--- a/src/HOL/Relation.thy Fri Mar 14 12:18:56 2008 +0100
+++ b/src/HOL/Relation.thy Fri Mar 14 19:57:12 2008 +0100
@@ -82,245 +82,245 @@
subsection {* The identity relation *}
lemma IdI [intro]: "(a, a) : Id"
- by (simp add: Id_def)
+by (simp add: Id_def)
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
- by (unfold Id_def) (iprover elim: CollectE)
+by (unfold Id_def) (iprover elim: CollectE)
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
- by (unfold Id_def) blast
+by (unfold Id_def) blast
lemma reflexive_Id: "reflexive Id"
- by (simp add: refl_def)
+by (simp add: refl_def)
lemma antisym_Id: "antisym Id"
-- {* A strange result, since @{text Id} is also symmetric. *}
- by (simp add: antisym_def)
+by (simp add: antisym_def)
lemma sym_Id: "sym Id"
- by (simp add: sym_def)
+by (simp add: sym_def)
lemma trans_Id: "trans Id"
- by (simp add: trans_def)
+by (simp add: trans_def)
subsection {* Diagonal: identity over a set *}
lemma diag_empty [simp]: "diag {} = {}"
- by (simp add: diag_def)
+by (simp add: diag_def)
lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
- by (simp add: diag_def)
+by (simp add: diag_def)
lemma diagI [intro!,noatp]: "a : A ==> (a, a) : diag A"
- by (rule diag_eqI) (rule refl)
+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)
+by (unfold diag_def) (iprover elim!: UN_E singletonE)
lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
- by blast
+by blast
lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
- by blast
+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
+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)
+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)
+by (iprover elim: rel_compE Pair_inject ssubst)
lemma R_O_Id [simp]: "R O Id = R"
- by fast
+by fast
lemma Id_O_R [simp]: "Id O R = R"
- by fast
+by fast
lemma rel_comp_empty1[simp]: "{} O R = {}"
- by blast
+by blast
lemma rel_comp_empty2[simp]: "R O {} = {}"
- by blast
+by blast
lemma O_assoc: "(R O S) O T = R O (S O T)"
- by blast
+by blast
lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
- by (unfold trans_def) blast
+by (unfold trans_def) blast
lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
- by blast
+by blast
lemma rel_comp_subset_Sigma:
"s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
- by blast
+by blast
subsection {* Reflexivity *}
lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
- by (unfold refl_def) (iprover intro!: ballI)
+by (unfold refl_def) (iprover intro!: ballI)
lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
- by (unfold refl_def) blast
+by (unfold refl_def) blast
lemma reflD1: "refl A r ==> (x, y) : r ==> x : A"
- by (unfold refl_def) blast
+by (unfold refl_def) blast
lemma reflD2: "refl A r ==> (x, y) : r ==> y : A"
- by (unfold refl_def) blast
+by (unfold refl_def) blast
lemma refl_Int: "refl A r ==> refl B s ==> refl (A \<inter> B) (r \<inter> s)"
- by (unfold refl_def) blast
+by (unfold refl_def) blast
lemma refl_Un: "refl A r ==> refl B s ==> refl (A \<union> B) (r \<union> s)"
- by (unfold refl_def) blast
+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
+by (unfold refl_def) fast
lemma refl_UNION:
"ALL x:S. refl (A x) (r x) \<Longrightarrow> refl (UNION S A) (UNION S r)"
- by (unfold refl_def) blast
+by (unfold refl_def) blast
lemma refl_diag: "refl A (diag A)"
- by (rule reflI [OF diag_subset_Times diagI])
+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
+by (unfold antisym_def) iprover
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
- by (unfold antisym_def) iprover
+by (unfold antisym_def) iprover
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
- by (unfold antisym_def) blast
+by (unfold antisym_def) blast
lemma antisym_empty [simp]: "antisym {}"
- by (unfold antisym_def) blast
+by (unfold antisym_def) blast
lemma antisym_diag [simp]: "antisym (diag A)"
- by (unfold antisym_def) blast
+by (unfold antisym_def) blast
subsection {* Symmetry *}
lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
- by (unfold sym_def) iprover
+by (unfold sym_def) iprover
lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
- by (unfold sym_def, blast)
+by (unfold sym_def, blast)
lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
- by (fast intro: symI dest: symD)
+by (fast intro: symI dest: symD)
lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
- by (fast intro: symI dest: symD)
+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)
+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)
+by (fast intro: symI dest: symD)
lemma sym_diag [simp]: "sym (diag A)"
- by (rule symI) clarify
+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
+by (unfold trans_def) iprover
lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
- by (unfold trans_def) iprover
+by (unfold trans_def) iprover
lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
- by (fast intro: transI elim: transD)
+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)
+by (fast intro: transI elim: transD)
lemma trans_diag [simp]: "trans (diag A)"
- by (fast intro: transI elim: transD)
+by (fast intro: transI elim: transD)
subsection {* Converse *}
lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
- by (simp add: converse_def)
+by (simp add: converse_def)
lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
- by (simp add: converse_def)
+by (simp add: converse_def)
lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
- by (simp add: converse_def)
+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)
+by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
lemma converse_converse [simp]: "(r^-1)^-1 = r"
- by (unfold converse_def) blast
+by (unfold converse_def) blast
lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
- by blast
+by blast
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
- by blast
+by blast
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
- by blast
+by blast
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
- by fast
+by fast
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
- by blast
+by blast
lemma converse_Id [simp]: "Id^-1 = Id"
- by blast
+by blast
lemma converse_diag [simp]: "(diag A)^-1 = diag A"
- by blast
+by blast
lemma refl_converse [simp]: "refl A (converse r) = refl A r"
- by (unfold refl_def) auto
+by (unfold refl_def) auto
lemma sym_converse [simp]: "sym (converse r) = sym r"
