(* Title: HOL/Relation.thy ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1996 University of Cambridge *) header {* Relations *} theory Relation imports Product_Type FixedPoint begin subsection {* Definitions *} definition converse :: "('a * 'b) set => ('b * 'a) set" ("(_^-1)" [1000] 999) where "r^-1 == {(y, x). (x, y) : r}" notation (xsymbols) converse ("(_\)" [1000] 999) definition rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 75) where "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}" definition Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90) where "r `` s == {y. EX x:s. (x,y):r}" definition Id :: "('a * 'a) set" where -- {* the identity relation *} "Id == {p. EX x. p = (x,x)}" definition diag :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *} "diag A == \x\A. {(x,x)}" definition Domain :: "('a * 'b) set => 'a set" where "Domain r == {x. EX y. (x,y):r}" definition Range :: "('a * 'b) set => 'b set" where "Range r == Domain(r^-1)" definition Field :: "('a * 'a) set => 'a set" where "Field r == Domain r \ Range r" definition refl :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *} "refl A r == r \ A \ A & (ALL x: A. (x,x) : r)" definition sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *} "sym r == ALL x y. (x,y): r --> (y,x): r" definition antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *} "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y" definition trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *} "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)" definition single_valued :: "('a * 'b) set => bool" where "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)" definition inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where "inv_image r f == {(x, y). (f x, f y) : r}" abbreviation reflexive :: "('a * 'a) set => bool" where -- {* reflexivity over a type *} "reflexive == refl UNIV" subsection {* The identity relation *} lemma IdI [intro]: "(a, a) : Id" by (simp add: Id_def) lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" by (unfold Id_def) (iprover elim: CollectE) lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" by (unfold Id_def) blast lemma reflexive_Id: "reflexive Id" by (simp add: refl_def) lemma antisym_Id: "antisym Id" -- {* A strange result, since @{text Id} is also symmetric. *} by (simp add: antisym_def) lemma sym_Id: "sym Id" by (simp add: sym_def) lemma trans_Id: "trans Id" by (simp add: trans_def) subsection {* Diagonal: identity over a set *} lemma diag_empty [simp]: "diag {} = {}" by (simp add: diag_def) lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A" by (simp add: diag_def) lemma diagI [intro!]: "a : A ==> (a, a) : diag A" by (rule diag_eqI) (rule refl) lemma diagE [elim!]: "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" -- {* The general elimination rule. *} by (unfold diag_def) (iprover elim!: UN_E singletonE) lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)" by blast lemma diag_subset_Times: "diag A \ A \ A" by blast subsection {* Composition of two relations *} lemma rel_compI [intro]: "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s" by (unfold rel_comp_def) blast lemma rel_compE [elim!]: "xz : r O s ==> (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r ==> P) ==> P" by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE) lemma rel_compEpair: "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P" by (iprover elim: rel_compE Pair_inject ssubst) lemma R_O_Id [simp]: "R O Id = R" by fast lemma Id_O_R [simp]: "Id O R = R" by fast lemma rel_comp_empty1[simp]: "{} O R = {}" by blast lemma rel_comp_empty2[simp]: "R O {} = {}" by blast lemma O_assoc: "(R O S) O T = R O (S O T)" by blast lemma trans_O_subset: "trans r ==> r O r \ r" by (unfold trans_def) blast lemma rel_comp_mono: "r' \ r ==> s' \ s ==> (r' O s') \ (r O s)" by blast lemma rel_comp_subset_Sigma: "s \ A \ B ==> r \ B \ C ==> (r O s) \ A \ C" by blast subsection {* Reflexivity *} lemma reflI: "r \ A \ A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r" by (unfold refl_def) (iprover intro!: ballI) lemma reflD: "refl A r ==> a : A ==> (a, a) : r" by (unfold refl_def) blast lemma reflD1: "refl A r ==> (x, y) : r ==> x : A" by (unfold refl_def) blast lemma reflD2: "refl A r ==> (x, y) : r ==> y : A" by (unfold refl_def) blast