹Monotonicity›

definition monotone_on :: "'a set ⇒ ('a ⇒ 'a ⇒ bool) ⇒ ('b ⇒ 'b ⇒ bool) ⇒ ('a ⇒ 'b) ⇒ bool"
where "monotone_on A orda ordb f ⟷ (∀x∈A. ∀y∈A. 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))"

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"

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)"

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

subsubsection ‹Specializations For @{class ord} Type Class And More›

context ord begin

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

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

lemma mono_on_def[no_atp]: "mono_on A f ⟷ (∀r s. r ∈ A ∧ s ∈ A ∧ r ≤ s ⟶ f r ≤ f s)"

lemma strict_mono_on_def[no_atp]:
"strict_mono_on A f ⟷ (∀r s. r ∈ A ∧ s ∈ A ∧ r < s ⟶ f r < f s)"

text ‹Lemmas @{thm [source] mono_on_def} and @{thm [source] strict_mono_on_def} are provided for
backward compatibility.›

lemma mono_onI:
"(⋀r s. r ∈ A ⟹ s ∈ A ⟹ r ≤ s ⟹ f r ≤ f s) ⟹ mono_on A f"
by (rule monotone_onI)

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 mono_onD: "⟦mono_on A f; r ∈ A; s ∈ A; r ≤ s⟧ ⟹ f r ≤ f s"
by (rule monotone_onD)

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_subset: "mono_on A f ⟹ B ⊆ A ⟹ mono_on B f"
by (rule monotone_on_subset)

end

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

context order begin

abbreviation mono :: "('a ⇒ 'b::order) ⇒ bool"
where "mono ≡ mono_on UNIV"

abbreviation strict_mono :: "('a ⇒ 'b::order) ⇒ bool"
where "strict_mono ≡ strict_mono_on UNIV"

abbreviation antimono :: "('a ⇒ 'b::order) ⇒ bool"
where "antimono ≡ monotone (≤) (λx y. y ≤ x)"

lemma mono_def[no_atp]: "mono f ⟷ (∀x y. x ≤ y ⟶ f x ≤ f y)"

lemma strict_mono_def[no_atp]: "strict_mono f ⟷ (∀x y. x < y ⟶ f x < f y)"

lemma antimono_def[no_atp]: "antimono f ⟷ (∀x y. x ≤ y ⟶ f x ≥ f y)"

text ‹Lemmas @{thm [source] mono_def}, @{thm [source] strict_mono_def}, and
@{thm [source] antimono_def} are provided for backward compatibility.›

lemma monoI [intro?]: "(⋀x y. x ≤ y ⟹ f x ≤ f y) ⟹ mono f"
by (rule monotoneI)

lemma strict_monoI [intro?]: "(⋀x y. x < y ⟹ f x < f y) ⟹ strict_mono f"
by (rule monotoneI)

lemma antimonoI [intro?]: "(⋀x y. x ≤ y ⟹ f x ≥ f y) ⟹ antimono f"
by (rule monotoneI)

lemma monoD [dest?]: "mono f ⟹ x ≤ y ⟹ f x ≤ f y"
by (rule monotoneD)

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

lemma antimonoD [dest?]: "antimono f ⟹ x ≤ y ⟹ f x ≥ f y"
by (rule monotoneD)

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

lemma antimonoE:
fixes f :: "'a ⇒ 'b::order"
assumes "antimono f"
assumes "x ≤ y"
obtains "f x ≥ f y"
proof
from assms show "f x ≥ f y" by (simp add: antimono_def)
qed

lemma mono_imp_mono_on: "mono f ⟹ 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 ≤ y"
show "f x ≤ f y"
proof (cases "x = y")
case True then show ?thesis by simp
next
case False with ‹x ≤ y› 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 ⇒ 'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x ≤ y"
proof
show "x ≤ y"
proof (rule ccontr)
assume "¬ x ≤ y"
then have "y ≤ x" by simp
with ‹mono f› obtain "f y ≤ f x" by (rule monoE)
with ‹f x < f y› show False by simp
qed
qed

lemma mono_strict_invE:
fixes f :: "'a ⇒ 'b::order"
assumes "mono f"
assumes "f x < f y"
obtains "x < y"
proof
show "x < y"
proof (rule ccontr)
assume "¬ x < y"
then have "y ≤ x" by simp
with ‹mono f› obtain "f y ≤ f x" by (rule monoE)
with ‹f x < f y› show False by simp
qed
qed

