wenzelm@47455: (* Title: HOL/Library/Quotient_Set.thy kaliszyk@44413: Author: Cezary Kaliszyk and Christian Urban kaliszyk@44413: *) kaliszyk@44413: kaliszyk@44413: header {* Quotient infrastructure for the set type *} kaliszyk@44413: kaliszyk@44413: theory Quotient_Set kaliszyk@44413: imports Main Quotient_Syntax kaliszyk@44413: begin kaliszyk@44413: huffman@47648: subsection {* Relator for set type *} huffman@47648: huffman@47648: definition set_rel :: "('a \ 'b \ bool) \ 'a set \ 'b set \ bool" huffman@47648: where "set_rel R = (\A B. (\x\A. \y\B. R x y) \ (\y\B. \x\A. R x y))" huffman@47648: huffman@47648: lemma set_relI: huffman@47648: assumes "\x. x \ A \ \y\B. R x y" huffman@47648: assumes "\y. y \ B \ \x\A. R x y" huffman@47648: shows "set_rel R A B" huffman@47648: using assms unfolding set_rel_def by simp huffman@47648: huffman@47648: lemma set_rel_conversep: "set_rel (conversep R) = conversep (set_rel R)" huffman@47648: unfolding set_rel_def by auto huffman@47648: huffman@47648: lemma set_rel_OO: "set_rel (R OO S) = set_rel R OO set_rel S" huffman@47648: apply (intro ext, rename_tac X Z) huffman@47648: apply (rule iffI) huffman@47648: apply (rule_tac b="{y. (\x\X. R x y) \ (\z\Z. S y z)}" in relcomppI) huffman@47648: apply (simp add: set_rel_def, fast) huffman@47648: apply (simp add: set_rel_def, fast) huffman@47648: apply (simp add: set_rel_def, fast) huffman@47648: done huffman@47648: huffman@47648: lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)" huffman@47648: unfolding set_rel_def fun_eq_iff by auto huffman@47648: huffman@47648: lemma reflp_set_rel: "reflp R \ reflp (set_rel R)" huffman@47648: unfolding reflp_def set_rel_def by fast huffman@47648: huffman@47648: lemma symp_set_rel: "symp R \ symp (set_rel R)" huffman@47648: unfolding symp_def set_rel_def by fast huffman@47648: huffman@47648: lemma transp_set_rel: "transp R \ transp (set_rel R)" huffman@47648: unfolding transp_def set_rel_def by fast huffman@47648: huffman@47648: lemma equivp_set_rel: "equivp R \ equivp (set_rel R)" huffman@47648: by (blast intro: equivpI reflp_set_rel symp_set_rel transp_set_rel huffman@47648: elim: equivpE) huffman@47648: huffman@47648: lemma right_total_set_rel [transfer_rule]: huffman@47648: "right_total A \ right_total (set_rel A)" huffman@47648: unfolding right_total_def set_rel_def huffman@47648: by (rule allI, rename_tac Y, rule_tac x="{x. \y\Y. A x y}" in exI, fast) huffman@47648: huffman@47648: lemma right_unique_set_rel [transfer_rule]: huffman@47648: "right_unique A \ right_unique (set_rel A)" huffman@47648: unfolding right_unique_def set_rel_def by fast huffman@47648: huffman@47648: lemma bi_total_set_rel [transfer_rule]: huffman@47648: "bi_total A \ bi_total (set_rel A)" huffman@47648: unfolding bi_total_def set_rel_def huffman@47648: apply safe huffman@47648: apply (rename_tac X, rule_tac x="{y. \x\X. A x y}" in exI, fast) huffman@47648: apply (rename_tac Y, rule_tac x="{x. \y\Y. A x y}" in exI, fast) huffman@47648: done huffman@47648: huffman@47648: lemma bi_unique_set_rel [transfer_rule]: huffman@47648: "bi_unique A \ bi_unique (set_rel A)" huffman@47648: unfolding bi_unique_def set_rel_def by fast huffman@47648: huffman@47648: subsection {* Transfer rules for transfer package *} huffman@47648: huffman@47648: subsubsection {* Unconditional transfer rules *} huffman@47648: huffman@47648: lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}" huffman@47648: unfolding set_rel_def by simp huffman@47648: huffman@47648: lemma