# Theory Quotient_FSet

theory Quotient_FSet
imports Multiset Quotient_List
```(*  Title:      HOL/Quotient_Examples/Quotient_FSet.thy
Author:     Cezary Kaliszyk, TU Munich
Author:     Christian Urban, TU Munich

Type of finite sets.
*)

(********************************************************************
WARNING: There is a formalization of 'a fset as a subtype of sets in
HOL/Library/FSet.thy using Lifting/Transfer. The user should use
that file rather than this file unless there are some very specific
reasons.
*********************************************************************)

theory Quotient_FSet
imports "HOL-Library.Multiset" "HOL-Library.Quotient_List"
begin

text ‹
The type of finite sets is created by a quotient construction
over lists. The definition of the equivalence:
›

definition
list_eq :: "'a list ⇒ 'a list ⇒ bool" (infix "≈" 50)
where
[simp]: "xs ≈ ys ⟷ set xs = set ys"

lemma list_eq_reflp:
"reflp list_eq"
by (auto intro: reflpI)

lemma list_eq_symp:
"symp list_eq"
by (auto intro: sympI)

lemma list_eq_transp:
"transp list_eq"
by (auto intro: transpI)

lemma list_eq_equivp:
"equivp list_eq"
by (auto intro: equivpI list_eq_reflp list_eq_symp list_eq_transp)

text ‹The ‹fset› type›

quotient_type
'a fset = "'a list" / "list_eq"
by (rule list_eq_equivp)

text ‹
Definitions for sublist, cardinality,
intersection, difference and respectful fold over
lists.
›

declare List.member_def [simp]

definition
sub_list :: "'a list ⇒ 'a list ⇒ bool"
where
[simp]: "sub_list xs ys ⟷ set xs ⊆ set ys"

definition
card_list :: "'a list ⇒ nat"
where
[simp]: "card_list xs = card (set xs)"

definition
inter_list :: "'a list ⇒ 'a list ⇒ 'a list"
where
[simp]: "inter_list xs ys = [x ← xs. x ∈ set xs ∧ x ∈ set ys]"

definition
diff_list :: "'a list ⇒ 'a list ⇒ 'a list"
where
[simp]: "diff_list xs ys = [x ← xs. x ∉ set ys]"

definition
rsp_fold :: "('a ⇒ 'b ⇒ 'b) ⇒ bool"
where
"rsp_fold f ⟷ (∀u v. f u ∘ f v = f v ∘ f u)"

lemma rsp_foldI:
"(⋀u v. f u ∘ f v = f v ∘ f u) ⟹ rsp_fold f"

lemma rsp_foldE:
assumes "rsp_fold f"
obtains "f u ∘ f v = f v ∘ f u"
using assms by (simp add: rsp_fold_def)

definition
fold_once :: "('a ⇒ 'b ⇒ 'b) ⇒ 'a list ⇒ 'b ⇒ 'b"
where
"fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"

lemma fold_once_default [simp]:
"¬ rsp_fold f ⟹ fold_once f xs = id"

lemma fold_once_fold_remdups:
"rsp_fold f ⟹ fold_once f xs = fold f (remdups xs)"

section ‹Quotient composition lemmas›

lemma list_all2_refl':
assumes q: "equivp R"
shows "(list_all2 R) r r"
by (rule list_all2_refl) (metis equivp_def q)

lemma compose_list_refl:
assumes q: "equivp R"
shows "(list_all2 R OOO op ≈) r r"
proof
have *: "r ≈ r" by (rule equivp_reflp[OF fset_equivp])
show "list_all2 R r r" by (rule list_all2_refl'[OF q])
with * show "(op ≈ OO list_all2 R) r r" ..
qed

lemma map_list_eq_cong: "b ≈ ba ⟹ map f b ≈ map f ba"
by (simp only: list_eq_def set_map)

lemma quotient_compose_list_g:
assumes q: "Quotient3 R Abs Rep"
and     e: "equivp R"
shows  "Quotient3 ((list_all2 R) OOO (op ≈))
(abs_fset ∘ (map Abs)) ((map Rep) ∘ rep_fset)"
unfolding Quotient3_def comp_def
proof (intro conjI allI)
fix a r s
show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
by (simp add: abs_o_rep[OF q] Quotient3_abs_rep[OF Quotient3_fset] List.map.id)
have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule list_all2_refl'[OF e])
have c: "(op ≈ OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
show "(list_all2 R OOO op ≈) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule list_all2_refl'[OF e]) (rule c)
show "(list_all2 R OOO op ≈) r s = ((list_all2 R OOO op ≈) r r ∧
(list_all2 R OOO op ≈) s s ∧ abs_fset (map Abs r) = abs_fset (map Abs s))"
proof (intro iffI conjI)
show "(list_all2 R OOO op ≈) r r" by (rule compose_list_refl[OF e])
show "(list_all2 R OOO op ≈) s s" by (rule compose_list_refl[OF e])
next
assume a: "(list_all2 R OOO op ≈) r s"
then have b: "map Abs r ≈ map Abs s"
