Theory Lift_FSet

theory Lift_FSet
imports Main
(*  Title:      HOL/Quotient_Examples/Lift_FSet.thy
Author: Brian Huffman, TU Munich
*)


header {* Lifting and transfer with a finite set type *}

theory Lift_FSet
imports Main
begin

subsection {* Equivalence relation and quotient type definition *}

definition list_eq :: "'a list => 'a list => bool"
where [simp]: "list_eq xs ys <-> set xs = set ys"

lemma reflp_list_eq: "reflp list_eq"
unfolding reflp_def by simp

lemma symp_list_eq: "symp list_eq"
unfolding symp_def by simp

lemma transp_list_eq: "transp list_eq"
unfolding transp_def by simp

lemma equivp_list_eq: "equivp list_eq"
by (intro equivpI reflp_list_eq symp_list_eq transp_list_eq)

context
begin
interpretation lifting_syntax .

lemma list_eq_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> list_all2 A ===> op =) list_eq list_eq"
unfolding list_eq_def [abs_def] by transfer_prover

quotient_type 'a fset = "'a list" / "list_eq" parametric list_eq_transfer
by (rule equivp_list_eq)

subsection {* Lifted constant definitions *}

lift_definition fnil :: "'a fset" ("{||}") is "[]" parametric Nil_transfer .

lift_definition fcons :: "'a => 'a fset => 'a fset" is Cons parametric Cons_transfer
by simp

lift_definition fappend :: "'a fset => 'a fset => 'a fset" is append parametric append_transfer
by simp

lift_definition fmap :: "('a => 'b) => 'a fset => 'b fset" is map parametric map_transfer
by simp

lift_definition ffilter :: "('a => bool) => 'a fset => 'a fset" is filter parametric filter_transfer
by simp

lift_definition fset :: "'a fset => 'a set" is set parametric set_transfer
by simp

text {* Constants with nested types (like concat) yield a more
complicated proof obligation. *}


lemma list_all2_cr_fset:
"list_all2 cr_fset xs ys <-> map abs_fset xs = ys"
unfolding cr_fset_def
apply safe
apply (erule list_all2_induct, simp, simp)
apply (simp add: list_all2_map2 List.list_all2_refl)
done

lemma abs_fset_eq_iff: "abs_fset xs = abs_fset ys <-> list_eq xs ys"
using Quotient_rel [OF Quotient_fset] by simp

lift_definition fconcat :: "'a fset fset => 'a fset" is concat parametric concat_transfer
proof -
fix xss yss :: "'a list list"
assume "(list_all2 cr_fset OO list_eq OO (list_all2 cr_fset)¯¯) xss yss"
then obtain uss vss where
"list_all2 cr_fset xss uss" and "list_eq uss vss" and
"list_all2 cr_fset yss vss" by clarsimp
hence "list_eq (map abs_fset xss) (map abs_fset yss)"
unfolding list_all2_cr_fset by simp
thus "list_eq (concat xss) (concat yss)"
apply (simp add: set_eq_iff image_def)
apply safe
apply (rename_tac xs, drule_tac x="abs_fset xs" in spec)
apply (drule iffD1, fast, clarsimp simp add: abs_fset_eq_iff, fast)
apply (rename_tac xs, drule_tac x="abs_fset xs" in spec)
apply (drule iffD2, fast, clarsimp simp add: abs_fset_eq_iff, fast)
done
qed

lemma member_transfer:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> list_all2 A ===> op=) (λx xs. x ∈ set xs) (λx xs. x ∈ set xs)"
by transfer_prover

end

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

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

lift_definition fmember :: "'a => 'a fset => bool" (infix "|∈|" 50) is "λx xs. x ∈ set xs"
parametric member_transfer by simp

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

lemma fmember_fmap[simp]: "a |∈| fmap f X = (∃b. b |∈| X ∧ a = f b)"
by transfer auto

text {* We can export code: *}

export_code fnil fcons fappend fmap ffilter fset fmember in SML

subsection {* Using transfer with type @{text "fset"} *}

text {* The correspondence relation @{text "cr_fset"} can only relate
@{text "list"} and @{text "fset"} types with the same element type.
To relate nested types like @{text "'a list list"} and
@{text "'a fset fset"}, we define a parameterized version of the
correspondence relation, @{text "pcr_fset"}. *}


thm pcr_fset_def

subsection {* Transfer examples *}

text {* The @{text "transfer"} method replaces equality on @{text
"fset"} with the @{text "list_eq"} relation on lists, which is
logically equivalent. *}


lemma "fmap f (fmap g xs) = fmap (f o g) xs"
apply transfer
apply simp
done

text {* The @{text "transfer'"} variant can replace equality on @{text
"fset"} with equality on @{text "list"}, which is logically stronger
but sometimes more convenient. *}


lemma "fmap f (fmap g xs) = fmap (f o g) xs"
using map_map [Transfer.transferred] .

lemma "ffilter p (fmap f xs) = fmap f (ffilter (p o f) xs)"
using filter_map [Transfer.transferred] .

lemma "ffilter p (ffilter q xs) = ffilter (λx. q x ∧ p x) xs"
using filter_filter [Transfer.transferred] .

lemma "fset (fcons x xs) = insert x (fset xs)"
using set.simps(2) [Transfer.transferred] .

lemma "fset (fappend xs ys) = fset xs ∪ fset ys"
using set_append [Transfer.transferred] .

lemma "fset (fconcat xss) = (\<Union>xs∈fset xss. fset xs)"
using set_concat [Transfer.transferred] .

lemma "∀x∈fset xs. f x = g x ==> fmap f xs = fmap g xs"
apply transfer
apply (simp cong: map_cong del: set_map)
done

lemma "fnil = fconcat xss <-> (∀xs∈fset xss. xs = fnil)"
apply transfer
apply simp
done

lemma "fconcat (fmap (λx. fcons x fnil) xs) = xs"
apply transfer
apply simp
done

lemma concat_map_concat: "concat (map concat xsss) = concat (concat xsss)"
by (induct xsss, simp_all)

lemma "fconcat (fmap fconcat xss) = fconcat (fconcat xss)"
using concat_map_concat [Transfer.transferred] .

end