insert_transfer [transfer_rule]: huffman@47648: "(A ===> set_rel A ===> set_rel A) insert insert" huffman@47648: unfolding fun_rel_def set_rel_def by auto huffman@47648: huffman@47648: lemma union_transfer [transfer_rule]: huffman@47648: "(set_rel A ===> set_rel A ===> set_rel A) union union" huffman@47648: unfolding fun_rel_def set_rel_def by auto huffman@47648: huffman@47648: lemma Union_transfer [transfer_rule]: huffman@47648: "(set_rel (set_rel A) ===> set_rel A) Union Union" huffman@47648: unfolding fun_rel_def set_rel_def by simp fast huffman@47648: huffman@47648: lemma image_transfer [transfer_rule]: huffman@47648: "((A ===> B) ===> set_rel A ===> set_rel B) image image" huffman@47648: unfolding fun_rel_def set_rel_def by simp fast huffman@47648: huffman@47660: lemma UNION_transfer [transfer_rule]: huffman@47660: "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION" huffman@47660: unfolding SUP_def [abs_def] by transfer_prover huffman@47660: huffman@47648: lemma Ball_transfer [transfer_rule]: huffman@47648: "(set_rel A ===> (A ===> op =) ===> op =) Ball Ball" huffman@47648: unfolding set_rel_def fun_rel_def by fast huffman@47648: huffman@47648: lemma Bex_transfer [transfer_rule]: huffman@47648: "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex" huffman@47648: unfolding set_rel_def fun_rel_def by fast huffman@47648: huffman@47648: lemma Pow_transfer [transfer_rule]: huffman@47648: "(set_rel A ===> set_rel (set_rel A)) Pow Pow" huffman@47648: apply (rule fun_relI, rename_tac X Y, rule set_relI) huffman@47648: apply (rename_tac X', rule_tac x="{y\Y. \x\X'. A x y}" in rev_bexI, clarsimp) huffman@47648: apply (simp add: set_rel_def, fast) huffman@47648: apply (rename_tac Y', rule_tac x="{x\X. \y\Y'. A x y}" in rev_bexI, clarsimp) huffman@47648: apply (simp add: set_rel_def, fast) huffman@47648: done huffman@47648: huffman@47648: subsubsection {* Rules requiring bi-unique or bi-total relations *} huffman@47648: huffman@47648: lemma member_transfer [transfer_rule]: huffman@47648: assumes "bi_unique A" huffman@47648: shows "(A ===> set_rel A ===> op =) (op \) (op \)" huffman@47648: using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast huffman@47648: huffman@47648: lemma Collect_transfer [transfer_rule]: huffman@47648: assumes "bi_total A" huffman@47648: shows "((A ===> op =) ===> set_rel A) Collect Collect" huffman@47648: using assms unfolding fun_rel_def set_rel_def bi_total_def by fast huffman@47648: huffman@47648: lemma inter_transfer [transfer_rule]: huffman@47648: assumes "bi_unique A" huffman@47648: shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter" huffman@47648: using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast huffman@47648: huffman@47680: lemma Diff_transfer [transfer_rule]: huffman@47680: assumes "bi_unique A" huffman@47680: shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)" huffman@47680: using assms unfolding fun_rel_def set_rel_def bi_unique_def huffman@47680: unfolding Ball_def Bex_def Diff_eq huffman@47680: by (safe, simp, metis, simp, metis) huffman@47680: huffman@47648: lemma subset_transfer [transfer_rule]: huffman@47648: assumes [transfer_rule]: "bi_unique A" huffman@47648: shows "(set_rel A ===> set_rel A ===> op =) (op \) (op \)" huffman@47648: unfolding subset_eq [abs_def] by transfer_prover huffman@47648: huffman@47648: lemma UNIV_transfer [transfer_rule]: huffman@47648: assumes "bi_total A" huffman@47648: shows "(set_rel A) UNIV UNIV" huffman@47648: using assms unfolding