author | krauss |
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permissions | -rw-r--r-- |
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(* Title: HOL/Relation.thy |
1983 | 2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1996 University of Cambridge |
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*) |
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header {* Relations *} |
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theory Relation |
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imports Finite_Set Datatype |
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(*FIXME order is important, otherwise merge problem for canonical interpretation of class monoid_mult wrt. power!*) |
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begin |
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subsection {* Definitions *} |
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definition |
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converse :: "('a * 'b) set => ('b * 'a) set" |
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("(_^-1)" [1000] 999) where |
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"r^-1 == {(y, x). (x, y) : r}" |
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notation (xsymbols) |
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converse ("(_\<inverse>)" [1000] 999) |
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definition |
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rel_comp :: "[('a * 'b) set, ('b * 'c) set] => ('a * 'c) set" |
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(infixr "O" 75) where |
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"r O s == {(x,z). EX y. (x, y) : r & (y, z) : s}" |
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definition |
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Image :: "[('a * 'b) set, 'a set] => 'b set" |
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(infixl "``" 90) where |
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"r `` s == {y. EX x:s. (x,y):r}" |
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definition |
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Id :: "('a * 'a) set" where -- {* the identity relation *} |
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"Id == {p. EX x. p = (x,x)}" |
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definition |
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Id_on :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *} |
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"Id_on A == \<Union>x\<in>A. {(x,x)}" |
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definition |
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Domain :: "('a * 'b) set => 'a set" where |
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"Domain r == {x. EX y. (x,y):r}" |
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definition |
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Range :: "('a * 'b) set => 'b set" where |
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"Range r == Domain(r^-1)" |
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definition |
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Field :: "('a * 'a) set => 'a set" where |
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"Field r == Domain r \<union> Range r" |
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definition |
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refl_on :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *} |
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"refl_on A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)" |
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abbreviation |
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refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *} |
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"refl == refl_on UNIV" |
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definition |
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sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *} |
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"sym r == ALL x y. (x,y): r --> (y,x): r" |
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definition |
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antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *} |
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"antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y" |
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definition |
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trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *} |
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"trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)" |
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definition |
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irrefl :: "('a * 'a) set => bool" where |
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"irrefl r \<equiv> \<forall>x. (x,x) \<notin> r" |
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definition |
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total_on :: "'a set => ('a * 'a) set => bool" where |
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"total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r" |
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abbreviation "total \<equiv> total_on UNIV" |
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definition |
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single_valued :: "('a * 'b) set => bool" where |
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"single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)" |
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definition |
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inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where |
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"inv_image r f == {(x, y). (f x, f y) : r}" |
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subsection {* The identity relation *} |
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lemma IdI [intro]: "(a, a) : Id" |
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by (simp add: Id_def) |
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lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" |
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by (unfold Id_def) (iprover elim: CollectE) |
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lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" |
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by (unfold Id_def) blast |
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lemma refl_Id: "refl Id" |
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by (simp add: refl_on_def) |
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lemma antisym_Id: "antisym Id" |
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-- {* A strange result, since @{text Id} is also symmetric. *} |
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by (simp add: antisym_def) |
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lemma sym_Id: "sym Id" |
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by (simp add: sym_def) |
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lemma trans_Id: "trans Id" |
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by (simp add: trans_def) |
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subsection {* Diagonal: identity over a set *} |
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lemma Id_on_empty [simp]: "Id_on {} = {}" |
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by (simp add: Id_on_def) |
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lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A" |
