# Theory Fun

Up to index of Isabelle/HOL

theory Fun
imports Complete_Lattices
(*  Title:      HOL/Fun.thy
Author: Tobias Nipkow, Cambridge University Computer Laboratory
*)

theory Fun
imports Complete_Lattices
keywords "enriched_type" :: thy_goal
begin

lemma apply_inverse:
"f x = u ==> (!!x. P x ==> g (f x) = x) ==> P x ==> x = g u"
by auto

subsection {* The Identity Function @{text 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"

subsection {* The Composition Operator @{text "f o g"} *}

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

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

notation (HTML output)
comp (infixl "o" 55)

lemma comp_apply [simp]: "(f o g) x = f (g x)"

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

lemma id_comp [simp]: "id o g = g"

lemma comp_id [simp]: "f o id = f"

lemma comp_eq_dest:
"a o b = c o d ==> a (b v) = c (d v)"

lemma comp_eq_elim:
"a o b = c o d ==> ((!!v. a (b v) = c (d v)) ==> R) ==> R"

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

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

lemma INF_comp:
"INFI A (g o f) = INFI (f ` A) g"

lemma SUP_comp:
"SUPR A (g o f) = SUPR (f ` A) g"

subsection {* The Forward Composition Operator @{text fcomp} *}

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

lemma fcomp_apply [simp]: "(f o> g) x = g (f x)"

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

lemma id_fcomp [simp]: "id o> g = g"

lemma fcomp_id [simp]: "f o> id = f"

code_const fcomp
(Eval infixl 1 "#>")

no_notation fcomp (infixl "o>" 60)

subsection {* Mapping functions *}

definition map_fun :: "('c => 'a) => ('b => 'd) => ('a => 'b) => 'c => 'd" where
"map_fun f g h = g o h o 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" where -- "injective"
"inj_on f A <-> (∀x∈A. ∀y∈A. f x = f y --> x = y)"

definition bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
"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 f ≡ inj_on f UNIV"

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

abbreviation
"bij f ≡ bij_betw f UNIV UNIV"

text{* The negated case: *}
translations
"¬ CONST surj f" <= "CONST range f ≠ CONST UNIV"

lemma injI:
assumes "!!x y. f x = f y ==> x = y"
shows "inj f"
using assms unfolding inj_on_def by auto

theorem range_ex1_eq: "inj f ==> b : range f = (EX! x. b = f x)"
by (unfold inj_on_def, 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)"

lemma inj_on_cong:
"(!! a. a : A ==> f a = g a) ==> inj_on f A = inj_on g A"
unfolding inj_on_def by auto

lemma inj_on_strict_subset:
"[| inj_on f B; A < B |] ==> f`A < f`B"
unfolding inj_on_def unfolding image_def by blast

lemma inj_comp:
"inj f ==> inj g ==> inj (f o 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_id[simp]: "inj_on id A"

lemma inj_on_id2[simp]: "inj_on (%x. 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 inj_on_INTER:
"[|I ≠ {}; !! i. i ∈ I ==> inj_on f (A i)|] ==> inj_on f (\<Inter> i ∈ I. A i)"
unfolding inj_on_def by blast

lemma inj_on_Inter:
"[|S ≠ {}; !! A. A ∈ S ==> inj_on f A|] ==> inj_on f (Inter S)"
unfolding inj_on_def by blast

lemma inj_on_UNION_chain:
assumes CH: "!! i j. [|i ∈ I; j ∈ I|] ==> A i ≤ A j ∨ A j ≤ A i" and
INJ: "!! i. i ∈ I ==> inj_on f (A i)"
shows "inj_on f (\<Union> i ∈ I. A i)"
proof -
{
fix i j x y
assume *: "i ∈ I" "j ∈ I" and **: "x ∈ A i" "y ∈ A j"
and ***: "f x = f y"
have "x = y"
proof -
{
assume "A i ≤ A j"
with ** have "x ∈ A j" by auto
with INJ * ** *** have ?thesis
}
moreover
{
assume "A j ≤ A i"
with ** have "y ∈ A i" by auto
with INJ * ** *** have ?thesis
}
ultimately show ?thesis using CH * by blast
qed
}
then show ?thesis by (unfold inj_on_def UNION_eq) auto
qed

