(* Title: HOL/Relation.thy ID: $Id: Relation.thy,v 1.34 2005/09/22 21:56:16 nipkow Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1996 University of Cambridge *) header {* Relations *} theory Relation imports Product_Type begin subsection {* Definitions *} constdefs converse :: "('a * 'b) set => ('b * 'a) set" ("(_^-1)" [1000] 999) "r^-1 == {(y, x). (x, y) : r}" syntax (xsymbols) converse :: "('a * 'b) set => ('b * 'a) set" ("(_\)" [1000] 999) constdefs rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 60) "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}" Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90) "r `` s == {y. EX x:s. (x,y):r}" Id :: "('a * 'a) set" -- {* the identity relation *} "Id == {p. EX x. p = (x,x)}" diag :: "'a set => ('a * 'a) set" -- {* diagonal: identity over a set *} "diag A == \x\A. {(x,x)}" Domain :: "('a * 'b) set => 'a set" "Domain r == {x. EX y. (x,y):r}" Range :: "('a * 'b) set => 'b set" "Range r == Domain(r^-1)" Field :: "('a * 'a) set => 'a set" "Field r == Domain r \ Range r" refl :: "['a set, ('a * 'a) set] => bool" -- {* reflexivity over a set *} "refl A r == r \ A \ A & (ALL x: A. (x,x) : r)" sym :: "('a * 'a) set => bool" -- {* symmetry predicate *} "sym r == ALL x y. (x,y): r --> (y,x): r" antisym:: "('a * 'a) set => bool" -- {* antisymmetry predicate *} "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y" trans :: "('a * 'a) set => bool" -- {* transitivity predicate *} "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)" single_valued :: "('a * 'b) set => bool" "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)" inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" "inv_image r f == {(x, y). (f x, f y) : r}" syntax reflexive :: "('a * 'a) set => bool" -- {* reflexivity over a type *} translations "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 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 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 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 subsection {* Symmetry and Transitivity *} lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r" by (unfold sym_def, blast) 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 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_Id [simp]: "Id^-1 = Id" by blast lemma converse_diag [simp]: "(diag A)^-1 = diag A" by blast lemma refl_converse: "refl A r ==> refl A (converse r)" by (unfold refl_def) blast lemma antisym_converse: "antisym (converse r) = antisym r" by (unfold antisym_def) blast lemma trans_converse: "trans (converse r) = trans r" by (unfold trans_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 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 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) 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 trans_inv_image: "trans r ==> trans (inv_image r f)" apply (unfold trans_def inv_image_def) apply (simp (no_asm)) apply blast done end