proof (elim relcomppE)
fix b ba
assume c: "list_all2 R r b"
assume d: "b ≈ ba"
assume e: "list_all2 R ba s"
have f: "map Abs r = map Abs b"
using Quotient3_rel[OF list_quotient3[OF q]] c by blast
have "map Abs ba = map Abs s"
using Quotient3_rel[OF list_quotient3[OF q]] e by blast
then have g: "map Abs s = map Abs ba" by simp
then show "map Abs r ≈ map Abs s" using d f map_list_eq_cong by simp
qed
then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
using Quotient3_rel[OF Quotient3_fset] by blast
next
assume a: "(list_all2 R OOO op ≈) r r ∧ (list_all2 R OOO op ≈) s s
∧ abs_fset (map Abs r) = abs_fset (map Abs s)"
then have s: "(list_all2 R OOO op ≈) s s" by simp
have d: "map Abs r ≈ map Abs s"
by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
have b: "map Rep (map Abs r) ≈ map Rep (map Abs s)"
by (rule map_list_eq_cong[OF d])
have y: "list_all2 R (map Rep (map Abs s)) s"
by (fact rep_abs_rsp_left[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of s]])
have c: "(op ≈ OO list_all2 R) (map Rep (map Abs r)) s"
by (rule relcomppI) (rule b, rule y)
have z: "list_all2 R r (map Rep (map Abs r))"
by (fact rep_abs_rsp[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of r]])
then show "(list_all2 R OOO op ≈) r s"
using a c relcomppI by simp
qed
qed

lemma quotient_compose_list[quot_thm]:
shows  "Quotient3 ((list_all2 op ≈) OOO (op ≈))
(abs_fset ∘ (map abs_fset)) ((map rep_fset) ∘ rep_fset)"
by (rule quotient_compose_list_g, rule Quotient3_fset, rule list_eq_equivp)

section ‹Quotient definitions for fsets›

subsection ‹Finite sets are a bounded, distributive lattice with minus›

instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
begin

quotient_definition
"bot :: 'a fset"
is "Nil :: 'a list" done

abbreviation
empty_fset  ("{||}")
where
"{||} ≡ bot :: 'a fset"

quotient_definition
"less_eq_fset :: ('a fset ⇒ 'a fset ⇒ bool)"
is "sub_list :: ('a list ⇒ 'a list ⇒ bool)" by simp

abbreviation
subset_fset :: "'a fset ⇒ 'a fset ⇒ bool" (infix "|⊆|" 50)
where
"xs |⊆| ys ≡ xs ≤ ys"

definition
less_fset :: "'a fset ⇒ 'a fset ⇒ bool"
where
"xs < ys ≡ xs ≤ ys ∧ xs ≠ (ys::'a fset)"

abbreviation
psubset_fset :: "'a fset ⇒ 'a fset ⇒ bool" (infix "|⊂|" 50)
where
"xs |⊂| ys ≡ xs < ys"

quotient_definition
"sup :: 'a fset ⇒ 'a fset ⇒ 'a fset"
is "append :: 'a list ⇒ 'a list ⇒ 'a list" by simp

abbreviation
union_fset (infixl "|∪|" 65)
where
"xs |∪| ys ≡ sup xs (ys::'a fset)"

quotient_definition
"inf :: 'a fset ⇒ 'a fset ⇒ 'a fset"
is "inter_list :: 'a list ⇒ 'a list ⇒ 'a list" by simp

abbreviation
inter_fset (infixl "|∩|" 65)
where
"xs |∩| ys ≡ inf xs (ys::'a fset)"

quotient_definition
"minus :: 'a fset ⇒ 'a fset ⇒ 'a fset"
is "diff_list :: 'a list ⇒ 'a list ⇒ 'a list" by fastforce

instance
proof
fix x y z :: "'a fset"
show "x |⊂| y ⟷ x |⊆| y ∧ ¬ y |⊆| x"
by (unfold less_fset_def, descending) auto
show "x |⊆| x" by (descending) (simp)
show "{||} |⊆| x" by (descending) (simp)
show "x |⊆| x |∪| y" by (descending) (simp)
show "y |⊆| x |∪| y" by (descending) (simp)
show "x |∩| y |⊆| x" by (descending) (auto)
show "x |∩| y |⊆| y" by (descending) (auto)
show "x |∪| (y |∩| z) = x |∪| y |∩| (x |∪| z)"
by (descending) (auto)
next
fix x y z :: "'a fset"
assume a: "x |⊆| y"
assume b: "y |⊆| z"
show "x |⊆| z" using a b by (descending) (simp)
next
fix x y :: "'a fset"
assume a: "x |⊆| y"
assume b: "y |⊆| x"
show "x = y" using a b by (descending) (auto)
next
fix x y z :: "'a fset"
assume a: "y |⊆| x"
assume b: "z |⊆| x"
show "y |∪| z |⊆| x" using a b by (descending) (simp)
next
fix x y z :: "'a fset"
assume a: "x |⊆| y"
assume b: "x |⊆| z"
show "x |⊆| y |∩| z" using a b by (descending) (auto)
qed

end

subsection ‹Other constants for fsets›

quotient_definition
"insert_fset :: 'a ⇒ 'a fset ⇒ 'a fset"
is "Cons" by auto

syntax
"_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")

translations
"{|x, xs|}" == "CONST insert_fset x {|xs|}"