lemma surj_id: "surj id"
by simp

lemma bij_id[simp]: "bij id"

lemma inj_onI:
"(!! 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"
by (unfold inj_on_def, blast)

lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)"
by (blast dest!: inj_onD)

lemma comp_inj_on:
"[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A"

lemma inj_on_imageI: "inj_on (g o f) A ==> inj_on g (f ` A)"
apply blast
done

lemma inj_on_image_iff: "[| ALL x:A. ALL 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"

apply(unfold inj_on_def)
apply blast
done

lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)"
by (unfold inj_on_def, blast)

lemma inj_singleton: "inj (%s. {s})"

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

lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
by (unfold inj_on_def, blast)

lemma inj_on_Un:
"inj_on f (A Un B) =
(inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"

apply(unfold inj_on_def)
apply (blast intro:sym)
done

lemma inj_on_insert[iff]:
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
apply(unfold inj_on_def)
apply (blast intro:sym)
done

lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
apply(unfold inj_on_def)
apply (blast)
done

lemma comp_inj_on_iff:
"inj_on f A ==> inj_on f' (f ` A) <-> inj_on (f' o f) A"

lemma inj_on_imageI2:
"inj_on (f' o f) A ==> inj_on f A"

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

lemma surjI: assumes *: "!! x. g (f x) = x" shows "surj g"
using *[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 o f)"
apply (simp add: comp_def surj_def, clarify)
apply (drule_tac x = y in spec, clarify)
apply (drule_tac x = x in spec, blast)
done

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

lemma bij_betw_empty1:
assumes "bij_betw f {} A"
shows "A = {}"
using assms unfolding bij_betw_def by blast

lemma bij_betw_empty2:
assumes "bij_betw f A {}"
shows "A = {}"
using assms unfolding bij_betw_def by blast

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

lemma bij_def: "bij f <-> inj f ∧ surj f"
unfolding bij_betw_def ..

lemma bijI: "[| inj f; surj f |] ==> bij f"

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 o f) A C"

lemma bij_comp: "bij f ==> bij g ==> bij (g o f)"
by (rule bij_betw_trans)

lemma bij_betw_comp_iff:
"bij_betw f A A' ==> bij_betw f' A' A'' <-> bij_betw (f' o f) A A''"

lemma bij_betw_comp_iff2:
assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A ≤ A'"
shows "bij_betw f A A' <-> bij_betw (f' o f) A A''"
using assms
assume *: "bij_betw (f' o f) A A''"
thus "bij_betw f A A'"
using IM
assume "inj_on (f' o f) A"
thus "inj_on f A" using inj_on_imageI2 by blast
next
fix a' assume **: "a' ∈ A'"
hence "f' a' ∈ A''" using BIJ unfolding bij_betw_def by auto
then obtain a where 1: "a ∈ A ∧ f'(f a) = f' a'" using *
unfolding bij_betw_def by force
hence "f a ∈ A'" using IM by auto
hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
thus "a' ∈ f ` A" using 1 by auto
qed
qed

lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX 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 = "%b a. a:A ∧ f a = b" let ?g = "%b. The (?P b)"
{ fix a b assume P: "?P b a"
hence ex1: "∃a. ?P b a" using s unfolding image_def by blast
hence uex1: "∃!a. ?P b a" by(blast dest:inj_onD[OF i])
hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
} note g = this
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" unfolding image_def by blast
from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
qed
moreover have "?g ` B = A"
proof(auto simp:image_def)
fix b assume "b:B"
with s obtain a where P: "?P b a" unfolding image_def by blast
thus "?g b ∈ A" using g[OF P] by auto
next
fix a assume "a:A"
then obtain b where P: "?P b a" using s unfolding image_def by blast
then have "b:B" using s unfolding image_def by blast
with g[OF P] show "∃b∈B. a = ?g b" by blast
qed
ultimately show ?thesis by(auto simp:bij_betw_def)
qed

lemma bij_betw_cong:
"(!! a. a ∈ A ==> f a = g a) ==> bij_betw f A A' = bij_betw g A A'"
unfolding bij_betw_def inj_on_def by force

