# HG changeset patch
# User nipkow
# Date 1205521032 -3600
# Node ID e324f8918c9837adff73c6de81d083dfe1bd3553
# Parent  73ac6430f5e73bffdc36d3fda5bf9068e5bf1777
Added lemmas

diff -r 73ac6430f5e7 -r e324f8918c98 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri Mar 14 12:18:56 2008 +0100
+++ b/src/HOL/IsaMakefile	Fri Mar 14 19:57:12 2008 +0100
@@ -170,7 +170,7 @@
 HOL-Complex: HOL $(OUT)/HOL-Complex
 
 $(OUT)/HOL-Complex: $(OUT)/HOL Complex/ROOT.ML $(SRC)/Tools/float.ML \
-  Library/Zorn.thy							\
+  Library/Zorn.thy Library/Order_Relation.thy				\
   Real/ContNotDenum.thy Real/float_arith.ML Real/Float.thy		\
   Real/Lubs.thy Real/PReal.thy Real/RComplete.thy 			\
   Real/Rational.thy Real/Real.thy Real/RealDef.thy Real/RealPow.thy	\
diff -r 73ac6430f5e7 -r e324f8918c98 src/HOL/Relation.thy
--- 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)
diff -r 73ac6430f5e7 -r e324f8918c98 src/HOL/Transitive_Closure.thy
--- a/src/HOL/Transitive_Closure.thy	Fri Mar 14 12:18:56 2008 +0100
+++ b/src/HOL/Transitive_Closure.thy	Fri Mar 14 19:57:12 2008 +0100
@@ -63,6 +63,18 @@
   reflcl  ("(_\<^sup>=)" [1000] 999)
 
 
+subsection {* Reflexive closure *}
+
+lemma reflexive_reflcl[simp]: "reflexive(r^=)"
+by(simp add:refl_def)
+
+lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
+by(simp add:antisym_def)
+
+lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
+unfolding trans_def by blast
+
+
 subsection {* Reflexive-transitive closure *}
 
 lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r Un Id)"
@@ -546,7 +558,7 @@
   by (unfold Domain_def) (blast dest: tranclD)
 
 lemma trancl_range [simp]: "Range (r^+) = Range r"
-  by (simp add: Range_def trancl_converse [symmetric])
+unfolding Range_def by(simp add: trancl_converse [symmetric])
 
 lemma Not_Domain_rtrancl:
     "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"