lemma strict_mono_eq:
assumes "strict_mono f"
shows "f x = f y ⟷ 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 ‹f x = f y› 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 ‹f x = f y› show ?thesis by simp
qed
qed simp

lemma strict_mono_less_eq:
assumes "strict_mono f"
shows "f x ≤ f y ⟷ x ≤ y"
proof
assume "x ≤ y"
with assms strict_mono_mono monoD show "f x ≤ f y" by auto
next
assume "f x ≤ f y"
show "x ≤ y" proof (rule ccontr)
assume "¬ x ≤ y" then have "y < x" by simp
with assms strict_monoD have "f y < f x" by auto
with ‹f x ≤ f y› show False by simp
qed
qed

lemma strict_mono_less:
assumes "strict_mono f"
shows "f x < f y ⟷ 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) ⇒ ('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 (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) ⇒ (_ :: preorder)"
assumes "strict_mono_on A f" "f x = f y" "x ∈ A" "y ∈ 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) ⇒ _ :: preorder) ⟹ mono_on A f"
by (rule mono_onI, rule strict_mono_on_leD)

lemma mono_compose: "mono Q ⟹ mono (λi x. Q i (f x))"
unfolding mono_def le_fun_def by auto

shows "mono ((+) a)"

lemma (in semilattice_inf) mono_inf: "mono f ⟹ f (A ⊓ B) ≤ f A ⊓ f B"
for f :: "'a ⇒ 'b::semilattice_inf"
by (auto simp add: mono_def intro: Lattices.inf_greatest)

lemma (in semilattice_sup) mono_sup: "mono f ⟹ f A ⊔ f B ≤ f (A ⊔ B)"
for f :: "'a ⇒ 'b::semilattice_sup"
by (auto simp add: mono_def intro: Lattices.sup_least)

lemma (in linorder) min_of_mono: "mono f ⟹ 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 ⟹ 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 ⟹ max (f x) (f y) = f (min x y)" and
min_of_antimono: "antimono f ⟹ 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 ⟹ inj_on f A"
by (auto intro!: inj_onI dest: strict_mono_eq)

lemma mono_Int: "mono f ⟹ f (A ∩ B) ⊆ f A ∩ f B"
by (fact mono_inf)

lemma mono_Un: "mono f ⟹ f A ∪ f B ⊆ f (A ∪ B)"
by (fact mono_sup)

subsubsection ‹Least value operator›

lemma Least_mono: "mono f ⟹ ∃x∈S. ∀y∈S. x ≤ y ⟹ (LEAST y. y ∈ f ` S) = f (LEAST x. x ∈ S)"
for f :: "'a::order ⇒ 'b::order"
― ‹Courtesy of Stephan Merz›
apply clarify
apply (erule_tac P = "λx. x ∈ S" in LeastI2_order)
apply fast
apply (rule LeastI2_order)
apply (auto elim: monoD intro!: order_antisym)
done

subsection ‹Setup›

subsubsection ‹Proof tools›

text ‹Simplify terms of the form ‹f(…,x:=y,…,x:=z,…)› to ‹f(…,x:=z,…)››

simproc_setup fun_upd2 ("f(v := w, x := y)") = ‹fn _ =>
let
fun gen_fun_upd NONE T _ _ = NONE
| gen_fun_upd (SOME f) T x y = SOME (Const (\<^const_name>‹fun_upd›, T) \$ f \$ x \$ y)
fun dest_fun_T1 (Type (_, T :: Ts)) = T
fun find_double (t as Const (\<^const_name>‹fun_upd›,T) \$ f \$ x \$ y) =
let
fun find (Const (\<^const_name>‹fun_upd›,T) \$ g \$ v \$ w) =
if v aconv x then SOME g else gen_fun_upd (find g) T v w
| find t = NONE
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end

val ss = simpset_of \<^context>

fun proc ctxt ct =
let
val t = Thm.term_of ct
in
(case find_double t of
(T, NONE) => NONE
| (T, SOME rhs) =>
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
(fn _ =>
resolve_tac ctxt [eq_reflection] 1 THEN
resolve_tac ctxt @{thms ext} 1 THEN
simp_tac (put_simpset ss ctxt) 1)))
end
in proc end
›

subsubsection ‹Functorial structure of types›

ML_file ‹Tools/functor.ML›

functor map_fun: map_fun

functor vimage

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
```