wenzelm@10358: (* Title: HOL/Relation.thy haftmann@46664: Author: Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen nipkow@1128: *) nipkow@1128: wenzelm@56790: header {* Relations -- as sets of pairs, and binary predicates *} berghofe@12905: nipkow@15131: theory Relation blanchet@54555: imports Finite_Set nipkow@15131: begin paulson@5978: haftmann@46694: text {* A preliminary: classical rules for reasoning on predicates *} haftmann@46664: noschinl@46882: declare predicate1I [Pure.intro!, intro!] noschinl@46882: declare predicate1D [Pure.dest, dest] haftmann@46664: declare predicate2I [Pure.intro!, intro!] haftmann@46664: declare predicate2D [Pure.dest, dest] haftmann@46767: declare bot1E [elim!] haftmann@46664: declare bot2E [elim!] haftmann@46664: declare top1I [intro!] haftmann@46664: declare top2I [intro!] haftmann@46664: declare inf1I [intro!] haftmann@46664: declare inf2I [intro!] haftmann@46664: declare inf1E [elim!] haftmann@46664: declare inf2E [elim!] haftmann@46664: declare sup1I1 [intro?] haftmann@46664: declare sup2I1 [intro?] haftmann@46664: declare sup1I2 [intro?] haftmann@46664: declare sup2I2 [intro?] haftmann@46664: declare sup1E [elim!] haftmann@46664: declare sup2E [elim!] haftmann@46664: declare sup1CI [intro!] haftmann@46664: declare sup2CI [intro!] haftmann@56742: declare Inf1_I [intro!] haftmann@46664: declare INF1_I [intro!] haftmann@56742: declare Inf2_I [intro!] haftmann@46664: declare INF2_I [intro!] haftmann@56742: declare Inf1_D [elim] haftmann@46664: declare INF1_D [elim] haftmann@56742: declare Inf2_D [elim] haftmann@46664: declare INF2_D [elim] haftmann@56742: declare Inf1_E [elim] haftmann@46664: declare INF1_E [elim] haftmann@56742: declare Inf2_E [elim] haftmann@46664: declare INF2_E [elim] haftmann@56742: declare Sup1_I [intro] haftmann@46664: declare SUP1_I [intro] haftmann@56742: declare Sup2_I [intro] haftmann@46664: declare SUP2_I [intro] haftmann@56742: declare Sup1_E [elim!] haftmann@46664: declare SUP1_E [elim!] haftmann@56742: declare Sup2_E [elim!] haftmann@46664: declare SUP2_E [elim!] haftmann@46664: haftmann@46694: subsection {* Fundamental *} haftmann@46664: haftmann@46694: subsubsection {* Relations as sets of pairs *} haftmann@46694: haftmann@46694: type_synonym 'a rel = "('a * 'a) set" haftmann@46694: haftmann@46694: lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *} haftmann@46694: "(\x y. (x, y) \ r \ (x, y) \ s) \ r \ s" haftmann@46694: by auto haftmann@46694: haftmann@46694: lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *} haftmann@46694: "(a, b) \ lfp f \ mono f \ haftmann@46694: (\a b. (a, b) \ f (lfp f \ {(x, y). P x y}) \ P a b) \ P a b" blanchet@55414: using lfp_induct_set [of "(a, b)" f "case_prod P"] by auto haftmann@46694: haftmann@46694: haftmann@46694: subsubsection {* Conversions between set and predicate relations *} haftmann@46664: haftmann@46833: lemma pred_equals_eq [pred_set_conv]: "(\x. x \ R) = (\x. x \ S) \ R = S" haftmann@46664: by (simp add: set_eq_iff fun_eq_iff) haftmann@46664: haftmann@46833: lemma pred_equals_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) = (\x y. (x, y) \ S) \ R = S" haftmann@46664: by (simp add: set_eq_iff fun_eq_iff) haftmann@46664: haftmann@46833: lemma pred_subset_eq [pred_set_conv]: "(\x. x \ R) \ (\x. x \ S) \ R \ S" haftmann@46664: by (simp add: subset_iff le_fun_def) haftmann@46664: haftmann@46833: lemma pred_subset_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) \ (\x y. (x, y) \ S) \ R \ S" haftmann@46664: by (simp add: subset_iff le_fun_def) haftmann@46664: noschinl@46883: lemma bot_empty_eq [pred_set_conv]: "\ = (\x. x \ {})" haftmann@46689: by (auto simp add: fun_eq_iff) haftmann@46689: noschinl@46883: lemma bot_empty_eq2 [pred_set_conv]: "\ = (\x y. (x, y) \ {})" haftmann@46664: by (auto simp add: fun_eq_iff) haftmann@46664: noschinl@46883: lemma top_empty_eq [pred_set_conv]: "\ = (\x. x \ UNIV)" noschinl@46883: by (auto simp add: fun_eq_iff) haftmann@46689: noschinl@46883: lemma top_empty_eq2 [pred_set_conv]: "\ = (\x y. (x, y) \ UNIV)" noschinl@46883: by (auto simp add: fun_eq_iff) haftmann@46664: haftmann@46664: lemma inf_Int_eq [pred_set_conv]: "(\x. x \ R) \ (\x. x \ S) = (\x. x \ R \ S)" haftmann@46664: by (simp add: inf_fun_def) haftmann@46664: haftmann@46664: lemma inf_Int_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) \ (\x y. (x, y) \ S) = (\x y. (x, y) \ R \ S)" haftmann@46664: by (simp add: inf_fun_def) haftmann@46664: haftmann@46664: lemma sup_Un_eq [pred_set_conv]: "(\x. x \ R) \ (\x. x \ S) = (\x. x \ R \ S)" haftmann@46664: by (simp add: sup_fun_def) haftmann@46664: haftmann@46664: lemma sup_Un_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) \ (\x y. (x, y) \ S) = (\x y. (x, y) \ R \ S)" haftmann@46664: by (simp add: sup_fun_def) haftmann@46664: haftmann@46981: lemma INF_INT_eq [pred_set_conv]: "(\i\S. (\x. x \ r i)) = (\x. x \ (\i\S. r i))" haftmann@46981: by (simp add: fun_eq_iff) haftmann@46981: haftmann@46981: lemma INF_INT_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ r i)) = (\x y. (x, y) \ (\i\S. r i))" haftmann@46981: by (simp add: fun_eq_iff) haftmann@46981: haftmann@46981: lemma SUP_UN_eq [pred_set_conv]: "(\i\S. (\x. x \ r i)) = (\x. x \ (\i\S. r i))" haftmann@46981: by (simp add: fun_eq_iff) haftmann@46981: haftmann@46981: lemma SUP_UN_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ r i)) = (\x y. (x, y) \ (\i\S. r i))" haftmann@46981: by (simp add: fun_eq_iff) haftmann@46981: haftmann@46833: lemma Inf_INT_eq [pred_set_conv]: "\S = (\x. x \ INTER S Collect)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: haftmann@46833: lemma INF_Int_eq [pred_set_conv]: "(\i\S. (\x. x \ i)) = (\x. x \ \S)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: blanchet@55414: lemma Inf_INT_eq2 [pred_set_conv]: "\S = (\x y. (x, y) \ INTER (case_prod ` S) Collect)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: haftmann@46833: lemma