lemma refl_Int: "refl A r ==> refl B s ==> refl (A \ B) (r \ s)" by (unfold refl_def) blast lemma refl_Un: "refl A r ==> refl B s ==> refl (A \ B) (r \ s)" by (unfold refl_def) blast lemma refl_INTER: "ALL x:S. refl (A x) (r x) ==> refl (INTER S A) (INTER S r)" by (unfold refl_def) fast lemma refl_UNION: "ALL x:S. refl (A x) (r x) \ refl (UNION S A) (UNION S r)" by (unfold refl_def) blast lemma refl_diag: "refl A (diag A)" by (rule reflI [OF diag_subset_Times diagI]) subsection {* Antisymmetry *} lemma antisymI: "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" by (unfold antisym_def) iprover lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" by (unfold antisym_def) iprover lemma antisym_subset: "r \ s ==> antisym s ==> antisym r" by (unfold antisym_def) blast lemma antisym_empty [simp]: "antisym {}" by (unfold antisym_def) blast lemma antisym_diag [simp]: "antisym (diag A)" by (unfold antisym_def) blast subsection {* Symmetry *} lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r" by (unfold sym_def) iprover lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r" by (unfold sym_def, blast) lemma sym_Int: "sym r ==> sym s ==> sym (r \ s)" by (fast intro: symI dest: symD) lemma sym_Un: "sym r ==> sym s ==> sym (r \ s)" by (fast intro: symI dest: symD) lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)" by (fast intro: symI dest: symD) lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)" by (fast intro: symI dest: symD) lemma sym_diag [simp]: "sym (diag A)" by (rule symI) clarify subsection {* Transitivity *} lemma transI: "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r" by (unfold trans_def) iprover lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r" by (unfold trans_def) iprover lemma trans_Int: "trans r ==> trans s ==> trans (r \ s)" by (fast intro: transI elim: transD) lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)" by (fast intro: transI elim: transD) lemma trans_diag [simp]: "trans (diag A)" by (fast intro: transI elim: transD) subsection {* Converse *} lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)" by (simp add: converse_def) lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1" by (simp add: converse_def) lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r" by (simp add: converse_def) lemma converseE [elim!]: "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P" -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} by (unfold converse_def) (iprover elim!: CollectE splitE bexE) lemma converse_converse [simp]: "(r^-1)^-1 = r" by (unfold converse_def) blast lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1" by blast lemma converse_Int: "(r \ s)^-1 = r^-1 \ s^-1" by blast lemma converse_Un: "(r \ s)^-1 = r^-1 \ s^-1" by blast lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)" by fast lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)" by blast lemma converse_Id [simp]: "Id^-1 = Id" by blast lemma converse_diag [simp]: "(diag A)^-1 = diag A" by blast lemma refl_converse [simp]: "refl A (converse r) = refl A r" by (unfold refl_def) auto lemma sym_converse [simp]: "sym (converse r) = sym r" by (unfold sym_def) blast lemma antisym_converse [simp]: "antisym (converse r) = antisym r" by (unfold antisym_def) blast lemma trans_converse [simp]: "trans (converse r) = trans r" by (unfold trans_def) blast lemma sym_conv_converse_eq: "sym r = (r^-1 = r)" by (unfold sym_def) fast lemma sym_Un_converse: "sym (r \ r^-1)" by (unfold sym_def) blast lemma sym_Int_converse: "sym (r \ r^-1)" by (unfold sym_def) blast subsection {* Domain *} lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)" by (unfold Domain_def) blast lemma DomainI [intro]: "(a, b) : r ==> a : Domain r" by (iprover intro!: iffD2 [OF Domain_iff]) lemma DomainE [elim!]: "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P" by (iprover dest!: iffD1 [OF Domain_iff]) lemma Domain_empty [simp]: "Domain {} = {}" by blast lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)" by blast lemma Domain_Id [simp]: "Domain Id = UNIV" by blast lemma Domain_diag [simp]: "Domain (diag A) = A" by blast lemma Domain_Un_eq: "Domain(A \ B) = Domain(A) \ Domain(B)" by blast lemma Domain_Int_subset: "Domain(A \ B) \ Domain(A) \ Domain(B)" by blast lemma Domain_Diff_subset: "Domain(A) - Domain(B) \ Domain(A - B)" by