set_rel_def bi_total_def by simp huffman@47648: huffman@47648: lemma Compl_transfer [transfer_rule]: huffman@47648: assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" huffman@47648: shows "(set_rel A ===> set_rel A) uminus uminus" huffman@47648: unfolding Compl_eq [abs_def] by transfer_prover huffman@47648: huffman@47648: lemma Inter_transfer [transfer_rule]: huffman@47648: assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" huffman@47648: shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter" huffman@47648: unfolding Inter_eq [abs_def] by transfer_prover huffman@47648: huffman@47648: lemma finite_transfer [transfer_rule]: huffman@47648: assumes "bi_unique A" huffman@47648: shows "(set_rel A ===> op =) finite finite" huffman@47648: apply (rule fun_relI, rename_tac X Y) huffman@47648: apply (rule iffI) huffman@47648: apply (subgoal_tac "Y \ (\x. THE y. A x y) ` X") huffman@47648: apply (erule finite_subset, erule finite_imageI) huffman@47648: apply (rule subsetI, rename_tac y) huffman@47648: apply (clarsimp simp add: set_rel_def) huffman@47648: apply (drule (1) bspec, clarify) huffman@47648: apply (rule image_eqI) huffman@47648: apply (rule the_equality [symmetric]) huffman@47648: apply assumption huffman@47648: apply (simp add: assms [unfolded bi_unique_def]) huffman@47648: apply assumption huffman@47648: apply (subgoal_tac "X \ (\y. THE x. A x y) ` Y") huffman@47648: apply (erule finite_subset, erule finite_imageI) huffman@47648: apply (rule subsetI, rename_tac x) huffman@47648: apply (clarsimp simp add: set_rel_def) huffman@47648: apply (drule (1) bspec, clarify) huffman@47648: apply (rule image_eqI) huffman@47648: apply (rule the_equality [symmetric]) huffman@47648: apply assumption huffman@47648: apply (simp add: assms [unfolded bi_unique_def]) huffman@47648: apply assumption huffman@47648: done huffman@47648: huffman@47648: subsection {* Setup for lifting package *} huffman@47648: huffman@47648: lemma Quotient_set: huffman@47648: assumes "Quotient R Abs Rep T" huffman@47648: shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)" huffman@47648: using assms unfolding Quotient_alt_def4 huffman@47648: apply (simp add: set_rel_OO set_rel_conversep) huffman@47648: apply (simp add: set_rel_def, fast) huffman@47648: done huffman@47648: huffman@47648: declare [[map set = (set_rel, Quotient_set)]] huffman@47648: huffman@47648: lemma set_invariant_commute [invariant_commute]: huffman@47648: "set_rel (Lifting.invariant P) = Lifting.invariant (\A. Ball A P)" huffman@47648: unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast huffman@47648: huffman@47648: subsection {* Contravariant set map (vimage) and set relator *} huffman@47626: huffman@47647: definition "vset_rel R xs ys \ \x y. R x y \ x \ xs \ y \ ys" huffman@47626: huffman@47647: lemma vset_rel_eq [id_simps]: huffman@47647: "vset_rel op = = op =" huffman@47647: by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff vset_rel_def) huffman@47626: huffman@47647: lemma vset_rel_equivp: huffman@47626: assumes e: "equivp R" huffman@47647: shows "vset_rel R xs ys \ xs = ys \ (\x y. x \ xs \ R x y \ y \ xs)" huffman@47647: unfolding vset_rel_def huffman@47626: using equivp_reflp[OF e] huffman@47626: by auto (metis, metis equivp_symp[OF e]) huffman@47626: kaliszyk@44413: lemma set_quotient [quot_thm]: kuncar@47308: assumes "Quotient3 R Abs Rep" huffman@47647: shows "Quotient3 (vset_rel R) (vimage Rep) (vimage Abs)" kuncar@47308: proof (rule Quotient3I) kuncar@47308: from assms