INF_Int_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ i)) = (\x y. (x, y) \ \S)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: haftmann@46833: lemma Sup_SUP_eq [pred_set_conv]: "\S = (\x. x \ UNION S Collect)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: haftmann@46833: lemma SUP_Sup_eq [pred_set_conv]: "(\i\S. (\x. x \ i)) = (\x. x \ \S)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: blanchet@55414: lemma Sup_SUP_eq2 [pred_set_conv]: "\S = (\x y. (x, y) \ UNION (case_prod ` S) Collect)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: haftmann@46833: lemma SUP_Sup_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ i)) = (\x y. (x, y) \ \S)" noschinl@46884: by (simp add: fun_eq_iff) haftmann@46833: haftmann@46694: subsection {* Properties of relations *} paulson@5978: haftmann@46692: subsubsection {* Reflexivity *} paulson@10786: haftmann@46752: definition refl_on :: "'a set \ 'a rel \ bool" haftmann@46752: where haftmann@46752: "refl_on A r \ r \ A \ A \ (\x\A. (x, x) \ r)" paulson@6806: haftmann@46752: abbreviation refl :: "'a rel \ bool" haftmann@46752: where -- {* reflexivity over a type *} haftmann@45137: "refl \ refl_on UNIV" nipkow@26297: haftmann@46752: definition reflp :: "('a \ 'a \ bool) \ bool" haftmann@46752: where huffman@47375: "reflp r \ (\x. r x x)" haftmann@46694: haftmann@46752: lemma reflp_refl_eq [pred_set_conv]: haftmann@46752: "reflp (\x y. (x, y) \ r) \ refl r" haftmann@46752: by (simp add: refl_on_def reflp_def) haftmann@46752: haftmann@46692: lemma refl_onI: "r \ A \ A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r" haftmann@46752: by (unfold refl_on_def) (iprover intro!: ballI) haftmann@46692: haftmann@46692: lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r" haftmann@46752: by (unfold refl_on_def) blast haftmann@46692: haftmann@46692: lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A" haftmann@46752: by (unfold refl_on_def) blast haftmann@46692: haftmann@46692: lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A" haftmann@46752: by (unfold refl_on_def) blast haftmann@46692: haftmann@46694: lemma reflpI: haftmann@46694: "(\x. r x x) \ reflp r" haftmann@46694: by (auto intro: refl_onI simp add: reflp_def) haftmann@46694: haftmann@46694: lemma reflpE: haftmann@46694: assumes "reflp r" haftmann@46694: obtains "r x x" haftmann@46694: using assms by (auto dest: refl_onD simp add: reflp_def) haftmann@46694: kuncar@47937: lemma reflpD: kuncar@47937: assumes "reflp r" kuncar@47937: shows "r x x" kuncar@47937: using assms by (auto elim: reflpE) kuncar@47937: haftmann@46692: lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \ B) (r \ s)" haftmann@46752: by (unfold refl_on_def) blast haftmann@46752: haftmann@46752: lemma reflp_inf: haftmann@46752: "reflp r \ reflp s \ reflp (r \ s)" haftmann@46752: by (auto intro: reflpI elim: reflpE) haftmann@46692: haftmann@46692: lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \ B) (r \ s)" haftmann@46752: by (unfold refl_on_def) blast haftmann@46752: haftmann@46752: lemma reflp_sup: haftmann@46752: "reflp r \ reflp s \ reflp (r \ s)" haftmann@46752: by (auto intro: reflpI elim: reflpE) haftmann@46692: haftmann@46692: lemma refl_on_INTER: haftmann@46692: "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)" haftmann@46752: by (unfold refl_on_def) fast haftmann@46692: haftmann@46692: lemma refl_on_UNION: haftmann@46692: "ALL x:S. refl_on (A x) (r x) \ refl_on (UNION S A) (UNION S r)" haftmann@46752: by (unfold refl_on_def) blast haftmann@46692: haftmann@46752: lemma refl_on_empty [simp]: "refl_on {} {}" haftmann@46752: by (simp add:refl_on_def) haftmann@46692: haftmann@46692: lemma refl_on_def' [nitpick_unfold, code]: haftmann@46752: "refl_on A r \ (\(x, y) \ r. x \ A \ y \ A) \ (\x \ A. (x, x) \ r)" haftmann@46752: by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2) haftmann@46692: haftmann@46692: haftmann@46694: subsubsection {* Irreflexivity *} paulson@6806: haftmann@46752: definition irrefl :: "'a rel \ bool" haftmann@46752: where haftmann@56545: "irrefl r \ (\a. (a, a) \ r)" haftmann@56545: haftmann@56545: definition irreflp :: "('a \ 'a \ bool) \ bool" haftmann@56545: where haftmann@56545: "irreflp R \ (\a. \ R a a)" haftmann@56545: haftmann@56545: lemma irreflp_irrefl_eq [pred_set_conv]: haftmann@56545: "irreflp (\a b. (a, b) \ R) \ irrefl R" haftmann@56545: by (simp add: irrefl_def irreflp_def) haftmann@56545: haftmann@56545: lemma irreflI: haftmann@56545: "(\a. (a, a) \ R) \ irrefl R" haftmann@56545: by (simp add: irrefl_def) haftmann@56545: haftmann@56545: lemma irreflpI: haftmann@56545: "(\a. \ R a a) \ irreflp R" haftmann@56545: by (fact irreflI [to_pred]) haftmann@46692: haftmann@46694: lemma irrefl_distinct [code]: haftmann@56545: "irrefl r \ (\(a, b) \ r. a \ b)" haftmann@46694: by (auto simp add: irrefl_def) haftmann@46692: haftmann@46692: haftmann@56545: subsubsection {* Asymmetry *} haftmann@56545: haftmann@56545: inductive asym :: "'a rel \ bool" haftmann@56545: where haftmann@56545: asymI: "irrefl R \ (\a b. (a, b) \ R \ (b, a) \ R) \ asym R" haftmann@56545: haftmann@56545: inductive asymp :: "('a \ 'a \ bool) \ bool" haftmann@56545: where haftmann@56545: asympI: "irreflp R \ (\a b. R a b \ \ R b a) \ asymp R" haftmann@56545: haftmann@56545: lemma asymp_asym_eq [pred_set_conv]: haftmann@56545: "asymp (\a b. (a, b) \ R) \ asym R" haftmann@56545: by (auto intro!: asymI asympI elim: asym.cases asymp.cases simp add: irreflp_irrefl_eq) haftmann@56545: haftmann@56545: haftmann@46692: subsubsection {* Symmetry *} haftmann@46692: haftmann@46752: definition sym :: "'a rel \ bool" haftmann@46752: where haftmann@46752: "sym r \ (\x y. (x, y) \ r \ (y, x) \ r)" haftmann@46752: haftmann@46752: definition symp :: "('a \ 'a \ bool) \ bool" haftmann@46752: where haftmann@46752: "symp r \ (\x y. r x y \ r y x)" haftmann@46692: haftmann@46752: lemma symp_sym_eq [pred_set_conv]: haftmann@46752: "symp (\x