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:
assumes "bij_betw f A B" "bij_betw f C D" "B ∩ D = {}"
shows "bij_betw f (A ∪ C) (B ∪ D)"
using assms unfolding bij_betw_def inj_on_Un image_Un by auto

lemma bij_betw_UNION_chain:
assumes CH: "!! i j. [|i ∈ I; j ∈ I|] ==> A i ≤ A j ∨ A j ≤ A i" and
BIJ: "!! i. i ∈ I ==> bij_betw f (A i) (A' i)"
shows "bij_betw f (\<Union> i ∈ I. A i) (\<Union> i ∈ I. A' i)"
proof (unfold bij_betw_def, auto)
have "!! i. i ∈ I ==> inj_on f (A i)"
using BIJ bij_betw_def[of f] by auto
thus "inj_on f (\<Union> i ∈ I. A i)"
using CH inj_on_UNION_chain[of I A f] by auto
next
fix i x
assume *: "i ∈ I" "x ∈ A i"
hence "f x ∈ A' i" using BIJ bij_betw_def[of f] by auto
thus "∃j ∈ I. f x ∈ A' j" using * by blast
next
fix i x'
assume *: "i ∈ I" "x' ∈ A' i"
hence "∃x ∈ A i. x' = f x" using BIJ bij_betw_def[of f] by blast
then have "∃j ∈ I. ∃x ∈ A j. x' = f x"
using * by blast
then show "x' ∈ f ` (\<Union>x∈I. A x)" by (simp add: image_def)
qed

lemma bij_betw_subset:
assumes BIJ: "bij_betw f A A'" and
SUB: "B ≤ A" and IM: "f ` B = B'"
shows "bij_betw f B B'"
using assms
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)

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"

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"
by (unfold inj_on_def, blast)

lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
apply (unfold bij_def)
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
done

lemma inj_on_Un_image_eq_iff: "inj_on f (A ∪ B) ==> f ` A = f ` B <-> A = B"
by(blast dest: inj_onD)

lemma inj_on_image_Int:
"[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B"
done

lemma inj_on_image_set_diff:
"[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B"
done

lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"

lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"

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)"

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

(*injectivity's required. Left-to-right inclusion holds even if A is empty*)
lemma image_INT:
"[| inj_on f C; ALL x:A. B x <= C; j:A |]
==> f ` (INTER A B) = (INT x:A. f ` B x)"

done

(*Compare with image_INT: no use of inj_on, and if f is surjective then
it doesn't matter whether A is empty*)

lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
apply (simp add: inj_on_def surj_def, blast)
done

lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
by auto

lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"

lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
apply (rule equalityI)
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
done

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. *}

apply (blast intro: the_equality [symmetric])
done

lemma inj_on_vimage_singleton:
"inj_on f A ==> f -` {a} ∩ A ⊆ {THE x. x ∈ A ∧ f x = a}"
by (auto simp add: inj_on_def intro: the_equality [symmetric])

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

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

subsection{*Function Updating*}

definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
"fun_upd f a b == % x. 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 sum_case.
A nice infix syntax could be defined (in Datatype.thy or below) by
notation
sum_case (infixr "'(+')"80)
*)

lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
apply (erule subst)
apply (rule_tac [2] ext, 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 (rule ext, auto)

lemma inj_on_fun_updI: "[| inj_on f A; y ∉ f`A |] ==> inj_on (f(x:=y)) A"
by (fastforce simp:inj_on_def image_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 o (g(x := y)) = (f o g)(x := f y)"
by auto

lemma UNION_fun_upd:
"UNION J (A(i:=B)) = (UNION (J-{i}) A ∪ (if i∈J then B else {}))"
by (auto split: if_splits)

subsection {* @{text 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"

subsection {* @{text swap} *}

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

lemma swap_self [simp]: "swap a a f = f"

lemma swap_commute: "swap a b f = swap b a f"
by (rule ext, simp add: fun_upd_def swap_def)

