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(* Title: HOL/Quotient_Examples/Lift_FSet.thy
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Author: Brian Huffman, TU Munich
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*)
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header {* Lifting and transfer with a finite set type *}
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theory Lift_FSet
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imports "~~/src/HOL/Library/Quotient_List"
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begin
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subsection {* Equivalence relation and quotient type definition *}
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definition list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
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where [simp]: "list_eq xs ys \<longleftrightarrow> set xs = set ys"
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lemma reflp_list_eq: "reflp list_eq"
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unfolding reflp_def by simp
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lemma symp_list_eq: "symp list_eq"
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unfolding symp_def by simp
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lemma transp_list_eq: "transp list_eq"
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unfolding transp_def by simp
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lemma equivp_list_eq: "equivp list_eq"
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by (intro equivpI reflp_list_eq symp_list_eq transp_list_eq)
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quotient_type 'a fset = "'a list" / "list_eq"
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by (rule equivp_list_eq)
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subsection {* Lifted constant definitions *}
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lift_definition fnil :: "'a fset" is "[]"
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by simp
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lift_definition fcons :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Cons
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by simp
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lift_definition fappend :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is append
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by simp
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lift_definition fmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" is map
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by simp
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lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is filter
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by simp
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lift_definition fset :: "'a fset \<Rightarrow> 'a set" is set
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by simp
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text {* Constants with nested types (like concat) yield a more
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complicated proof obligation. *}
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lemma list_all2_cr_fset:
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"list_all2 cr_fset xs ys \<longleftrightarrow> map abs_fset xs = ys"
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unfolding cr_fset_def
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apply safe
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apply (erule list_all2_induct, simp, simp)
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apply (simp add: list_all2_map2 List.list_all2_refl)
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done
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lemma abs_fset_eq_iff: "abs_fset xs = abs_fset ys \<longleftrightarrow> list_eq xs ys"
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using Quotient_rel [OF Quotient_fset] by simp
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lift_definition fconcat :: "'a fset fset \<Rightarrow> 'a fset" is concat
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proof -
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fix xss yss :: "'a list list"
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assume "(list_all2 cr_fset OO list_eq OO (list_all2 cr_fset)\<inverse>\<inverse>) xss yss"
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then obtain uss vss where
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"list_all2 cr_fset xss uss" and "list_eq uss vss" and
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"list_all2 cr_fset yss vss" by clarsimp
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hence "list_eq (map abs_fset xss) (map abs_fset yss)"
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unfolding list_all2_cr_fset by simp
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thus "list_eq (concat xss) (concat yss)"
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apply (simp add: set_eq_iff image_def)
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apply safe
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apply (rename_tac xs, drule_tac x="abs_fset xs" in spec)
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apply (drule iffD1, fast, clarsimp simp add: abs_fset_eq_iff, fast)
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apply (rename_tac xs, drule_tac x="abs_fset xs" in spec)
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apply (drule iffD2, fast, clarsimp simp add: abs_fset_eq_iff, fast)
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done
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qed
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text {* Note that the generated transfer rule contains a composition
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of relations. The transfer rule is not yet very useful in this form. *}
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lemma "(list_all2 cr_fset OO cr_fset ===> cr_fset) concat fconcat"
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by (fact fconcat.transfer)
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subsection {* Using transfer with type @{text "fset"} *}
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text {* The correspondence relation @{text "cr_fset"} can only relate
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@{text "list"} and @{text "fset"} types with the same element type.
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To relate nested types like @{text "'a list list"} and
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@{text "'a fset fset"}, we define a parameterized version of the
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correspondence relation, @{text "cr_fset'"}. *}
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definition cr_fset' :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b fset \<Rightarrow> bool"
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where "cr_fset' R = list_all2 R OO cr_fset"
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lemma right_unique_cr_fset' [transfer_rule]:
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"right_unique A \<Longrightarrow> right_unique (cr_fset' A)"
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unfolding cr_fset'_def
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by (intro right_unique_OO right_unique_list_all2 fset.right_unique)
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lemma right_total_cr_fset' [transfer_rule]:
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"right_total A \<Longrightarrow> right_total (cr_fset' A)"
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unfolding cr_fset'_def
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by (intro right_total_OO right_total_list_all2 fset.right_total)
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lemma bi_total_cr_fset' [transfer_rule]:
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"bi_total A \<Longrightarrow> bi_total (cr_fset' A)"
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unfolding cr_fset'_def
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by (intro bi_total_OO bi_total_list_all2 fset.bi_total)
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text {* Transfer rules for @{text "cr_fset'"} can be derived from the
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existing transfer rules for @{text "cr_fset"} together with the
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transfer rules for the polymorphic raw constants. *}
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text {* Note that the proofs below all have a similar structure and
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could potentially be automated. *}
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lemma fnil_transfer [transfer_rule]:
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"(cr_fset' A) [] fnil"
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unfolding cr_fset'_def
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apply (rule relcomppI)
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apply (rule Nil_transfer)
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apply (rule fnil.transfer)
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done
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lemma fcons_transfer [transfer_rule]:
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"(A ===> cr_fset' A ===> cr_fset' A) Cons fcons"
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unfolding cr_fset'_def
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (rule relcomppI)
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apply (erule (1) Cons_transfer [THEN fun_relD, THEN fun_relD])