y. (x, y) \ r) \ sym r" haftmann@46752: by (simp add: sym_def symp_def) haftmann@46692: haftmann@46752: lemma symI: haftmann@46752: "(\a b. (a, b) \ r \ (b, a) \ r) \ sym r" haftmann@46752: by (unfold sym_def) iprover haftmann@46694: haftmann@46694: lemma sympI: haftmann@46752: "(\a b. r a b \ r b a) \ symp r" haftmann@46752: by (fact symI [to_pred]) haftmann@46752: haftmann@46752: lemma symE: haftmann@46752: assumes "sym r" and "(b, a) \ r" haftmann@46752: obtains "(a, b) \ r" haftmann@46752: using assms by (simp add: sym_def) haftmann@46694: haftmann@46694: lemma sympE: haftmann@46752: assumes "symp r" and "r b a" haftmann@46752: obtains "r a b" haftmann@46752: using assms by (rule symE [to_pred]) haftmann@46752: haftmann@46752: lemma symD: haftmann@46752: assumes "sym r" and "(b, a) \ r" haftmann@46752: shows "(a, b) \ r" haftmann@46752: using assms by (rule symE) haftmann@46694: haftmann@46752: lemma sympD: haftmann@46752: assumes "symp r" and "r b a" haftmann@46752: shows "r a b" haftmann@46752: using assms by (rule symD [to_pred]) haftmann@46752: haftmann@46752: lemma sym_Int: haftmann@46752: "sym r \ sym s \ sym (r \ s)" haftmann@46752: by (fast intro: symI elim: symE) haftmann@46692: haftmann@46752: lemma symp_inf: haftmann@46752: "symp r \ symp s \ symp (r \ s)" haftmann@46752: by (fact sym_Int [to_pred]) haftmann@46752: haftmann@46752: lemma sym_Un: haftmann@46752: "sym r \ sym s \ sym (r \ s)" haftmann@46752: by (fast intro: symI elim: symE) haftmann@46752: haftmann@46752: lemma symp_sup: haftmann@46752: "symp r \ symp s \ symp (r \ s)" haftmann@46752: by (fact sym_Un [to_pred]) haftmann@46692: haftmann@46752: lemma sym_INTER: haftmann@46752: "\x\S. sym (r x) \ sym (INTER S r)" haftmann@46752: by (fast intro: symI elim: symE) haftmann@46752: haftmann@46982: lemma symp_INF: haftmann@56218: "\x\S. symp (r x) \ symp (INFIMUM S r)" haftmann@46982: by (fact sym_INTER [to_pred]) haftmann@46692: haftmann@46752: lemma sym_UNION: haftmann@46752: "\x\S. sym (r x) \ sym (UNION S r)" haftmann@46752: by (fast intro: symI elim: symE) haftmann@46752: haftmann@46982: lemma symp_SUP: haftmann@56218: "\x\S. symp (r x) \ symp (SUPREMUM S r)" haftmann@46982: by (fact sym_UNION [to_pred]) haftmann@46692: haftmann@46692: haftmann@46694: subsubsection {* Antisymmetry *} haftmann@46694: haftmann@46752: definition antisym :: "'a rel \ bool" haftmann@46752: where haftmann@46752: "antisym r \ (\x y. (x, y) \ r \ (y, x) \ r \ x = y)" haftmann@46752: haftmann@46752: abbreviation antisymP :: "('a \ 'a \ bool) \ bool" haftmann@46752: where haftmann@46752: "antisymP r \ antisym {(x, y). r x y}" haftmann@46694: haftmann@46694: lemma antisymI: haftmann@46694: "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" haftmann@46752: by (unfold antisym_def) iprover haftmann@46694: haftmann@46694: lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" haftmann@46752: by (unfold antisym_def) iprover haftmann@46694: haftmann@46694: lemma antisym_subset: "r \ s ==> antisym s ==> antisym r" haftmann@46752: by (unfold antisym_def) blast haftmann@46694: haftmann@46694: lemma antisym_empty [simp]: "antisym {}" haftmann@46752: by (unfold antisym_def) blast haftmann@46694: haftmann@46694: haftmann@46692: subsubsection {* Transitivity *} haftmann@46692: haftmann@46752: definition trans :: "'a rel \ bool" haftmann@46752: where haftmann@46752: "trans r \ (\x y z. (x, y) \ r \ (y, z) \ r \ (x, z) \ r)" haftmann@46752: haftmann@46752: definition transp :: "('a \ 'a \ bool) \ bool" haftmann@46752: where haftmann@46752: "transp r \ (\x y z. r x y \ r y z \ r x z)" haftmann@46752: haftmann@46752: lemma transp_trans_eq [pred_set_conv]: haftmann@46752: "transp (\x y. (x, y) \ r) \ trans r" haftmann@46752: by (simp add: trans_def transp_def) haftmann@46752: haftmann@46752: abbreviation transP :: "('a \ 'a \ bool) \ bool" haftmann@46752: where -- {* FIXME drop *} haftmann@46752: "transP r \ trans {(x, y). r x y}" paulson@5978: haftmann@46692: lemma transI: haftmann@46752: "(\x y z. (x, y) \ r \ (y, z) \ r \ (x, z) \ r) \ trans r" haftmann@46752: by (unfold trans_def) iprover haftmann@46694: haftmann@46694: lemma transpI: haftmann@46694: "(\x y z. r x y \ r y z \ r x z) \ transp r" haftmann@46752: by (fact transI [to_pred]) haftmann@46752: haftmann@46752: lemma transE: haftmann@46752: assumes "trans r" and "(x, y) \ r" and "(y, z) \ r" haftmann@46752: obtains "(x, z) \ r" haftmann@46752: using assms by (unfold trans_def) iprover haftmann@46752: haftmann@46694: lemma transpE: haftmann@46694: assumes "transp r" and "r x y" and "r y z" haftmann@46694: obtains "r x z" haftmann@46752: using assms by (rule transE [to_pred]) haftmann@46752: haftmann@46752: lemma transD: haftmann@46752: assumes "trans r" and "(x, y) \ r" and "(y, z) \ r" haftmann@46752: shows "(x, z) \ r" haftmann@46752: using assms by (rule transE) haftmann@46752: haftmann@46752: lemma transpD: haftmann@46752: assumes "transp r" and "r x y" and "r y z" haftmann@46752: shows "r x z" haftmann@46752: using assms by (rule transD [to_pred]) haftmann@46694: haftmann@46752: lemma trans_Int: haftmann@46752: "trans r \ trans s \ trans (r \ s)" haftmann@46752: by (fast intro: transI elim: transE) haftmann@46692: haftmann@46752: lemma transp_inf: haftmann@46752: "transp r \ transp s \ transp (r \ s)" haftmann@46752: by (fact trans_Int [to_pred]) haftmann@46752: haftmann@46752: lemma trans_INTER: haftmann@46752: "\x\S. trans (r x) \ trans (INTER S r)" haftmann@46752: by (fast intro: transI elim: transD) haftmann@46752: haftmann@46752: (* FIXME thm trans_INTER [to_pred] *) haftmann@46692: haftmann@46694: lemma trans_join [code]: haftmann@46694: "trans r \ (\(x, y1) \ r. \(y2, z) \ r. y1 = y2 \ (x, z) \ r)" haftmann@46694: by (auto simp add: trans_def) haftmann@46692: haftmann@46752: lemma transp_trans: haftmann@46752: "transp r \ trans {(x, y). r x y}" haftmann@46752: by (simp add: trans_def transp_def) haftmann@46752: haftmann@46692: haftmann@46692: subsubsection {* Totality *} haftmann@46692: haftmann@46752: definition total_on :: "'a set \ 'a rel \ bool" haftmann@46752: where haftmann@46752: "total_on A r \ (\x\A. \y\A. x \ y \ (x, y) \ r \ (y, x) \ r)" nipkow@29859: nipkow@29859: abbreviation "total \ total_on UNIV" nipkow@29859: haftmann@46752: lemma total_on_empty [simp]: "total_on {} r" haftmann@46752: by (simp add: total_on_def) haftmann@46692: haftmann@46692: haftmann@46692: subsubsection {* Single valued relations *} haftmann@46692: haftmann@46752: definition single_valued :: "('a \ 'b) set \ bool" haftmann@46752: where haftmann@46752: "single_valued r \ (\x y. (x, y) \ r \ (\z. (x, z) \ r \ y = z))" haftmann@46692: haftmann@46694: abbreviation single_valuedP :: "('a \ 'b \ bool) \ bool" where haftmann@46694: "single_valuedP r \ single_valued {(x, y). r x y}" haftmann@46694: haftmann@46752: lemma single_valuedI: haftmann@46752: "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r" haftmann@46752: by (unfold single_valued_def) haftmann@46752: haftmann@46752: lemma single_valuedD: haftmann@46752: "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" haftmann@46752: by (simp add: single_valued_def) haftmann@46752: nipkow@57111: lemma single_valued_empty[simp]: "single_valued {}" nipkow@52392: by(simp add: single_valued_def) nipkow@52392: haftmann@46692: lemma single_valued_subset: haftmann@46692: "r \ s ==> single_valued s ==> single_valued r" haftmann@46752: by (unfold single_valued_def) blast oheimb@11136: berghofe@12905: haftmann@46694: subsection {* Relation operations *} haftmann@46694: haftmann@46664: subsubsection {* The identity relation *} berghofe@12905: haftmann@46752: definition Id :: "'a rel" haftmann@46752: where bulwahn@48253: [code del]: "Id = {p. \x. p = (x, x)}" haftmann@46692: berghofe@12905: lemma IdI [intro]: "(a, a) : Id" haftmann@46752: by (simp add: Id_def) berghofe@12905: berghofe@12905: lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" haftmann@46752: by (unfold Id_def) (iprover elim: CollectE) berghofe@12905: berghofe@12905: lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" haftmann@46752: by (unfold Id_def) blast berghofe@12905: nipkow@30198: lemma refl_Id: "refl Id" haftmann@46752: by (simp add: refl_on_def) berghofe@12905: berghofe@12905: lemma antisym_Id: "antisym Id" berghofe@12905: -- {* A strange result, since @{text Id} is also symmetric. *} haftmann@46752: by (simp add: antisym_def) berghofe@12905: huffman@19228: lemma sym_Id: "sym Id" haftmann@46752: by (simp add: sym_def) huffman@19228: berghofe@12905: lemma trans_Id: "trans Id" haftmann@46752: by (simp add: trans_def) berghofe@12905: haftmann@46692: lemma single_valued_Id [simp]: "single_valued Id" haftmann@46692: by (unfold single_valued_def) blast haftmann@46692: haftmann@46692: lemma irrefl_diff_Id [simp]: "irrefl (r - Id)" haftmann@46692: by (simp add:irrefl_def) haftmann@46692: haftmann@46692: lemma trans_diff_Id: "trans r \ antisym r \ trans (r - Id)" haftmann@46692: unfolding antisym_def trans_def by blast haftmann@46692: haftmann@46692: lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r" haftmann@46692: by (simp add: total_on_def) haftmann@46692: berghofe@12905: haftmann@46664: subsubsection {* Diagonal: identity over a set *} berghofe@12905: haftmann@46752: definition Id_on :: "'a set \ 'a rel" haftmann@46752: where haftmann@46752: "Id_on A = (\x\A. {(x, x)})" haftmann@46692: nipkow@30198: lemma Id_on_empty [simp]: "Id_on {} = {}" haftmann@46752: by (simp add: Id_on_def) paulson@13812: nipkow@30198: lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A" haftmann@46752: by (simp add: Id_on_def) berghofe@12905: blanchet@54147: lemma Id_onI [intro!]: "a : A ==> (a, a) : Id_on A" haftmann@46752: by (rule Id_on_eqI) (rule refl) berghofe@12905: nipkow@30198: lemma Id_onE [elim!]: nipkow@30198: "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" wenzelm@12913: -- {* The general elimination rule. *} haftmann@46752: by (unfold Id_on_def) (iprover elim!: UN_E singletonE) berghofe@12905: nipkow@30198: lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)" haftmann@46752: by blast berghofe@12905: haftmann@45967: lemma Id_on_def' [nitpick_unfold]: haftmann@44278: "Id_on {x. A x} = Collect (\(x, y). x = y \ A x)" haftmann@46752: by auto bulwahn@40923: nipkow@30198: lemma Id_on_subset_Times: "Id_on A \ A \ A" haftmann@46752: by blast berghofe@12905: haftmann@46692: lemma refl_on_Id_on: "refl_on A (Id_on A)" haftmann@46752: by (rule refl_onI [OF Id_on_subset_Times Id_onI]) haftmann@46692: haftmann@46692: lemma antisym_Id_on [simp]: "antisym (Id_on A)" haftmann@46752: by (unfold antisym_def) blast haftmann@46692: haftmann@46692: lemma sym_Id_on [simp]: "sym (Id_on A)" haftmann@46752: by (rule symI) clarify haftmann@46692: haftmann@46692: lemma trans_Id_on [simp]: "trans (Id_on A)" haftmann@46752: by (fast intro: transI elim: transD) haftmann@46692: haftmann@46692: lemma single_valued_Id_on [simp]: "single_valued (Id_on A)" haftmann@46692: by (unfold single_valued_def) blast haftmann@46692: berghofe@12905: haftmann@46694: subsubsection {* Composition *} berghofe@12905: griff@47433: inductive_set relcomp :: "('a \ 'b) set \ ('b \ 'c) set \ ('a \ 'c) set" (infixr "O" 75) haftmann@46752: for r :: "('a \ 'b) set" and s :: "('b \ 'c) set" haftmann@46694: where griff@47433: relcompI [intro]: "(a, b) \ r \ (b, c) \ s \ (a, c) \ r O s" haftmann@46692: griff@47434: notation relcompp (infixr "OO" 75) berghofe@12905: griff@47434: lemmas relcomppI = relcompp.intros berghofe@12905: haftmann@46752: text {* haftmann@46752: For historic reasons, the elimination rules are not wholly corresponding. haftmann@46752: Feel free to consolidate this. haftmann@46752: *} haftmann@46694: griff@47433: inductive_cases relcompEpair: "(a, c) \ r O s" griff@47434: inductive_cases relcomppE [elim!]: "(r OO s) a c" haftmann@46694: griff@47433: lemma relcompE [elim!]: "xz \ r O s \ haftmann@46752: (\x y z. xz = (x, z) \ (x, y) \ r \ (y, z) \ s \ P) \ P" griff@47433: by (cases xz) (simp, erule relcompEpair, iprover) haftmann@46752: haftmann@46752: lemma R_O_Id [simp]: haftmann@46752: "R O Id = R" haftmann@46752: by fast haftmann@46694: haftmann@46752: lemma Id_O_R [simp]: haftmann@46752: "Id O R = R" haftmann@46752: by fast haftmann@46752: griff@47433: lemma relcomp_empty1 [simp]: haftmann@46752: "{} O R = {}" haftmann@46752: by blast berghofe@12905: griff@47434: lemma relcompp_bot1 [simp]: noschinl@46883: "\ OO R = \" griff@47433: by (fact relcomp_empty1 [to_pred]) berghofe@12905: griff@47433: lemma relcomp_empty2 [simp]: haftmann@46752: "R O {} = {}" haftmann@46752: by blast berghofe@12905: griff@47434: lemma relcompp_bot2 [simp]: noschinl@46883: "R OO \ = \" griff@47433: by (fact relcomp_empty2 [to_pred]) krauss@23185: haftmann@46752: lemma O_assoc: haftmann@46752: "(R O S) O T = R O (S O T)" haftmann@46752: by blast haftmann@46752: noschinl@46883: griff@47434: lemma relcompp_assoc: haftmann@46752: "(r OO s) OO t = r OO (s OO t)" haftmann@46752: by (fact O_assoc [to_pred]) krauss@23185: haftmann@46752: lemma trans_O_subset: haftmann@46752: "trans r \ r O r \ r" haftmann@46752: by (unfold trans_def) blast haftmann@46752: griff@47434: lemma transp_relcompp_less_eq: haftmann@46752: "transp r \ r OO r \ r " haftmann@46752: by (fact trans_O_subset [to_pred]) berghofe@12905: griff@47433: lemma relcomp_mono: haftmann@46752: "r' \ r \ s' \ s \ r' O s' \ r O s" haftmann@46752: by blast berghofe@12905: griff@47434: lemma relcompp_mono: haftmann@46752: "r' \ r \ s' \ s \ r' OO s' \ r OO s " griff@47433: by (fact relcomp_mono [to_pred]) berghofe@12905: griff@47433: lemma relcomp_subset_Sigma: haftmann@46752: "r \ A \ B \ s \ B \ C \ r O s \ A \ C" haftmann@46752: by blast haftmann@46752: griff@47433: lemma relcomp_distrib [simp]: haftmann@46752: "R O (S \ T) = (R O S) \ (R O T)" haftmann@46752: by auto berghofe@12905: griff@47434: lemma relcompp_distrib [simp]: haftmann@46752: "R OO (S \ T) = R OO S \ R OO T" griff@47433: by (fact relcomp_distrib [to_pred]) haftmann@46752: griff@47433: lemma relcomp_distrib2 [simp]: haftmann@46752: "(S \ T) O R = (S O R) \ (T O R)" haftmann@46752: by auto krauss@28008: griff@47434: lemma relcompp_distrib2 [simp]: haftmann@46752: "(S \ T) OO R = S OO R \ T OO R" griff@47433: by (fact relcomp_distrib2 [to_pred]) haftmann@46752: griff@47433: lemma relcomp_UNION_distrib: haftmann@46752: "s O UNION I r = (\i\I. s O r i) " haftmann@46752: by auto krauss@28008: griff@47433: (* FIXME thm relcomp_UNION_distrib [to_pred] *) krauss@36772: griff@47433: lemma relcomp_UNION_distrib2: haftmann@46752: "UNION I r O s = (\i\I. r i O s) " haftmann@46752: by auto haftmann@46752: griff@47433: (* FIXME thm relcomp_UNION_distrib2 [to_pred] *) krauss@36772: griff@47433: lemma single_valued_relcomp: haftmann@46752: "single_valued r \ single_valued s \ single_valued (r O s)" haftmann@46752: by (unfold single_valued_def) blast haftmann@46752: griff@47433: lemma relcomp_unfold: haftmann@46752: "r O s = {(x, z). \y. (x, y) \ r \ (y, z) \ s}" haftmann@46752: by (auto simp add: set_eq_iff) berghofe@12905: blanchet@55083: lemma eq_OO: "op= OO R = R" blanchet@55083: by blast blanchet@55083: haftmann@46664: haftmann@46664: subsubsection {* Converse *} wenzelm@12913: haftmann@46752: inductive_set converse :: "('a \ 'b) set \ ('b \ 'a) set" ("(_^-1)" [1000] 999) haftmann@46752: for r :: "('a \ 'b) set" haftmann@46752: where haftmann@46752: "(a, b) \ r \ (b, a) \ r^-1" haftmann@46692: haftmann@46692: notation (xsymbols) haftmann@46692: converse ("(_\)" [1000] 999) haftmann@46692: haftmann@46752: notation haftmann@46752: conversep ("(_^--1)" [1000] 1000) haftmann@46694: haftmann@46694: notation (xsymbols) haftmann@46694: conversep ("(_\\)" [1000] 1000) haftmann@46694: haftmann@46752: lemma converseI [sym]: haftmann@46752: "(a, b) \ r \ (b, a) \ r\" haftmann@46752: by (fact converse.intros) haftmann@46752: haftmann@46752: lemma conversepI (* CANDIDATE [sym] *): haftmann@46752: "r a b \ r\\ b a" haftmann@46752: by (fact conversep.intros) haftmann@46752: haftmann@46752: lemma converseD [sym]: haftmann@46752: "(a, b) \ r\ \ (b, a) \ r" haftmann@46752: by (erule converse.cases) iprover haftmann@46752: haftmann@46752: lemma conversepD (* CANDIDATE [sym] *): haftmann@46752: "r\\ b a \ r a b" haftmann@46752: by (fact converseD [to_pred]) haftmann@46752: haftmann@46752: lemma converseE [elim!]: haftmann@46752: -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} haftmann@46752: "yx \ r\ \ (\x y. yx = (y, x) \ (x, y) \ r \ P) \ P" haftmann@46752: by (cases yx) (simp, erule converse.cases, iprover) haftmann@46694: noschinl@46882: lemmas conversepE [elim!] = conversep.cases haftmann@46752: haftmann@46752: lemma converse_iff [iff]: haftmann@46752: "(a, b) \ r\ \ (b, a) \ r" haftmann@46752: by (auto intro: converseI) haftmann@46752: haftmann@46752: lemma conversep_iff [iff]: haftmann@46752: "r\\ a b = r b a" haftmann@46752: by (fact converse_iff [to_pred]) haftmann@46694: haftmann@46752: lemma converse_converse [simp]: haftmann@46752: "(r\)\ = r" haftmann@46752: by (simp add: set_eq_iff) haftmann@46694: haftmann@46752: lemma conversep_conversep [simp]: haftmann@46752: "(r\\)\\ = r" haftmann@46752: by (fact converse_converse [to_pred]) haftmann@46752: nipkow@53680: lemma converse_empty[simp]: "{}\ = {}" nipkow@53680: by auto nipkow@53680: nipkow@53680: lemma converse_UNIV[simp]: "UNIV\ = UNIV" nipkow@53680: by auto nipkow@53680: griff@47433: lemma converse_relcomp: "(r O s)^-1 = s^-1 O r^-1" haftmann@46752: by blast haftmann@46694: griff@47434: lemma converse_relcompp: "(r OO s)^--1 = s^--1 OO r^--1" griff@47434: by (iprover intro: order_antisym conversepI relcomppI griff@47434: elim: relcomppE dest: conversepD) haftmann@46694: haftmann@46752: lemma converse_Int: "(r \ s)^-1 = r^-1 \ s^-1" haftmann@46752: by blast haftmann@46752: haftmann@46694: lemma converse_meet: "(r \ s)^--1 = r^--1 \ s^--1" haftmann@46694: by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) haftmann@46694: haftmann@46752: lemma converse_Un: "(r \ s)^-1 = r^-1 \ s^-1" haftmann@46752: by blast haftmann@46752: haftmann@46694: lemma converse_join: "(r \ s)^--1 = r^--1 \ s^--1" haftmann@46694: by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) haftmann@46694: huffman@19228: lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)" haftmann@46752: by fast huffman@19228: huffman@19228: lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)" haftmann@46752: by blast huffman@19228: traytel@52749: lemma converse_mono[simp]: "r^-1 \ s ^-1 \ r \ s" traytel@52749: by auto traytel@52749: traytel@52749: lemma conversep_mono[simp]: "r^--1 \ s ^--1 \ r \ s" traytel@52749: by (fact converse_mono[to_pred]) traytel@52749: traytel@52749: lemma converse_inject[simp]: "r^-1 = s ^-1 \ r = s" traytel@52730: by auto traytel@52730: traytel@52749: lemma conversep_inject[simp]: "r^--1 = s ^--1 \ r = s" traytel@52749: by (fact converse_inject[to_pred]) traytel@52749: traytel@52749: lemma converse_subset_swap: "r \ s ^-1 = (r ^-1 \ s)" traytel@52749: by auto traytel@52749: traytel@52749: lemma conversep_le_swap: "r \ s ^--1 = (r ^--1 \ s)" traytel@52749: by (fact converse_subset_swap[to_pred]) traytel@52730: berghofe@12905: lemma converse_Id [simp]: "Id^-1 = Id" haftmann@46752: by blast berghofe@12905: nipkow@30198: lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A" haftmann@46752: by blast berghofe@12905: nipkow@30198: lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r" haftmann@46752: by (unfold refl_on_def) auto berghofe@12905: huffman@19228: lemma sym_converse [simp]: "sym (converse r) = sym r" haftmann@46752: by (unfold sym_def) blast huffman@19228: huffman@19228: lemma antisym_converse [simp]: "antisym (converse r) = antisym r" haftmann@46752: by (unfold antisym_def) blast berghofe@12905: huffman@19228: lemma trans_converse [simp]: "trans (converse r) = trans r" haftmann@46752: by (unfold trans_def) blast berghofe@12905: huffman@19228: lemma sym_conv_converse_eq: "sym r = (r^-1 = r)" haftmann@46752: by (unfold sym_def) fast huffman@19228: huffman@19228: lemma sym_Un_converse: "sym (r \ r^-1)" haftmann@46752: by (unfold sym_def) blast huffman@19228: huffman@19228: lemma sym_Int_converse: "sym (r \ r^-1)" haftmann@46752: by (unfold sym_def) blast huffman@19228: haftmann@46752: lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r" haftmann@46752: by (auto simp: total_on_def) nipkow@29859: traytel@52749: lemma finite_converse [iff]: "finite (r^-1) = finite r" traytel@54611: unfolding converse_def conversep_iff using [[simproc add: finite_Collect]] traytel@54611: by (auto elim: finite_imageD simp: inj_on_def) wenzelm@12913: haftmann@46752: lemma conversep_noteq [simp]: "(op \)^--1 = op \" haftmann@46752: by (auto simp add: fun_eq_iff) haftmann@46752: haftmann@46752: lemma conversep_eq [simp]: "(op =)^--1 = op =" haftmann@46752: by (auto simp add: fun_eq_iff) haftmann@46752: nipkow@53680: lemma converse_unfold [code]: haftmann@46752: "r\ = {(y, x). (x, y) \ r}" haftmann@46752: by (simp add: set_eq_iff) haftmann@46752: haftmann@46692: haftmann@46692: subsubsection {* Domain, range and field *} haftmann@46692: haftmann@46767: inductive_set Domain :: "('a \ 'b) set \ 'a set" haftmann@46767: for r :: "('a \ 'b) set" haftmann@46752: where haftmann@46767: DomainI [intro]: "(a, b) \ r \ a \ Domain r" haftmann@46767: haftmann@46767: abbreviation (input) "DomainP \ Domainp" haftmann@46767: haftmann@46767: lemmas DomainPI = Domainp.DomainI haftmann@46767: haftmann@46767: inductive_cases DomainE [elim!]: "a \ Domain r" haftmann@46767: inductive_cases DomainpE [elim!]: "Domainp r a" haftmann@46692: haftmann@46767: inductive_set Range :: "('a \ 'b) set \ 'b set" haftmann@46767: for r :: "('a \ 'b) set" haftmann@46752: where haftmann@46767: RangeI [intro]: "(a, b) \ r \ b \ Range r" haftmann@46767: haftmann@46767: abbreviation (input) "RangeP \ Rangep" haftmann@46767: haftmann@46767: lemmas RangePI = Rangep.RangeI haftmann@46767: haftmann@46767: inductive_cases RangeE [elim!]: "b \ Range r" haftmann@46767: inductive_cases RangepE [elim!]: "Rangep r b" haftmann@46692: haftmann@46752: definition Field :: "'a rel \ 'a set" haftmann@46752: where haftmann@46692: "Field r = Domain r \ Range r" berghofe@12905: haftmann@46694: lemma Domain_fst [code]: haftmann@46694: "Domain r = fst ` r" haftmann@46767: by force haftmann@46767: haftmann@46767: lemma Range_snd [code]: haftmann@46767: "Range r = snd ` r" haftmann@46767: by force haftmann@46767: haftmann@46767: lemma fst_eq_Domain: "fst ` R = Domain R" haftmann@46767: by force haftmann@46767: haftmann@46767: lemma snd_eq_Range: "snd ` R = Range R" haftmann@46767: by force haftmann@46694: haftmann@46694: lemma Domain_empty [simp]: "Domain {} = {}" haftmann@46767: by auto haftmann@46767: haftmann@46767: lemma Range_empty [simp]: "Range {} = {}" haftmann@46767: by auto haftmann@46767: haftmann@46767: lemma Field_empty [simp]: "Field {} = {}" haftmann@46767: by (simp add: Field_def) haftmann@46694: haftmann@46694: lemma Domain_empty_iff: "Domain r = {} \ r = {}" haftmann@46694: by auto haftmann@46694: haftmann@46767: lemma Range_empty_iff: "Range r = {} \ r = {}" haftmann@46767: by auto haftmann@46767: noschinl@46882: lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)" haftmann@46767: by blast haftmann@46767: noschinl@46882: lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \ Field r" noschinl@46884: by (auto simp add: Field_def) haftmann@46767: haftmann@46767: lemma Domain_iff: "a \ Domain r \ (\y. (a, y) \ r)" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Range_iff: "a \ Range r \ (\y. (y, a) \ r)" haftmann@46694: by blast haftmann@46694: haftmann@46694: lemma Domain_Id [simp]: "Domain Id = UNIV" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Range_Id [simp]: "Range Id = UNIV" haftmann@46767: by blast haftmann@46767: haftmann@46694: lemma Domain_Id_on [simp]: "Domain (Id_on A) = A" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Range_Id_on [simp]: "Range (Id_on A) = A" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Domain_Un_eq: "Domain (A \ B) = Domain A \ Domain B" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Range_Un_eq: "Range (A \ B) = Range A \ Range B" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Field_Un [simp]: "Field (r \ s) = Field r \ Field s" haftmann@46767: by (auto simp: Field_def) haftmann@46767: haftmann@46767: lemma Domain_Int_subset: "Domain (A \ B) \ Domain A \ Domain B" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Range_Int_subset: "Range (A \ B) \ Range A \ Range B" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Domain_Diff_subset: "Domain A - Domain B \ Domain (A - B)" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Range_Diff_subset: "Range A - Range B \ Range (A - B)" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Domain_Union: "Domain (\S) = (\A\S. Domain A)" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Range_Union: "Range (\S) = (\A\S. Range A)" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Field_Union [simp]: "Field (\R) = \(Field ` R)" haftmann@46767: by (auto simp: Field_def) haftmann@46767: haftmann@46752: lemma Domain_converse [simp]: "Domain (r\) = Range r" haftmann@46752: by auto haftmann@46694: haftmann@46767: lemma Range_converse [simp]: "Range (r\) = Domain r" haftmann@46694: by blast haftmann@46694: haftmann@46767: lemma Field_converse [simp]: "Field (r\) = Field r" haftmann@46767: by (auto simp: Field_def) haftmann@46767: haftmann@46767: lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}" haftmann@46767: by auto haftmann@46767: haftmann@46767: lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}" haftmann@46767: by auto haftmann@46767: haftmann@46767: lemma finite_Domain: "finite r \ finite (Domain r)" noschinl@46884: by (induct set: finite) auto haftmann@46767: haftmann@46767: lemma finite_Range: "finite r \ finite (Range r)" noschinl@46884: by (induct set: finite) auto haftmann@46767: haftmann@46767: lemma finite_Field: "finite r \ finite (Field r)" haftmann@46767: by (simp add: Field_def finite_Domain finite_Range) haftmann@46767: haftmann@46767: lemma Domain_mono: "r \ s \ Domain r \ Domain s" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma Range_mono: "r \ s \ Range r \ Range s" haftmann@46767: by blast haftmann@46767: haftmann@46767: lemma mono_Field: "r \ s \ Field r \ Field s" haftmann@46767: by (auto simp: Field_def Domain_def Range_def) haftmann@46767: haftmann@46767: lemma Domain_unfold: haftmann@46767: "Domain r = {x. \y. (x, y) \ r}" haftmann@46767: by blast haftmann@46694: berghofe@12905: haftmann@46664: subsubsection {* Image of a set under a relation *} berghofe@12905: nipkow@50420: definition Image :: "('a \ 'b) set \ 'a set \ 'b set" (infixr "``" 90) haftmann@46752: where haftmann@46752: "r `` s = {y. \x\s. (x, y) \ r}" haftmann@46692: wenzelm@12913: lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" haftmann@46752: by (simp add: Image_def) berghofe@12905: wenzelm@12913: lemma Image_singleton: "r``{a} = {b. (a, b) : r}" haftmann@46752: by (simp add: Image_def) berghofe@12905: wenzelm@12913: lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" haftmann@46752: by (rule Image_iff [THEN trans]) simp berghofe@12905: blanchet@54147: lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A" haftmann@46752: by (unfold Image_def) blast berghofe@12905: berghofe@12905: lemma ImageE [elim!]: haftmann@46752: "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" haftmann@46752: by (unfold Image_def) (iprover elim!: CollectE bexE) berghofe@12905: berghofe@12905: lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" berghofe@12905: -- {* This version's more effective when we already have the required @{text a} *} haftmann@46752: by blast berghofe@12905: berghofe@12905: lemma Image_empty [simp]: "R``{} = {}" haftmann@46752: by blast berghofe@12905: berghofe@12905: lemma Image_Id [simp]: "Id `` A = A" haftmann@46752: by blast berghofe@12905: nipkow@30198: lemma Image_Id_on [simp]: "Id_on A `` B = A \ B" haftmann@46752: by blast paulson@13830: paulson@13830: lemma Image_Int_subset: "R `` (A \ B) \ R `` A \ R `` B" haftmann@46752: by blast berghofe@12905: paulson@13830: lemma Image_Int_eq: haftmann@46767: "single_valued (converse R) ==> R `` (A \ B) = R `` A \ R `` B" haftmann@46767: by (simp add: single_valued_def, blast) berghofe@12905: paulson@13830: lemma Image_Un: "R `` (A \ B) = R `` A \ R `` B" haftmann@46752: by blast berghofe@12905: paulson@13812: lemma Un_Image: "(R \ S) `` A = R `` A \ S `` A" haftmann@46752: by blast paulson@13812: wenzelm@12913: lemma Image_subset: "r \ A \ B ==> r``C \ B" haftmann@46752: by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) berghofe@12905: paulson@13830: lemma Image_eq_UN: "r``B = (\y\ B. r``{y})" berghofe@12905: -- {* NOT suitable for rewriting *} haftmann@46752: by blast berghofe@12905: wenzelm@12913: lemma Image_mono: "r' \ r ==> A' \ A ==> (r' `` A') \ (r `` A)" haftmann@46752: by blast berghofe@12905: paulson@13830: lemma Image_UN: "(r `` (UNION A B)) = (\x\A. r `` (B x))" haftmann@46752: by blast paulson@13830: hoelzl@54410: lemma UN_Image: "(\i\I. X i) `` S = (\i\I. X i `` S)" hoelzl@54410: by auto hoelzl@54410: paulson@13830: lemma Image_INT_subset: "(r `` INTER A B) \ (\x\A. r `` (B x))" haftmann@46752: by blast berghofe@12905: paulson@13830: text{*Converse inclusion requires some assumptions*} paulson@13830: lemma Image_INT_eq: paulson@13830: "[|single_valued (r\); A\{}|] ==> r `` INTER A B = (\x\A. r `` B x)" paulson@13830: apply (rule equalityI) paulson@13830: apply (rule Image_INT_subset) paulson@13830: apply (simp add: single_valued_def, blast) paulson@13830: done berghofe@12905: wenzelm@12913: lemma Image_subset_eq: "(r``A \ B) = (A \ - ((r^-1) `` (-B)))" haftmann@46752: by blast berghofe@12905: haftmann@46692: lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}" haftmann@46752: by auto berghofe@12905: hoelzl@54410: lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\x\X \ A. B x)" hoelzl@54410: by auto hoelzl@54410: hoelzl@54410: lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)" hoelzl@54410: by auto berghofe@12905: haftmann@46664: subsubsection {* Inverse image *} berghofe@12905: haftmann@46752: definition inv_image :: "'b rel \ ('a \ 'b) \ 'a rel" haftmann@46752: where haftmann@46752: "inv_image r f = {(x, y). (f x, f y) \ r}" haftmann@46692: haftmann@46752: definition inv_imagep :: "('b \ 'b \ bool) \ ('a \ 'b) \ 'a \ 'a \ bool" haftmann@46752: where haftmann@46694: "inv_imagep r f = (\x y. r (f x) (f y))" haftmann@46694: haftmann@46694: lemma [pred_set_conv]: "inv_imagep (\x y. (x, y) \ r) f = (\x y. (x, y) \ inv_image r f)" haftmann@46694: by (simp add: inv_image_def inv_imagep_def) haftmann@46694: huffman@19228: lemma sym_inv_image: "sym r ==> sym (inv_image r f)" haftmann@46752: by (unfold sym_def inv_image_def) blast huffman@19228: wenzelm@12913: lemma trans_inv_image: "trans r ==> trans (inv_image r f)" berghofe@12905: apply (unfold trans_def inv_image_def) berghofe@12905: apply (simp (no_asm)) berghofe@12905: apply blast berghofe@12905: done berghofe@12905: krauss@32463: lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)" krauss@32463: by (auto simp:inv_image_def) krauss@32463: krauss@33218: lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f" haftmann@46752: unfolding inv_image_def converse_unfold by auto krauss@33218: haftmann@46664: lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)" haftmann@46664: by (simp add: inv_imagep_def) haftmann@46664: haftmann@46664: haftmann@46664: subsubsection {* Powerset *} haftmann@46664: haftmann@46752: definition Powp :: "('a \ bool) \ 'a set \ bool" haftmann@46752: where haftmann@46664: "Powp A = (\B. \x \ B. A x)" haftmann@46664: haftmann@46664: lemma Powp_Pow_eq [pred_set_conv]: "Powp (\x. x \ A) = (\x. x \ Pow A)" haftmann@46664: by (auto simp add: Powp_def fun_eq_iff) haftmann@46664: haftmann@46664: lemmas Powp_mono [mono] = Pow_mono [to_pred] haftmann@46664: kuncar@48620: subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *} kuncar@48620: kuncar@48620: lemma Id_on_fold: kuncar@48620: assumes "finite A" kuncar@48620: shows "Id_on A = Finite_Set.fold (\x. Set.insert (Pair x x)) {} A" kuncar@48620: proof - kuncar@48620: interpret comp_fun_commute "\x. Set.insert (Pair x x)" by default auto kuncar@48620: show ?thesis using assms unfolding Id_on_def by (induct A) simp_all kuncar@48620: qed kuncar@48620: kuncar@48620: lemma comp_fun_commute_Image_fold: kuncar@48620: "comp_fun_commute (\(x,y) A. if x \ S then Set.insert y A else A)" kuncar@48620: proof - kuncar@48620: interpret comp_fun_idem Set.insert kuncar@48620: by (fact comp_fun_idem_insert) kuncar@48620: show ?thesis kuncar@48620: by default (auto simp add: fun_eq_iff comp_fun_commute split:prod.split) kuncar@48620: qed kuncar@48620: kuncar@48620: lemma Image_fold: kuncar@48620: assumes "finite R" kuncar@48620: shows "R `` S = Finite_Set.fold (\(x,y) A. if x \ S then Set.insert y A else A) {} R" kuncar@48620: proof - kuncar@48620: interpret comp_fun_commute "(\(x,y) A. if x \ S then Set.insert y A else A)" kuncar@48620: by (rule comp_fun_commute_Image_fold) kuncar@48620: have *: "\x F. Set.insert x F `` S = (if fst x \ S then Set.insert (snd x) (F `` S) else (F `` S))" traytel@52749: by (force intro: rev_ImageI) kuncar@48620: show ?thesis using assms by (induct R) (auto simp: *) kuncar@48620: qed kuncar@48620: kuncar@48620: lemma insert_relcomp_union_fold: kuncar@48620: assumes "finite S" kuncar@48620: shows "{x} O S \ X = Finite_Set.fold (\(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S" kuncar@48620: proof - kuncar@48620: interpret comp_fun_commute "\(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'" kuncar@48620: proof - kuncar@48620: interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) kuncar@48620: show "comp_fun_commute (\(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')" kuncar@48620: by default (auto simp add: fun_eq_iff split:prod.split) kuncar@48620: qed kuncar@48620: have *: "{x} O S = {(x', z). x' = fst x \ (snd x,z) \ S}" by (auto simp: relcomp_unfold intro!: exI) kuncar@48620: show ?thesis unfolding * kuncar@48620: using `finite S` by (induct S) (auto split: prod.split) kuncar@48620: qed kuncar@48620: kuncar@48620: lemma insert_relcomp_fold: kuncar@48620: assumes "finite S" kuncar@48620: shows "Set.insert x R O S = kuncar@48620: Finite_Set.fold (\(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S" kuncar@48620: proof - kuncar@48620: have "Set.insert x R O S = ({x} O S) \ (R O S)" by auto kuncar@48620: then show ?thesis by (auto simp: insert_relcomp_union_fold[OF assms]) kuncar@48620: qed kuncar@48620: kuncar@48620: lemma comp_fun_commute_relcomp_fold: kuncar@48620: assumes "finite S" kuncar@48620: shows "comp_fun_commute (\(x,y) A. kuncar@48620: Finite_Set.fold (\(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)" kuncar@48620: proof - kuncar@48620: have *: "\a b A. kuncar@48620: Finite_Set.fold (\(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \ A" kuncar@48620: by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong) kuncar@48620: show ?thesis by default (auto simp: *) kuncar@48620: qed kuncar@48620: kuncar@48620: lemma relcomp_fold: kuncar@48620: assumes "finite R" kuncar@48620: assumes "finite S" kuncar@48620: shows "R O S = Finite_Set.fold kuncar@48620: (\(x,y) A. Finite_Set.fold (\(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R" traytel@52749: using assms by (induct R) traytel@52749: (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold kuncar@48620: cong: if_cong) kuncar@48620: nipkow@1128: end