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by (simp add: Id_on_def) |
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lemma Id_onI [intro!,noatp]: "a : A ==> (a, a) : Id_on A" |
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by (rule Id_on_eqI) (rule refl) |
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lemma Id_onE [elim!]: |
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"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" |
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-- {* The general elimination rule. *} |
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by (unfold Id_on_def) (iprover elim!: UN_E singletonE) |
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lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)" |
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by blast |
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lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A" |
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by blast |
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subsection {* Composition of two relations *} |
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lemma rel_compI [intro]: |
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"(a, b) : r ==> (b, c) : s ==> (a, c) : r O s" |
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by (unfold rel_comp_def) blast |
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lemma rel_compE [elim!]: "xz : r O s ==> |
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(!!x y z. xz = (x, z) ==> (x, y) : r ==> (y, z) : s ==> P) ==> P" |
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by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE) |
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lemma rel_compEpair: |
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"(a, c) : r O s ==> (!!y. (a, y) : r ==> (y, c) : s ==> P) ==> P" |
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by (iprover elim: rel_compE Pair_inject ssubst) |
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lemma R_O_Id [simp]: "R O Id = R" |
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by fast |
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lemma Id_O_R [simp]: "Id O R = R" |
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by fast |
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lemma rel_comp_empty1[simp]: "{} O R = {}" |
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by blast |
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lemma rel_comp_empty2[simp]: "R O {} = {}" |
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by blast |
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lemma O_assoc: "(R O S) O T = R O (S O T)" |
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by blast |
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lemma trans_O_subset: "trans r ==> r O r \<subseteq> r" |
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by (unfold trans_def) blast |
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lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)" |
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by blast |
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lemma rel_comp_subset_Sigma: |
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"r \<subseteq> A \<times> B ==> s \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C" |
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by blast |
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lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)" |
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by auto |
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lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)" |
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by auto |
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subsection {* Reflexivity *} |
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lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r" |
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by (unfold refl_on_def) (iprover intro!: ballI) |
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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r" |
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by (unfold refl_on_def) blast |
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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A" |
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by (unfold refl_on_def) blast |
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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A" |
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by (unfold refl_on_def) blast |
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lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)" |
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by (unfold refl_on_def) blast |
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lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)" |
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by (unfold refl_on_def) blast |
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lemma refl_on_INTER: |
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"ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)" |
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by (unfold refl_on_def) fast |
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lemma refl_on_UNION: |
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"ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)" |
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by (unfold refl_on_def) blast |
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lemma refl_on_empty[simp]: "refl_on {} {}" |
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by(simp add:refl_on_def) |
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lemma refl_on_Id_on: "refl_on A (Id_on A)" |
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by (rule refl_onI [OF Id_on_subset_Times Id_onI]) |
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subsection {* Antisymmetry *} |
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lemma antisymI: |
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"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" |
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by (unfold antisym_def) iprover |
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lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" |
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by (unfold antisym_def) iprover |
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lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r" |
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by (unfold antisym_def) blast |
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lemma antisym_empty [simp]: "antisym {}" |
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by (unfold antisym_def) blast |
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lemma antisym_Id_on [simp]: "antisym (Id_on A)" |
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by (unfold antisym_def) blast |
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subsection {* Symmetry *} |
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lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r" |
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by (unfold sym_def) iprover |
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lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r" |
|
26271 | 246 |
by (unfold sym_def, blast) |
12905 | 247 |
|
19228 | 248 |
lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)" |
26271 | 249 |
by (fast intro: symI dest: symD) |
19228 | 250 |