- by (unfold sym_def) blast
+by (unfold sym_def) blast
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
- by (unfold antisym_def) blast
+by (unfold antisym_def) blast
lemma trans_converse [simp]: "trans (converse r) = trans r"
- by (unfold trans_def) blast
+by (unfold trans_def) blast
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
- by (unfold sym_def) fast
+by (unfold sym_def) fast
lemma sym_Un_converse: "sym (r \<union> r^-1)"
- by (unfold sym_def) blast
+by (unfold sym_def) blast
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
- by (unfold sym_def) blast
+by (unfold sym_def) blast
subsection {* Domain *}
@@ -328,87 +328,110 @@
declare Domain_def [noatp]
lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
- by (unfold Domain_def) blast
+by (unfold Domain_def) blast
lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
- by (iprover intro!: iffD2 [OF Domain_iff])
+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])
+by (iprover dest!: iffD1 [OF Domain_iff])
lemma Domain_empty [simp]: "Domain {} = {}"
- by blast
+by blast
lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
- by blast
+by blast
lemma Domain_Id [simp]: "Domain Id = UNIV"
- by blast
+by blast
lemma Domain_diag [simp]: "Domain (diag A) = A"
- by blast
+by blast
lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
- by blast
+by blast
lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
- by blast
+by blast
lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
- by blast
+by blast
lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
- by blast
+by blast
+
+lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
+by(auto simp:Range_def)
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
- by blast
+by blast
lemma fst_eq_Domain: "fst ` R = Domain R";
- apply auto
- apply (rule image_eqI, auto)
- done
+by (auto intro!:image_eqI)
subsection {* Range *}
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
- by (simp add: Domain_def Range_def)
+by (simp add: Domain_def Range_def)
lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
- by (unfold Range_def) (iprover intro!: converseI DomainI)
+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)
+by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
lemma Range_empty [simp]: "Range {} = {}"
- by blast
+by blast
lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
- by blast
+by blast
lemma Range_Id [simp]: "Range Id = UNIV"
- by blast
+by blast
lemma Range_diag [simp]: "Range (diag A) = A"
- by auto
+by auto
lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
- by blast
+by blast
lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
- by blast
+by blast
lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
- by blast
+by blast
lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
- by blast
+by blast
+
+lemma Range_converse[simp]: "Range(r^-1) = Domain r"
+by blast
lemma snd_eq_Range: "snd ` R = Range R";
- apply auto
- apply (rule image_eqI, auto)
- done
+by (auto intro!:image_eqI)
+
+
+subsection {* Field *}
+
+lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
+by(auto simp:Field_def Domain_def Range_def)
+
+lemma Field_empty[simp]: "Field {} = {}"
+by(auto simp:Field_def)
+
+lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
+by(auto simp:Field_def)
+
+lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
+by(auto simp:Field_def)
+
+lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
+by(auto simp:Field_def)
+
+lemma Field_converse[simp]: "Field(r^-1) = Field r"
+by(auto simp:Field_def)
subsection {* Image of a set under a relation *}
@@ -416,62 +439,62 @@
declare Image_def [noatp]
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
- by (simp add: Image_def)
+by (simp add: Image_def)
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
- by (simp add: Image_def)
+by (simp add: Image_def)
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
- by (rule Image_iff [THEN trans]) simp
+by (rule Image_iff [THEN trans]) simp
lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A"
- by (unfold Image_def) blast
+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)
+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
+by blast
lemma Image_empty [simp]: "R``{} = {}"
- by blast
+by blast
lemma Image_Id [simp]: "Id `` A = A"
- by blast
+by blast
lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
- by blast
+by blast
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
- by blast
+by blast
lemma Image_Int_eq:
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
- by (simp add: single_valued_def, blast)
+by (simp add: single_valued_def, blast)
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
- by blast
+by blast
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
- by blast
+by blast
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
- by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
+by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
-- {* NOT suitable for rewriting *}
- by blast
+by blast
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
- by blast
+by blast
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
- by blast
+by blast
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
- by blast
+by blast
text{*Converse inclusion requires some assumptions*}
lemma Image_INT_eq:
@@ -482,50 +505,50 @@
done
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
- by blast
+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)
+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)
+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
+by (unfold single_valued_def) blast
lemma single_valued_subset:
"r \<subseteq> s ==> single_valued s ==> single_valued r"
- by (unfold single_valued_def) blast
+by (unfold single_valued_def) blast
lemma single_valued_Id [simp]: "single_valued Id"
- by (unfold single_valued_def) blast
+by (unfold single_valued_def) blast
lemma single_valued_diag [simp]: "single_valued (diag A)"
- by (unfold single_valued_def) blast
+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
+by auto
lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
- by auto
+by auto
lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
- by auto
+by auto
subsection {* Inverse image *}
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
- by (unfold sym_def inv_image_def) blast
+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)