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

lemma swap_triple:
assumes "a ≠ c" and "b ≠ c"
shows "swap a b (swap b c (swap a b f)) = swap a c f"
using assms by (simp add: fun_eq_iff swap_def)

lemma comp_swap: "f o swap a b g = swap a b (f o g)"
by (rule ext, simp add: fun_upd_def swap_def)

lemma swap_image_eq [simp]:
assumes "a ∈ A" "b ∈ A" shows "swap a b f ` A = f ` A"
proof -
have subset: "!!f. swap a b f ` A ⊆ f ` A"
using assms by (auto simp: image_iff swap_def)
then have "swap a b (swap a b f) ` A ⊆ (swap a b f) ` A" .
with subset[of f] show ?thesis by auto
qed

lemma inj_on_imp_inj_on_swap:
"[|inj_on f A; a ∈ A; b ∈ A|] ==> inj_on (swap a b f) A"
by (simp add: inj_on_def swap_def, blast)

lemma inj_on_swap_iff [simp]:
assumes A: "a ∈ A" "b ∈ A" shows "inj_on (swap a b f) A <-> inj_on f A"
proof
assume "inj_on (swap a b f) A"
with A have "inj_on (swap a b (swap a b f)) A"
by (iprover intro: inj_on_imp_inj_on_swap)
thus "inj_on f A" by simp
next
assume "inj_on f A"
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
qed

lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
by simp

lemma surj_swap_iff [simp]: "surj (swap a b f) <-> surj f"
by simp

lemma bij_betw_swap_iff [simp]:
"[| x ∈ A; y ∈ A |] ==> bij_betw (swap x y f) A B <-> bij_betw f A B"
by (auto simp: bij_betw_def)

lemma bij_swap_iff [simp]: "bij (swap a b f) <-> bij f"
by simp

hide_const (open) swap

subsection {* Inversion of injective functions *}

definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
"the_inv_into A f == %x. 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"
apply blast
done

lemma f_the_inv_into_f:
"inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y"
apply (rule the1I2)
apply(blast dest: inj_onD)
apply blast
done

lemma the_inv_into_into:
"[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
apply (rule the1I2)
apply(blast dest: inj_onD)
apply blast
done

lemma the_inv_into_onto[simp]:
"inj_on f A ==> the_inv_into A f ` (f ` A) = A"
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])

lemma the_inv_into_f_eq:
"[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
apply (erule subst)
apply (erule the_inv_into_f_f, assumption)
done

lemma the_inv_into_comp:
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
the_inv_into A (f o g) x = (the_inv_into A g o 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: image_def the_inv_into_f_f)

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

abbreviation the_inv :: "('a => 'b) => ('b => 'a)" where
"the_inv f ≡ the_inv_into UNIV f"

lemma the_inv_f_f:
assumes "inj f"
shows "the_inv f (f x) = x" using assms UNIV_I
by (rule the_inv_into_f_f)

"¬(∃f. f ` A = Pow A)"
proof clarify
fix f assume "f ` A = Pow A" hence *: "Pow A ≤ f ` A" by blast
let ?X = "{a ∈ A. a ∉ f a}"
have "?X ∈ Pow A" unfolding Pow_def by auto
with * obtain x where "x ∈ A ∧ f x = ?X" by blast
thus False by best
qed

subsection {* Setup *}

subsubsection {* Proof tools *}

text {* simplifies 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

fun proc ss ct =
let
val ctxt = Simplifier.the_context ss
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 _ =>
rtac eq_reflection 1 THEN
rtac ext 1 THEN
simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
end
in proc end
*}

subsubsection {* Code generator *}

code_const "op o"
(SML infixl 5 "o")

code_const "id"

subsubsection {* Functorial structure of types *}

ML_file "Tools/enriched_type.ML"

enriched_type map_fun: map_fun

enriched_type 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 image_compose = image_comp
lemmas vimage_compose = vimage_comp

end