blast lemma Domain_Union: "Domain (Union S) = (\A\S. Domain A)" by blast lemma Domain_mono: "r \ s ==> Domain r \ Domain s" by blast lemma fst_eq_Domain: "fst ` R = Domain R"; apply auto apply (rule image_eqI, auto) done subsection {* Range *} lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)" by (simp add: Domain_def Range_def) lemma RangeI [intro]: "(a, b) : r ==> b : Range r" by (unfold Range_def) (iprover intro!: converseI DomainI) lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P" by (unfold Range_def) (iprover elim!: DomainE dest!: converseD) lemma Range_empty [simp]: "Range {} = {}" by blast lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)" by blast lemma Range_Id [simp]: "Range Id = UNIV" by blast lemma Range_diag [simp]: "Range (diag A) = A" by auto lemma Range_Un_eq: "Range(A \ B) = Range(A) \ Range(B)" by blast lemma Range_Int_subset: "Range(A \ B) \ Range(A) \ Range(B)" by blast lemma Range_Diff_subset: "Range(A) - Range(B) \ Range(A - B)" by blast lemma Range_Union: "Range (Union S) = (\A\S. Range A)" by blast lemma snd_eq_Range: "snd ` R = Range R"; apply auto apply (rule image_eqI, auto) done subsection {* Image of a set under a relation *} lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" by (simp add: Image_def) lemma Image_singleton: "r``{a} = {b. (a, b) : r}" by (simp add: Image_def) lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" by (rule Image_iff [THEN trans]) simp lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A" by (unfold Image_def) blast lemma ImageE [elim!]: "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" by (unfold Image_def) (iprover elim!: CollectE bexE) lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" -- {* This version's more effective when we already have the required @{text a} *} by blast lemma Image_empty [simp]: "R``{} = {}" by blast lemma Image_Id [simp]: "Id `` A = A" by blast lemma Image_diag [simp]: "diag A `` B = A \ B" by blast lemma Image_Int_subset: "R `` (A \ B) \ R `` A \ R `` B" by blast lemma Image_Int_eq: "single_valued (converse R) ==> R `` (A \ B) = R `` A \ R `` B" by (simp add: single_valued_def, blast) lemma Image_Un: "R `` (A \ B) = R `` A \ R `` B" by blast lemma Un_Image: "(R \ S) `` A = R `` A \ S `` A" by blast lemma Image_subset: "r \ A \ B ==> r``C \ B" by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) lemma Image_eq_UN: "r``B = (\y\ B. r``{y})" -- {* NOT suitable for rewriting *} by blast lemma Image_mono: "r' \ r ==> A' \ A ==> (r' `` A') \ (r `` A)" by blast lemma Image_UN: "(r `` (UNION A B)) = (\x\A. r `` (B x))" by blast lemma Image_INT_subset: "(r `` INTER A B) \ (\x\A. r `` (B x))" by blast text{*Converse inclusion requires some assumptions*} lemma Image_INT_eq: "[|single_valued (r\); A\{}|] ==> r `` INTER A B = (\x\A. r `` B x)" apply (rule equalityI) apply (rule Image_INT_subset) apply (simp add: single_valued_def, blast) done lemma Image_subset_eq: "(r``A \ B) = (A \ - ((r^-1) `` (-B)))" by blast subsection {* Single valued relations *} lemma single_valuedI: "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r" by (unfold single_valued_def) lemma single_valuedD: "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" by (simp add: single_valued_def) lemma single_valued_rel_comp: "single_valued r ==> single_valued s ==> single_valued (r O s)" by (unfold single_valued_def) blast lemma single_valued_subset: "r \ s ==> single_valued s ==> single_valued r" by (unfold single_valued_def) blast lemma single_valued_Id [simp]: "single_valued Id" by (unfold single_valued_def) blast lemma single_valued_diag [simp]: "single_valued (diag A)" by (unfold single_valued_def) blast subsection {* Graphs given by @{text Collect} *} lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}" by auto lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}" by auto lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}" by auto subsection {* Inverse image *} lemma sym_inv_image: "sym r ==> sym (inv_image r f)" by (unfold sym_def inv_image_def) blast lemma trans_inv_image: "trans r ==> trans (inv_image r f)" apply (unfold trans_def inv_image_def) apply (simp (no_asm)) apply blast done subsection {* Version of @{text lfp_induct} for binary relations *} lemmas lfp_induct2 = lfp_induct_set [of "(a, b)", split_format (complete)] end