"{|x|}"     == "CONST insert_fset x {||}"

quotient_definition
fset_member
where
"fset_member :: 'a fset ⇒ 'a ⇒ bool" is "List.member" by fastforce

abbreviation
in_fset :: "'a ⇒ 'a fset ⇒ bool" (infix "|∈|" 50)
where
"x |∈| S ≡ fset_member S x"

abbreviation
notin_fset :: "'a ⇒ 'a fset ⇒ bool" (infix "|∉|" 50)
where
"x |∉| S ≡ ¬ (x |∈| S)"

subsection ‹Other constants on the Quotient Type›

quotient_definition
"card_fset :: 'a fset ⇒ nat"
is card_list by simp

quotient_definition
"map_fset :: ('a ⇒ 'b) ⇒ 'a fset ⇒ 'b fset"
is map by simp

quotient_definition
"remove_fset :: 'a ⇒ 'a fset ⇒ 'a fset"
is removeAll by simp

quotient_definition
"fset :: 'a fset ⇒ 'a set"
is "set" by simp

lemma fold_once_set_equiv:
assumes "xs ≈ ys"
shows "fold_once f xs = fold_once f ys"
proof (cases "rsp_fold f")
case False then show ?thesis by simp
next
case True
then have "⋀x y. x ∈ set (remdups xs) ⟹ y ∈ set (remdups xs) ⟹ f x ∘ f y = f y ∘ f x"
by (rule rsp_foldE)
moreover from assms have "mset (remdups xs) = mset (remdups ys)"
ultimately have "fold f (remdups xs) = fold f (remdups ys)"
by (rule fold_multiset_equiv)
with True show ?thesis by (simp add: fold_once_fold_remdups)
qed

quotient_definition
"fold_fset :: ('a ⇒ 'b ⇒ 'b) ⇒ 'a fset ⇒ 'b ⇒ 'b"
is fold_once by (rule fold_once_set_equiv)

lemma concat_rsp_pre:
assumes a: "list_all2 op ≈ x x'"
and     b: "x' ≈ y'"
and     c: "list_all2 op ≈ y' y"
and     d: "∃x∈set x. xa ∈ set x"
shows "∃x∈set y. xa ∈ set x"
proof -
obtain xb where e: "xb ∈ set x" and f: "xa ∈ set xb" using d by auto
have "∃y. y ∈ set x' ∧ xb ≈ y" by (rule list_all2_find_element[OF e a])
then obtain ya where h: "ya ∈ set x'" and i: "xb ≈ ya" by auto
have "ya ∈ set y'" using b h by simp
then have "∃yb. yb ∈ set y ∧ ya ≈ yb" using c by (rule list_all2_find_element)
then show ?thesis using f i by auto
qed

quotient_definition
"concat_fset :: ('a fset) fset ⇒ 'a fset"
is concat
proof (elim relcomppE)
fix a b ba bb
assume a: "list_all2 op ≈ a ba"
with list_symp [OF list_eq_symp] have a': "list_all2 op ≈ ba a" by (rule sympE)
assume b: "ba ≈ bb"
with list_eq_symp have b': "bb ≈ ba" by (rule sympE)
assume c: "list_all2 op ≈ bb b"
with list_symp [OF list_eq_symp] have c': "list_all2 op ≈ b bb" by (rule sympE)
have "∀x. (∃xa∈set a. x ∈ set xa) = (∃xa∈set b. x ∈ set xa)"
proof
fix x
show "(∃xa∈set a. x ∈ set xa) = (∃xa∈set b. x ∈ set xa)"
proof
assume d: "∃xa∈set a. x ∈ set xa"
show "∃xa∈set b. x ∈ set xa" by (rule concat_rsp_pre[OF a b c d])
next
assume e: "∃xa∈set b. x ∈ set xa"
show "∃xa∈set a. x ∈ set xa" by (rule concat_rsp_pre[OF c' b' a' e])
qed
qed
then show "concat a ≈ concat b" by auto
qed

quotient_definition
"filter_fset :: ('a ⇒ bool) ⇒ 'a fset ⇒ 'a fset"
is filter by force

subsection ‹Compositional respectfulness and preservation lemmas›

lemma Nil_rsp2 [quot_respect]:
shows "(list_all2 op ≈ OOO op ≈) Nil Nil"
by (rule compose_list_refl, rule list_eq_equivp)

lemma Cons_rsp2 [quot_respect]:
shows "(op ≈ ===> list_all2 op ≈ OOO op ≈ ===> list_all2 op ≈ OOO op ≈) Cons Cons"
apply (auto intro!: rel_funI)
apply (rule_tac b="x # b" in relcomppI)
apply auto
apply (rule_tac b="x # ba" in relcomppI)
apply auto
done

lemma Nil_prs2 [quot_preserve]:
assumes "Quotient3 R Abs Rep"
shows "(Abs ∘ map f) [] = Abs []"
by simp

lemma Cons_prs2 [quot_preserve]:
assumes q: "Quotient3 R1 Abs1 Rep1"
and     r: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> (map Rep1 ∘ Rep2) ---> (Abs2 ∘ map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])

lemma append_prs2 [quot_preserve]:
assumes q: "Quotient3 R1 Abs1 Rep1"
and     r: "Quotient3 R2 Abs2 Rep2"
shows "((map Rep1 ∘ Rep2) ---> (map Rep1 ∘ Rep2) ---> (Abs2 ∘ map Abs1)) op @ =
(Rep2 ---> Rep2 ---> Abs2) op @"
by (simp add: fun_eq_iff abs_o_rep[OF q] List.map.id)

lemma list_all2_app_l:
assumes a: "reflp R"
and b: "list_all2 R l r"
shows "list_all2 R (z @ l) (z @ r)"
using a b by (induct z) (auto elim: reflpE)