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apply (erule fcons.transfer [THEN fun_relD, THEN fun_relD, OF refl])
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done
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lemma fappend_transfer [transfer_rule]:
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"(cr_fset' A ===> cr_fset' A ===> cr_fset' A) append fappend"
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unfolding cr_fset'_def
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (rule relcomppI)
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apply (erule (1) append_transfer [THEN fun_relD, THEN fun_relD])
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apply (erule (1) fappend.transfer [THEN fun_relD, THEN fun_relD])
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done
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lemma fmap_transfer [transfer_rule]:
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"((A ===> B) ===> cr_fset' A ===> cr_fset' B) map fmap"
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unfolding cr_fset'_def
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (rule relcomppI)
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apply (erule (1) map_transfer [THEN fun_relD, THEN fun_relD])
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apply (erule fmap.transfer [THEN fun_relD, THEN fun_relD, OF refl])
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done
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lemma ffilter_transfer [transfer_rule]:
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"((A ===> op =) ===> cr_fset' A ===> cr_fset' A) filter ffilter"
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unfolding cr_fset'_def
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (rule relcomppI)
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apply (erule (1) filter_transfer [THEN fun_relD, THEN fun_relD])
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apply (erule ffilter.transfer [THEN fun_relD, THEN fun_relD, OF refl])
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done
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lemma fset_transfer [transfer_rule]:
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"(cr_fset' A ===> set_rel A) set fset"
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unfolding cr_fset'_def
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (drule fset.transfer [THEN fun_relD])
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apply (erule subst)
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apply (erule set_transfer [THEN fun_relD])
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done
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lemma fconcat_transfer [transfer_rule]:
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"(cr_fset' (cr_fset' A) ===> cr_fset' A) concat fconcat"
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unfolding cr_fset'_def
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unfolding list_all2_OO
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (rule relcomppI)
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apply (erule concat_transfer [THEN fun_relD])
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apply (rule fconcat.transfer [THEN fun_relD])
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apply (erule (1) relcomppI)
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done
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lemma list_eq_transfer [transfer_rule]:
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assumes [transfer_rule]: "bi_unique A"
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shows "(list_all2 A ===> list_all2 A ===> op =) list_eq list_eq"
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unfolding list_eq_def [abs_def] by transfer_prover
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lemma fset_eq_transfer [transfer_rule]:
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assumes "bi_unique A"
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shows "(cr_fset' A ===> cr_fset' A ===> op =) list_eq (op =)"
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unfolding cr_fset'_def
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apply (intro fun_relI)
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apply (elim relcomppE)
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apply (rule trans)
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apply (erule (1) list_eq_transfer [THEN fun_relD, THEN fun_relD, OF assms])
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apply (erule (1) fset.rel_eq_transfer [THEN fun_relD, THEN fun_relD])
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done
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text {* We don't need the original transfer rules any more: *}
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lemmas [transfer_rule del] =
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fset.bi_total fset.right_total fset.right_unique
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fnil.transfer fcons.transfer fappend.transfer fmap.transfer
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ffilter.transfer fset.transfer fconcat.transfer fset.rel_eq_transfer
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subsection {* Transfer examples *}
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text {* The @{text "transfer"} method replaces equality on @{text
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"fset"} with the @{text "list_eq"} relation on lists, which is
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logically equivalent. *}
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lemma "fmap f (fmap g xs) = fmap (f \<circ> g) xs"
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apply transfer
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apply simp
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done
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text {* The @{text "transfer'"} variant can replace equality on @{text
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"fset"} with equality on @{text "list"}, which is logically stronger
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but sometimes more convenient. *}
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lemma "fmap f (fmap g xs) = fmap (f \<circ> g) xs"
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apply transfer'
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apply (rule map_map)
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done
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lemma "ffilter p (fmap f xs) = fmap f (ffilter (p \<circ> f) xs)"
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apply transfer'
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apply (rule filter_map)
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done
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lemma "ffilter p (ffilter q xs) = ffilter (\<lambda>x. q x \<and> p x) xs"
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apply transfer'
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apply (rule filter_filter)
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done
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lemma "fset (fcons x xs) = insert x (fset xs)"
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apply transfer
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apply (rule set.simps)
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done
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lemma "fset (fappend xs ys) = fset xs \<union> fset ys"
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apply transfer
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apply (rule set_append)
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done
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lemma "fset (fconcat xss) = (\<Union>xs\<in>fset xss. fset xs)"
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apply transfer
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apply (rule set_concat)
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done
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lemma "\<forall>x\<in>fset xs. f x = g x \<Longrightarrow> fmap f xs = fmap g xs"
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apply transfer
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apply (simp cong: map_cong del: set_map)
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done
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lemma "fnil = fconcat xss \<longleftrightarrow> (\<forall>xs\<in>fset xss. xs = fnil)"
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apply transfer
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apply simp
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done
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lemma "fconcat (fmap (\<lambda>x. fcons x fnil) xs) = xs"
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apply transfer'
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apply simp
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done
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lemma concat_map_concat: "concat (map concat xsss) = concat (concat xsss)"
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by (induct xsss, simp_all)
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lemma "fconcat (fmap fconcat xss) = fconcat (fconcat xss)"
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apply transfer'
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apply (rule concat_map_concat)
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done
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end
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