have "\x. Abs (Rep x) = x" by (rule Quotient3_abs_rep) kaliszyk@44413: then show "\xs. Rep -` (Abs -` xs) = xs" kaliszyk@44413: unfolding vimage_def by auto kaliszyk@44413: next huffman@47647: show "\xs. vset_rel R (Abs -` xs) (Abs -` xs)" huffman@47647: unfolding vset_rel_def vimage_def kuncar@47308: by auto (metis Quotient3_rel_abs[OF assms])+ kaliszyk@44413: next kaliszyk@44413: fix r s huffman@47647: show "vset_rel R r s = (vset_rel R r r \ vset_rel R s s \ Rep -` r = Rep -` s)" huffman@47647: unfolding vset_rel_def vimage_def set_eq_iff kuncar@47308: by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+ kaliszyk@44413: qed kaliszyk@44413: huffman@47647: declare [[mapQ3 set = (vset_rel, set_quotient)]] kuncar@47094: kaliszyk@44413: lemma empty_set_rsp[quot_respect]: huffman@47647: "vset_rel R {} {}" huffman@47647: unfolding vset_rel_def by simp kaliszyk@44413: kaliszyk@44413: lemma collect_rsp[quot_respect]: kuncar@47308: assumes "Quotient3 R Abs Rep" huffman@47647: shows "((R ===> op =) ===> vset_rel R) Collect Collect" huffman@47647: by (intro fun_relI) (simp add: fun_rel_def vset_rel_def) kaliszyk@44413: kaliszyk@44413: lemma collect_prs[quot_preserve]: kuncar@47308: assumes "Quotient3 R Abs Rep" kaliszyk@44413: shows "((Abs ---> id) ---> op -` Rep) Collect = Collect" kaliszyk@44413: unfolding fun_eq_iff kuncar@47308: by (simp add: Quotient3_abs_rep[OF assms]) kaliszyk@44413: kaliszyk@44413: lemma union_rsp[quot_respect]: kuncar@47308: assumes "Quotient3 R Abs Rep" huffman@47647: shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \ op \" huffman@47647: by (intro fun_relI) (simp add: vset_rel_def) kaliszyk@44413: kaliszyk@44413: lemma union_prs[quot_preserve]: kuncar@47308: assumes "Quotient3 R Abs Rep" kaliszyk@44413: shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \ = op \" kaliszyk@44413: unfolding fun_eq_iff kuncar@47308: by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) kaliszyk@44413: kaliszyk@44413: lemma diff_rsp[quot_respect]: kuncar@47308: assumes "Quotient3 R Abs Rep" huffman@47647: shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op - op -" huffman@47647: by (intro fun_relI) (simp add: vset_rel_def) kaliszyk@44413: kaliszyk@44413: lemma diff_prs[quot_preserve]: kuncar@47308: assumes "Quotient3 R Abs Rep" kaliszyk@44413: shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op - = op -" kaliszyk@44413: unfolding fun_eq_iff kuncar@47308: by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]] vimage_Diff) kaliszyk@44413: kaliszyk@44413: lemma inter_rsp[quot_respect]: kuncar@47308: assumes "Quotient3 R Abs Rep" huffman@47647: shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \ op \" huffman@47647: by (intro fun_relI) (auto simp add: vset_rel_def) kaliszyk@44413: kaliszyk@44413: lemma inter_prs[quot_preserve]: kuncar@47308: assumes "Quotient3 R Abs Rep" kaliszyk@44413: shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \ = op \" kaliszyk@44413: unfolding fun_eq_iff kuncar@47308: by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) kaliszyk@44413: kaliszyk@44459: lemma mem_prs[quot_preserve]: kuncar@47308: assumes "Quotient3 R Abs Rep" kaliszyk@44459: shows "(Rep ---> op -` Abs ---> id) op \ = op \" kuncar@47308: by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms]) kaliszyk@44459: haftmann@45970: lemma mem_rsp[quot_respect]: huffman@47647: shows "(R ===> vset_rel R ===> op =) op \ op \" huffman@47647: by (intro fun_relI) (simp add: vset_rel_def) kaliszyk@44459: kaliszyk@44413: end