lemma append_rsp2_pre0:
assumes a:"list_all2 op ≈ x x'"
shows "list_all2 op ≈ (x @ z) (x' @ z)"
using a apply (induct x x' rule: list_induct2')
by simp_all (rule list_all2_refl'[OF list_eq_equivp])

lemma append_rsp2_pre1:
assumes a:"list_all2 op ≈ x x'"
shows "list_all2 op ≈ (z @ x) (z @ x')"
using a apply (induct x x' arbitrary: z rule: list_induct2')
apply (rule list_all2_refl'[OF list_eq_equivp])
apply (simp_all del: list_eq_def)
apply (rule list_all2_app_l)
done

lemma append_rsp2_pre:
assumes "list_all2 op ≈ x x'"
and "list_all2 op ≈ z z'"
shows "list_all2 op ≈ (x @ z) (x' @ z')"
using assms by (rule list_all2_appendI)

lemma compositional_rsp3:
assumes "(R1 ===> R2 ===> R3) C C" and "(R4 ===> R5 ===> R6) C C"
shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
by (auto intro!: rel_funI)
(metis (full_types) assms rel_funE relcomppI)

lemma append_rsp2 [quot_respect]:
"(list_all2 op ≈ OOO op ≈ ===> list_all2 op ≈ OOO op ≈ ===> list_all2 op ≈ OOO op ≈) append append"
by (intro compositional_rsp3)
(auto intro!: rel_funI simp add: append_rsp2_pre)

lemma map_rsp2 [quot_respect]:
"((op ≈ ===> op ≈) ===> list_all2 op ≈ OOO op ≈ ===> list_all2 op ≈ OOO op ≈) map map"
proof (auto intro!: rel_funI)
fix f f' :: "'a list ⇒ 'b list"
fix xa ya x y :: "'a list list"
assume fs: "(op ≈ ===> op ≈) f f'" and x: "list_all2 op ≈ xa x" and xy: "set x = set y" and y: "list_all2 op ≈ y ya"
have a: "(list_all2 op ≈) (map f xa) (map f x)"
using x
by (induct xa x rule: list_induct2')
(simp_all, metis fs rel_funE list_eq_def)
have b: "set (map f x) = set (map f y)"
using xy fs
by (induct x y rule: list_induct2')
(simp_all, metis image_insert)
have c: "(list_all2 op ≈) (map f y) (map f' ya)"
using y fs
by (induct y ya rule: list_induct2')
(simp_all, metis apply_rsp' list_eq_def)
show "(list_all2 op ≈ OOO op ≈) (map f xa) (map f' ya)"
by (metis a b c list_eq_def relcomppI)
qed

lemma map_prs2 [quot_preserve]:
shows "((abs_fset ---> rep_fset) ---> (map rep_fset ∘ rep_fset) ---> abs_fset ∘ map abs_fset) map = (id ---> rep_fset ---> abs_fset) map"
(simp only: map_map[symmetric] map_prs_aux[OF Quotient3_fset Quotient3_fset])

section ‹Lifted theorems›

subsection ‹fset›

lemma fset_simps [simp]:
shows "fset {||} = {}"
and   "fset (insert_fset x S) = insert x (fset S)"
by (descending, simp)+

lemma finite_fset [simp]:
shows "finite (fset S)"
by (descending) (simp)

lemma fset_cong:
shows "fset S = fset T ⟷ S = T"
by (descending) (simp)

lemma filter_fset [simp]:
shows "fset (filter_fset P xs) = Collect P ∩ fset xs"
by (descending) (auto)

lemma remove_fset [simp]:
shows "fset (remove_fset x xs) = fset xs - {x}"
by (descending) (simp)

lemma inter_fset [simp]:
shows "fset (xs |∩| ys) = fset xs ∩ fset ys"
by (descending) (auto)

lemma union_fset [simp]:
shows "fset (xs |∪| ys) = fset xs ∪ fset ys"
by (lifting set_append)

lemma minus_fset [simp]:
shows "fset (xs - ys) = fset xs - fset ys"
by (descending) (auto)

subsection ‹in_fset›

lemma in_fset:
shows "x |∈| S ⟷ x ∈ fset S"
by descending simp

lemma notin_fset:
shows "x |∉| S ⟷ x ∉ fset S"

lemma notin_empty_fset:
shows "x |∉| {||}"

lemma fset_eq_iff:
shows "S = T ⟷ (∀x. (x |∈| S) = (x |∈| T))"
by descending auto

lemma none_in_empty_fset:
shows "(∀x. x |∉| S) ⟷ S = {||}"
by descending simp

subsection ‹insert_fset›

lemma in_insert_fset_iff [simp]:
shows "x |∈| insert_fset y S ⟷ x = y ∨ x |∈| S"
by descending simp

lemma
shows insert_fsetI1: "x |∈| insert_fset x S"
and   insert_fsetI2: "x |∈| S ⟹ x |∈| insert_fset y S"
by simp_all

lemma insert_absorb_fset [simp]:
shows "x |∈| S ⟹ insert_fset x S = S"
by (descending) (auto)

lemma empty_not_insert_fset[simp]:
shows "{||} ≠ insert_fset x S"
and   "insert_fset x S ≠ {||}"
by (descending, simp)+

lemma insert_fset_left_comm:
shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
by (descending) (auto)

lemma insert_fset_left_idem:
shows "insert_fset x (insert_fset x S) = insert_fset x S"
