# Theory Lift_FSet

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theory Lift_FSet
imports Quotient_List
`(*  Title:      HOL/Quotient_Examples/Lift_FSet.thy    Author:     Brian Huffman, TU Munich*)header {* Lifting and transfer with a finite set type *}theory Lift_FSetimports "~~/src/HOL/Library/Quotient_List"beginsubsection {* 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 simplemma symp_list_eq: "symp list_eq"  unfolding symp_def by simplemma transp_list_eq: "transp list_eq"  unfolding transp_def by simplemma equivp_list_eq: "equivp list_eq"  by (intro equivpI reflp_list_eq symp_list_eq transp_list_eq)quotient_type 'a fset = "'a list" / "list_eq"  by (rule equivp_list_eq)subsection {* Lifted constant definitions *}lift_definition fnil :: "'a fset" is "[]"  by simplift_definition fcons :: "'a => 'a fset => 'a fset" is Cons  by simplift_definition fappend :: "'a fset => 'a fset => 'a fset" is append  by simplift_definition fmap :: "('a => 'b) => 'a fset => 'b fset" is map  by simplift_definition ffilter :: "('a => bool) => 'a fset => 'a fset" is filter  by simplift_definition fset :: "'a fset => 'a set" is set  by simptext {* 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)  donelemma abs_fset_eq_iff: "abs_fset xs = abs_fset ys <-> list_eq xs ys"  using Quotient_rel [OF Quotient_fset] by simplift_definition fconcat :: "'a fset fset => 'a fset" is concatproof -  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)    doneqedtext {* We can export code: *}export_code fnil fcons fappend fmap ffilter fset in SMLtext {* Note that the generated transfer rule contains a composition  of relations. The transfer rule is not yet very useful in this form. *}lemma "(list_all2 cr_fset OO cr_fset ===> cr_fset) concat fconcat"  by (fact fconcat.transfer)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_deflemma right_unique_pcr_fset [transfer_rule]:  "right_unique A ==> right_unique (pcr_fset A)"  unfolding pcr_fset_def  by (intro right_unique_OO right_unique_list_all2 fset.right_unique)lemma right_total_pcr_fset [transfer_rule]:  "right_total A ==> right_total (pcr_fset A)"  unfolding pcr_fset_def  by (intro right_total_OO right_total_list_all2 fset.right_total)lemma bi_total_pcr_fset [transfer_rule]:  "bi_total A ==> bi_total (pcr_fset A)"  unfolding pcr_fset_def  by (intro bi_total_OO bi_total_list_all2 fset.bi_total)text {* Transfer rules for @{text "pcr_fset"} can be derived from the  existing transfer rules for @{text "cr_fset"} together with the  transfer rules for the polymorphic raw constants. *}text {* Note that the proofs below all have a similar structure and  could potentially be automated. *}lemma fnil_transfer [transfer_rule]:  "(pcr_fset A) [] fnil"  unfolding pcr_fset_def  apply (rule relcomppI)  apply (rule Nil_transfer)  apply (rule fnil.transfer)  donelemma fcons_transfer [transfer_rule]:  "(A ===> pcr_fset A ===> pcr_fset A) Cons fcons"  unfolding pcr_fset_def  apply (intro fun_relI)  apply (elim relcomppE)  apply (rule relcomppI)  apply (erule (1) Cons_transfer [THEN fun_relD, THEN fun_relD])  apply (erule fcons.transfer [THEN fun_relD, THEN fun_relD, OF refl])  donelemma fappend_transfer [transfer_rule]:  "(pcr_fset A ===> pcr_fset A ===> pcr_fset A) append fappend"  unfolding pcr_fset_def  apply (intro fun_relI)  apply (elim relcomppE)  apply (rule relcomppI)  apply (erule (1) append_transfer [THEN fun_relD, THEN fun_relD])  apply (erule (1) fappend.transfer [THEN fun_relD, THEN fun_relD])  donelemma fmap_transfer [transfer_rule]:  "((A ===> B) ===> pcr_fset A ===> pcr_fset B) map fmap"  unfolding pcr_fset_def  apply (intro fun_relI)  apply (elim relcomppE)  apply (rule relcomppI)  apply (erule (1) map_transfer [THEN fun_relD, THEN fun_relD])  apply (erule fmap.transfer [THEN fun_relD, THEN fun_relD, unfolded relator_eq, OF refl])  donelemma ffilter_transfer [transfer_rule]:  "((A ===> op =) ===> pcr_fset A ===> pcr_fset A) filter ffilter"  unfolding pcr_fset_def  apply (intro fun_relI)  apply (elim relcomppE)  apply (rule relcomppI)  apply (erule (1) filter_transfer [THEN fun_relD, THEN fun_relD])  apply (erule ffilter.transfer [THEN fun_relD, THEN fun_relD, unfolded relator_eq, OF refl])  donelemma fset_transfer [transfer_rule]:  "(pcr_fset A ===> set_rel A) set fset"  unfolding pcr_fset_def  apply (intro fun_relI)  apply (elim relcomppE)  apply (drule fset.transfer [THEN fun_relD, unfolded relator_eq])  apply (erule subst)  apply (erule set_transfer [THEN fun_relD])  donelemma fconcat_transfer [transfer_rule]:  "(pcr_fset (pcr_fset A) ===> pcr_fset A) concat fconcat"  unfolding pcr_fset_def  unfolding list_all2_OO  apply (intro fun_relI)  apply (elim relcomppE)  apply (rule relcomppI)  apply (erule concat_transfer [THEN fun_relD])  apply (rule fconcat.transfer [THEN fun_relD])  apply (erule (1) relcomppI)  donelemma 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_proverlemma fset_eq_transfer [transfer_rule]:  assumes "bi_unique A"  shows "(pcr_fset A ===> pcr_fset A ===> op =) list_eq (op =)"  unfolding pcr_fset_def  apply (intro fun_relI)  apply (elim relcomppE)  apply (rule trans)  apply (erule (1) list_eq_transfer [THEN fun_relD, THEN fun_relD, OF assms])  apply (erule (1) fset.rel_eq_transfer [THEN fun_relD, THEN fun_relD])  donetext {* We don't need the original transfer rules any more: *}lemmas [transfer_rule del] =  fset.bi_total fset.right_total fset.right_unique  fnil.transfer fcons.transfer fappend.transfer fmap.transfer  ffilter.transfer fset.transfer fconcat.transfer fset.rel_eq_transfersubsection {* 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  donetext {* 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"  apply transfer'  apply (rule map_map)  donelemma "ffilter p (fmap f xs) = fmap f (ffilter (p o f) xs)"  apply transfer'  apply (rule filter_map)  donelemma "ffilter p (ffilter q xs) = ffilter (λx. q x ∧ p x) xs"  apply transfer'  apply (rule filter_filter)  donelemma "fset (fcons x xs) = insert x (fset xs)"  apply transfer  apply (rule set.simps)  donelemma "fset (fappend xs ys) = fset xs ∪ fset ys"  apply transfer  apply (rule set_append)  donelemma "fset (fconcat xss) = (\<Union>xs∈fset xss. fset xs)"  apply transfer  apply (rule set_concat)  donelemma "∀x∈fset xs. f x = g x ==> fmap f xs = fmap g xs"  apply transfer  apply (simp cong: map_cong del: set_map)  donelemma "fnil = fconcat xss <-> (∀xs∈fset xss. xs = fnil)"  apply transfer  apply simp  donelemma "fconcat (fmap (λx. fcons x fnil) xs) = xs"  apply transfer'  apply simp  donelemma concat_map_concat: "concat (map concat xsss) = concat (concat xsss)"  by (induct xsss, simp_all)lemma "fconcat (fmap fconcat xss) = fconcat (fconcat xss)"  apply transfer'  apply (rule concat_map_concat)  doneend`