by (descending) (auto)

lemma singleton_fset_eq[simp]:
shows "{|x|} = {|y|} ⟷ x = y"
by (descending) (auto)

lemma in_fset_mdef:
shows "x |∈| F ⟷ x |∉| (F - {|x|}) ∧ F = insert_fset x (F - {|x|})"
by (descending) (auto)

subsection ‹union_fset›

lemmas [simp] =
sup_bot_left[where 'a="'a fset"]
sup_bot_right[where 'a="'a fset"]

lemma union_insert_fset [simp]:
shows "insert_fset x S |∪| T = insert_fset x (S |∪| T)"
by (lifting append.simps(2))

lemma singleton_union_fset_left:
shows "{|a|} |∪| S = insert_fset a S"
by simp

lemma singleton_union_fset_right:
shows "S |∪| {|a|} = insert_fset a S"
by (subst sup.commute) simp

lemma in_union_fset:
shows "x |∈| S |∪| T ⟷ x |∈| S ∨ x |∈| T"
by (descending) (simp)

subsection ‹minus_fset›

lemma minus_in_fset:
shows "x |∈| (xs - ys) ⟷ x |∈| xs ∧ x |∉| ys"
by (descending) (simp)

lemma minus_insert_fset:
shows "insert_fset x xs - ys = (if x |∈| ys then xs - ys else insert_fset x (xs - ys))"
by (descending) (auto)

lemma minus_insert_in_fset[simp]:
shows "x |∈| ys ⟹ insert_fset x xs - ys = xs - ys"

lemma minus_insert_notin_fset[simp]:
shows "x |∉| ys ⟹ insert_fset x xs - ys = insert_fset x (xs - ys)"

lemma in_minus_fset:
shows "x |∈| F - S ⟹ x |∉| S"
unfolding in_fset minus_fset
by blast

lemma notin_minus_fset:
shows "x |∈| S ⟹ x |∉| F - S"
unfolding in_fset minus_fset
by blast

subsection ‹remove_fset›

lemma in_remove_fset:
shows "x |∈| remove_fset y S ⟷ x |∈| S ∧ x ≠ y"
by (descending) (simp)

lemma notin_remove_fset:
shows "x |∉| remove_fset x S"
by (descending) (simp)

lemma notin_remove_ident_fset:
shows "x |∉| S ⟹ remove_fset x S = S"
by (descending) (simp)

lemma remove_fset_cases:
shows "S = {||} ∨ (∃x. x |∈| S ∧ S = insert_fset x (remove_fset x S))"
by (descending) (auto simp add: insert_absorb)

subsection ‹inter_fset›

lemma inter_empty_fset_l:
shows "{||} |∩| S = {||}"
by simp

lemma inter_empty_fset_r:
shows "S |∩| {||} = {||}"
by simp

lemma inter_insert_fset:
shows "insert_fset x S |∩| T = (if x |∈| T then insert_fset x (S |∩| T) else S |∩| T)"
by (descending) (auto)

lemma in_inter_fset:
shows "x |∈| (S |∩| T) ⟷ x |∈| S ∧ x |∈| T"
by (descending) (simp)

subsection ‹subset_fset and psubset_fset›

lemma subset_fset:
shows "xs |⊆| ys ⟷ fset xs ⊆ fset ys"
by (descending) (simp)

lemma psubset_fset:
shows "xs |⊂| ys ⟷ fset xs ⊂ fset ys"
unfolding less_fset_def
by (descending) (auto)

lemma subset_insert_fset:
shows "(insert_fset x xs) |⊆| ys ⟷ x |∈| ys ∧ xs |⊆| ys"
by (descending) (simp)

lemma subset_in_fset:
shows "xs |⊆| ys = (∀x. x |∈| xs ⟶ x |∈| ys)"
by (descending) (auto)

lemma subset_empty_fset:
shows "xs |⊆| {||} ⟷ xs = {||}"
by (descending) (simp)

lemma not_psubset_empty_fset:
shows "¬ xs |⊂| {||}"
by (metis fset_simps(1) psubset_fset not_psubset_empty)

subsection ‹map_fset›

lemma map_fset_simps [simp]:
shows "map_fset f {||} = {||}"
and   "map_fset f (insert_fset x S) = insert_fset (f x) (map_fset f S)"
by (descending, simp)+

lemma map_fset_image [simp]:
shows "fset (map_fset f S) = f ` (fset S)"
by (descending) (simp)

lemma inj_map_fset_cong:
shows "inj f ⟹ map_fset f S = map_fset f T ⟷ S = T"
by (descending) (metis inj_vimage_image_eq list_eq_def set_map)

lemma map_union_fset:
shows "map_fset f (S |∪| T) = map_fset f S |∪| map_fset f T"
by (descending) (simp)

lemma in_fset_map_fset[simp]: "a |∈| map_fset f X = (∃b. b |∈| X ∧ a = f b)"
by descending auto

subsection ‹card_fset›

lemma card_fset:
shows "card_fset xs = card (fset xs)"
by (descending) (simp)

lemma card_insert_fset_iff [simp]:
shows "card_fset (insert_fset x S) = (if x |∈| S then card_fset S else Suc (card_fset S))"

lemma card_fset_0[simp]:
shows "card_fset S = 0 ⟷ S = {||}"
by (descending) (simp)

lemma card_empty_fset[simp]:
shows "card_fset {||} = 0"

lemma card_fset_1:
shows "card_fset S = 1 ⟷ (∃x. S = {|x|})"
by (descending) (auto simp add: card_Suc_eq)

lemma card_fset_gt_0:
shows "x ∈ fset S ⟹ 0 < card_fset S"
by (descending) (auto simp add: card_gt_0_iff)

lemma card_notin_fset:
shows "(x |∉| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
by simp

lemma card_fset_Suc:
shows "card_fset S = Suc n ⟹ ∃x T. x |∉| T ∧ S = insert_fset x T ∧ card_fset T = n"
apply(descending)
apply(auto dest!: card_eq_SucD)
by (metis Diff_insert_absorb set_removeAll)

lemma card_remove_fset_iff [simp]:
shows "card_fset (remove_fset y S) = (if y |∈| S then card_fset S - 1 else card_fset S)"
by (descending) (simp)

lemma card_Suc_exists_in_fset:
shows "card_fset S = Suc n ⟹ ∃a. a |∈| S"
by (drule card_fset_Suc) (auto)

lemma in_card_fset_not_0:
shows "a |∈| A ⟹ card_fset A ≠ 0"
by (descending) (auto)

lemma card_fset_mono:
shows "xs |⊆| ys ⟹ card_fset xs ≤ card_fset ys"
unfolding card_fset psubset_fset

lemma card_subset_fset_eq:
shows "xs |⊆| ys ⟹ card_fset ys ≤ card_fset xs ⟹ xs = ys"
unfolding card_fset subset_fset
by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)

lemma psubset_card_fset_mono:
shows "xs |⊂| ys ⟹ card_fset xs < card_fset ys"
unfolding card_fset subset_fset
by (metis finite_fset psubset_fset psubset_card_mono)

lemma card_union_inter_fset:
shows "card_fset xs + card_fset ys = card_fset (xs |∪| ys) + card_fset (xs |∩| ys)"
unfolding card_fset union_fset inter_fset
by (rule card_Un_Int[OF finite_fset finite_fset])

lemma card_union_disjoint_fset:
shows "xs |∩| ys = {||} ⟹ card_fset (xs |∪| ys) = card_fset xs + card_fset ys"
unfolding card_fset union_fset
apply (rule card_Un_disjoint[OF finite_fset finite_fset])
by (metis inter_fset fset_simps(1))

lemma card_remove_fset_less1:
shows "x |∈| xs ⟹ card_fset (remove_fset x xs) < card_fset xs"
unfolding card_fset in_fset remove_fset
by (rule card_Diff1_less[OF finite_fset])

lemma card_remove_fset_less2:
shows "x |∈| xs ⟹ y |∈| xs ⟹ card_fset (remove_fset y (remove_fset x xs)) < card_fset xs"
unfolding card_fset remove_fset in_fset
by (rule card_Diff2_less[OF finite_fset])

lemma card_remove_fset_le1:
shows "card_fset (remove_fset x xs) ≤ card_fset xs"
unfolding remove_fset card_fset
by (rule card_Diff1_le[OF finite_fset])

lemma card_psubset_fset:
shows "ys |⊆| xs ⟹ card_fset ys < card_fset xs ⟹ ys |⊂| xs"
unfolding card_fset psubset_fset subset_fset
by (rule card_psubset[OF finite_fset])

lemma card_map_fset_le:
shows "card_fset (map_fset f xs) ≤ card_fset xs"
unfolding card_fset map_fset_image
by (rule card_image_le[OF finite_fset])

lemma card_minus_insert_fset[simp]:
assumes "a |∈| A" and "a |∉| B"
shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
using assms
unfolding in_fset card_fset minus_fset

lemma card_minus_subset_fset:
assumes "B |⊆| A"
shows "card_fset (A - B) = card_fset A - card_fset B"
using assms
unfolding subset_fset card_fset minus_fset
by (rule card_Diff_subset[OF finite_fset])

lemma card_minus_fset:
shows "card_fset (A - B) = card_fset A - card_fset (A |∩| B)"
unfolding inter_fset card_fset minus_fset
by (rule card_Diff_subset_Int) (simp)

subsection ‹concat_fset›

lemma concat_empty_fset [simp]:
shows "concat_fset {||} = {||}"
by descending simp

lemma concat_insert_fset [simp]:
shows "concat_fset (insert_fset x S) = x |∪| concat_fset S"
by descending simp

lemma concat_union_fset [simp]:
shows "concat_fset (xs |∪| ys) = concat_fset xs |∪| concat_fset ys"
by descending simp

lemma map_concat_fset:
shows "map_fset f (concat_fset xs) = concat_fset (map_fset (map_fset f) xs)"
by (lifting map_concat)

subsection ‹filter_fset›

lemma subset_filter_fset:
"filter_fset P xs |⊆| filter_fset Q xs = (∀ x. x |∈| xs ⟶ P x ⟶ Q x)"
by descending auto

lemma eq_filter_fset:
"(filter_fset P xs = filter_fset Q xs) = (∀x. x |∈| xs ⟶ P x = Q x)"
by descending auto

lemma psubset_filter_fset:
"(⋀x. x |∈| xs ⟹ P x ⟹ Q x) ⟹ (x |∈| xs & ¬ P x & Q x) ⟹
filter_fset P xs |⊂| filter_fset Q xs"
unfolding less_fset_def by (auto simp add: subset_filter_fset eq_filter_fset)

subsection ‹fold_fset›

lemma fold_empty_fset:
"fold_fset f {||} = id"

lemma fold_insert_fset: "fold_fset f (insert_fset a A) =
(if rsp_fold f then if a |∈| A then fold_fset f A else fold_fset f A ∘ f a else id)"

lemma remdups_removeAll:
"remdups (removeAll x xs) = remove1 x (remdups xs)"
by (induct xs) auto

lemma member_commute_fold_once:
assumes "rsp_fold f"
and "x ∈ set xs"
shows "fold_once f xs = fold_once f (removeAll x xs) ∘ f x"
proof -
from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) ∘ f x"
by (auto intro!: fold_remove1_split elim: rsp_foldE)
then show ?thesis using ‹rsp_fold f› by (simp add: fold_once_fold_remdups remdups_removeAll)
qed

lemma in_commute_fold_fset:
"rsp_fold f ⟹ h |∈| b ⟹ fold_fset f b = fold_fset f (remove_fset h b) ∘ f h"

subsection ‹Choice in fsets›

lemma fset_choice:
assumes a: "∀x. x |∈| A ⟶ (∃y. P x y)"
shows "∃f. ∀x. x |∈| A ⟶ P x (f x)"
using a
apply(descending)
using finite_set_choice

section ‹Induction and Cases rules for fsets›

lemma fset_exhaust [case_names empty insert, cases type: fset]:
assumes empty_fset_case: "S = {||} ⟹ P"
and     insert_fset_case: "⋀x S'. S = insert_fset x S' ⟹ P"
shows "P"
using assms by (lifting list.exhaust)

lemma fset_induct [case_names empty insert]:
assumes empty_fset_case: "P {||}"
and     insert_fset_case: "⋀x S. P S ⟹ P (insert_fset x S)"
shows "P S"
using assms
by (descending) (blast intro: list.induct)

lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
assumes empty_fset_case: "P {||}"
and     insert_fset_case: "⋀x S. ⟦x |∉| S; P S⟧ ⟹ P (insert_fset x S)"
shows "P S"
proof(induct S rule: fset_induct)
case empty
show "P {||}" using empty_fset_case by simp
next
case (insert x S)
have "P S" by fact
then show "P (insert_fset x S)" using insert_fset_case
by (cases "x |∈| S") (simp_all)
qed

lemma fset_card_induct:
assumes empty_fset_case: "P {||}"
and     card_fset_Suc_case: "⋀S T. Suc (card_fset S) = (card_fset T) ⟹ P S ⟹ P T"
shows "P S"
proof (induct S)
case empty
show "P {||}" by (rule empty_fset_case)
next
case (insert x S)
have h: "P S" by fact
have "x |∉| S" by fact
then have "Suc (card_fset S) = card_fset (insert_fset x S)"
using card_fset_Suc by auto
then show "P (insert_fset x S)"
using h card_fset_Suc_case by simp
qed

lemma fset_raw_strong_cases:
obtains "xs = []"
| ys x where "¬ List.member ys x" and "xs ≈ x # ys"
proof (induct xs)
case Nil
then show thesis by simp
next
case (Cons a xs)
have a: "⟦xs = [] ⟹ thesis; ⋀x ys. ⟦¬ List.member ys x; xs ≈ x # ys⟧ ⟹ thesis⟧ ⟹ thesis"
by (rule Cons(1))
have b: "⋀x' ys'. ⟦¬ List.member ys' x'; a # xs ≈ x' # ys'⟧ ⟹ thesis" by fact
have c: "xs = [] ⟹ thesis" using b
apply(simp)
by (metis list.set(1) emptyE empty_subsetI)
have "⋀x ys. ⟦¬ List.member ys x; xs ≈ x # ys⟧ ⟹ thesis"
proof -
fix x :: 'a
fix ys :: "'a list"
assume d:"¬ List.member ys x"
assume e:"xs ≈ x # ys"
show thesis
proof (cases "x = a")
assume h: "x = a"
then have f: "¬ List.member ys a" using d by simp
have g: "a # xs ≈ a # ys" using e h by auto
show thesis using b f g by simp
next
assume h: "x ≠ a"
then have f: "¬ List.member (a # ys) x" using d by auto
have g: "a # xs ≈ x # (a # ys)" using e h by auto
show thesis using b f g by (simp del: List.member_def)
qed
qed
then show thesis using a c by blast
qed

lemma fset_strong_cases:
obtains "xs = {||}"
| ys x where "x |∉| ys" and "xs = insert_fset x ys"
by (lifting fset_raw_strong_cases)

lemma fset_induct2:
"P {||} {||} ⟹
(⋀x xs. x |∉| xs ⟹ P (insert_fset x xs) {||}) ⟹
(⋀y ys. y |∉| ys ⟹ P {||} (insert_fset y ys)) ⟹
(⋀x xs y ys. ⟦P xs ys; x |∉| xs; y |∉| ys⟧ ⟹ P (insert_fset x xs) (insert_fset y ys)) ⟹
P xsa ysa"
apply (induct xsa arbitrary: ysa)
apply (induct_tac x rule: fset_induct_stronger)
apply simp_all
apply (induct_tac xa rule: fset_induct_stronger)
apply simp_all
done

text ‹Extensionality›

lemma fset_eqI:
assumes "⋀x. x ∈ fset A ⟷ x ∈ fset B"
shows "A = B"
using assms proof (induct A arbitrary: B)
case empty then show ?case
by (auto simp add: in_fset none_in_empty_fset [symmetric] sym)
next
case (insert x A)
from insert.prems insert.hyps(1) have "⋀z. z ∈ fset A ⟷ z ∈ fset (B - {|x|})"
then have A: "A = B - {|x|}" by (rule insert.hyps(2))
with insert.prems [symmetric, of x] have "x |∈| B" by (simp add: in_fset)
with A show ?case by (metis in_fset_mdef)
qed

subsection ‹alternate formulation with a different decomposition principle
and a proof of equivalence›

inductive
list_eq2 :: "'a list ⇒ 'a list ⇒ bool" ("_ ≈2 _")
where
"(a # b # xs) ≈2 (b # a # xs)"
| "[] ≈2 []"
| "xs ≈2 ys ⟹ ys ≈2 xs"
| "(a # a # xs) ≈2 (a # xs)"
| "xs ≈2 ys ⟹ (a # xs) ≈2 (a # ys)"
| "xs1 ≈2 xs2 ⟹ xs2 ≈2 xs3 ⟹ xs1 ≈2 xs3"

lemma list_eq2_refl:
shows "xs ≈2 xs"
by (induct xs) (auto intro: list_eq2.intros)

lemma cons_delete_list_eq2:
shows "(a # (removeAll a A)) ≈2 (if List.member A a then A else a # A)"
apply (induct A)
apply (case_tac "List.member (aa # A) a")
apply (simp_all)
apply (case_tac [!] "a = aa")
apply (simp_all)
apply (case_tac "List.member A a")
apply (auto)[2]
apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
done

lemma member_delete_list_eq2:
assumes a: "List.member r e"
shows "(e # removeAll e r) ≈2 r"
using a cons_delete_list_eq2[of e r]
by simp

lemma list_eq2_equiv:
"(l ≈ r) ⟷ (list_eq2 l r)"
proof
show "list_eq2 l r ⟹ l ≈ r" by (induct rule: list_eq2.induct) auto
next
{
fix n
assume a: "card_list l = n" and b: "l ≈ r"
have "l ≈2 r"
using a b
proof (induct n arbitrary: l r)
case 0
have "card_list l = 0" by fact
then have "∀x. ¬ List.member l x" by auto
then have z: "l = []" by auto
then have "r = []" using ‹l ≈ r› by simp
then show ?case using z list_eq2_refl by simp
next
case (Suc m)
have b: "l ≈ r" by fact
have d: "card_list l = Suc m" by fact
then have "∃a. List.member l a"
apply(simp)
apply(drule card_eq_SucD)
apply(blast)
done
then obtain a where e: "List.member l a" by auto
then have e': "List.member r a" using list_eq_def [simplified List.member_def [symmetric], of l r] b
by auto
have f: "card_list (removeAll a l) = m" using e d by (simp)
have g: "removeAll a l ≈ removeAll a r" using remove_fset.rsp b by simp
have "(removeAll a l) ≈2 (removeAll a r)" by (rule Suc.hyps[OF f g])
then have h: "(a # removeAll a l) ≈2 (a # removeAll a r)" by (rule list_eq2.intros(5))
have i: "l ≈2 (a # removeAll a l)"
by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
have "l ≈2 (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
then show ?case using list_eq2.intros(6)[OF _ member_delete_list_eq2[OF e']] by simp
qed
}
then show "l ≈ r ⟹ l ≈2 r" by blast
qed

(* We cannot write it as "assumes .. shows" since Isabelle changes
the quantifiers to schematic variables and reintroduces them in
a different order *)
lemma fset_eq_cases:
"⟦a1 = a2;
⋀a b xs. ⟦a1 = insert_fset a (insert_fset b xs); a2 = insert_fset b (insert_fset a xs)⟧ ⟹ P;
⟦a1 = {||}; a2 = {||}⟧ ⟹ P; ⋀xs ys. ⟦a1 = ys; a2 = xs; xs = ys⟧ ⟹ P;
⋀a xs. ⟦a1 = insert_fset a (insert_fset a xs); a2 = insert_fset a xs⟧ ⟹ P;
⋀xs ys a. ⟦a1 = insert_fset a xs; a2 = insert_fset a ys; xs = ys⟧ ⟹ P;
⋀xs1 xs2 xs3. ⟦a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3⟧ ⟹ P⟧
⟹ P"
by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])

lemma fset_eq_induct:
assumes "x1 = x2"
and "⋀a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
and "P {||} {||}"
and "⋀xs ys. ⟦xs = ys; P xs ys⟧ ⟹ P ys xs"
and "⋀a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
and "⋀xs ys a. ⟦xs = ys; P xs ys⟧ ⟹ P (insert_fset a xs) (insert_fset a ys)"
and "⋀xs1 xs2 xs3. ⟦xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3⟧ ⟹ P xs1 xs3"
shows "P x1 x2"
using assms
by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])

ML ‹
fun dest_fsetT (Type (@{type_name fset}, [T])) = T
| dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
›

no_notation
list_eq (infix "≈" 50) and
list_eq2 (infix "≈2" 50)

end
```