|
251 |
lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)" |
|
26271 | 252 |
by (fast intro: symI dest: symD) |
19228 | 253 |
|
254 |
lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)" |
|
26271 | 255 |
by (fast intro: symI dest: symD) |
19228 | 256 |
|
257 |
lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)" |
|
26271 | 258 |
by (fast intro: symI dest: symD) |
19228 | 259 |
|
30198 | 260 |
lemma sym_Id_on [simp]: "sym (Id_on A)" |
26271 | 261 |
by (rule symI) clarify |
19228 | 262 |
|
263 |
||
264 |
subsection {* Transitivity *} |
|
265 |
||
12905 | 266 |
lemma transI: |
267 |
"(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r" |
|
26271 | 268 |
by (unfold trans_def) iprover |
12905 | 269 |
|
270 |
lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r" |
|
26271 | 271 |
by (unfold trans_def) iprover |
12905 | 272 |
|
19228 | 273 |
lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)" |
26271 | 274 |
by (fast intro: transI elim: transD) |
19228 | 275 |
|
276 |
lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)" |
|
26271 | 277 |
by (fast intro: transI elim: transD) |
19228 | 278 |
|
30198 | 279 |
lemma trans_Id_on [simp]: "trans (Id_on A)" |
26271 | 280 |
by (fast intro: transI elim: transD) |
19228 | 281 |
|
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lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)" |
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283 |
unfolding antisym_def trans_def by blast |
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|
284 |
|
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285 |
subsection {* Irreflexivity *} |
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|
286 |
|
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287 |
lemma irrefl_diff_Id[simp]: "irrefl(r-Id)" |
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288 |
by(simp add:irrefl_def) |
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|
289 |
|
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290 |
subsection {* Totality *} |
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|
291 |
|
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292 |
lemma total_on_empty[simp]: "total_on {} r" |
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293 |
by(simp add:total_on_def) |
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|
294 |
|
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295 |
lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r" |
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296 |
by(simp add: total_on_def) |
12905 | 297 |
|
12913 | 298 |
subsection {* Converse *} |
299 |
||
300 |
lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)" |
|
26271 | 301 |
by (simp add: converse_def) |
12905 | 302 |
|
13343 | 303 |
lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1" |
26271 | 304 |
by (simp add: converse_def) |
12905 | 305 |
|
13343 | 306 |
lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r" |
26271 | 307 |
by (simp add: converse_def) |
12905 | 308 |
|
309 |
lemma converseE [elim!]: |
|
310 |
"yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P" |
|
12913 | 311 |
-- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} |
26271 | 312 |
by (unfold converse_def) (iprover elim!: CollectE splitE bexE) |
12905 | 313 |
|
314 |
lemma converse_converse [simp]: "(r^-1)^-1 = r" |
|
26271 | 315 |
by (unfold converse_def) blast |
12905 | 316 |
|
317 |
lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1" |
|
26271 | 318 |
by blast |
12905 | 319 |
|
19228 | 320 |
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1" |
26271 | 321 |
by blast |
19228 | 322 |
|
323 |
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1" |
|
26271 | 324 |
by blast |
19228 | 325 |
|
326 |
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)" |
|
26271 | 327 |
by fast |
19228 | 328 |
|
329 |
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)" |
|
26271 | 330 |
by blast |
19228 | 331 |
|
12905 | 332 |
lemma converse_Id [simp]: "Id^-1 = Id" |
26271 | 333 |
by blast |
12905 | 334 |
|
30198 | 335 |
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A" |
26271 | 336 |
by blast |
12905 | 337 |
|
30198 | 338 |
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r" |
339 |
by (unfold refl_on_def) auto |
|
12905 | 340 |
|
19228 | 341 |
lemma sym_converse [simp]: "sym (converse r) = sym r" |
26271 | 342 |
by (unfold sym_def) blast |
19228 | 343 |
|
344 |
lemma antisym_converse [simp]: "antisym (converse r) = antisym r" |
|
26271 | 345 |
by (unfold antisym_def) blast |
12905 | 346 |
|
19228 | 347 |
lemma trans_converse [simp]: "trans (converse r) = trans r" |
26271 | 348 |
by (unfold trans_def) blast |
12905 | 349 |
|
19228 | 350 |
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)" |
26271 | 351 |
by (unfold sym_def) fast |
19228 | 352 |
|
353 |
lemma sym_Un_converse: "sym (r \<union> r^-1)" |
|
26271 | 354 |
by (unfold sym_def) blast |
19228 | 355 |
|
356 |
lemma sym_Int_converse: "sym (r \<inter> r^-1)" |
|
26271 | 357 |
by (unfold sym_def) blast |
19228 | 358 |
|
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359 |
lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r" |
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360 |
by (auto simp: total_on_def) |
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361 |
|
12913 | 362 |
|
12905 | 363 |
subsection {* Domain *} |
364 |
||
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365 |
declare Domain_def [noatp] |
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366 |
|
12905 | 367 |
lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)" |
26271 | 368 |
by (unfold Domain_def) blast |
12905 | 369 |
|
370 |
lemma DomainI [intro]: "(a, b) : r ==> a : Domain r" |
|
26271 | 371 |
by (iprover intro!: iffD2 [OF Domain_iff]) |
12905 | 372 |
|
373 |
lemma DomainE [elim!]: |
|
374 |
"a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P" |
|
26271 | 375 |
by (iprover dest!: iffD1 [OF Domain_iff]) |
12905 | 376 |
|
377 |
lemma Domain_empty [simp]: "Domain {} = {}" |
|
26271 | 378 |
by blast |
12905 | 379 |
|
380 |
lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)" |
|
26271 | 381 |
by blast |
12905 | 382 |
|
383 |
lemma Domain_Id [simp]: "Domain Id = UNIV" |
|
26271 | 384 |
by blast |
12905 | 385 |
|
30198 | 386 |
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A" |
26271 | 387 |
by blast |
12905 | 388 |
|
13830 | 389 |
lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)" |
26271 | 390 |
by blast |
12905 | 391 |
|
13830 | 392 |
lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)" |
26271 | 393 |
by blast |
12905 | 394 |
|
12913 | 395 |
lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)" |
26271 | 396 |
by blast |
12905 | 397 |
|
13830 | 398 |
lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)" |
26271 | 399 |
by blast |
400 |
||
401 |
lemma Domain_converse[simp]: "Domain(r^-1) = Range r" |
|
402 |
by(auto simp:Range_def) |
|
12905 | 403 |
|
12913 | 404 |
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s" |
26271 | 405 |
by blast |
12905 | 406 |
|
22172 | 407 |
lemma fst_eq_Domain: "fst ` R = Domain R"; |
26271 | 408 |
by (auto intro!:image_eqI) |
22172 | 409 |
|
29609 | 410 |
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)" |
411 |
by auto |
|
412 |
||
413 |
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)" |
|
414 |
by auto |
|
415 |
||
12905 | 416 |
|
417 |
subsection {* Range *} |
|
418 |
||
419 |
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)" |
|
26271 | 420 |
by (simp add: Domain_def Range_def) |
12905 | 421 |
|
422 |
lemma RangeI [intro]: "(a, b) : r ==> b : Range r" |
|
26271 | 423 |
by (unfold Range_def) (iprover intro!: converseI DomainI) |
12905 | 424 |
|
425 |
lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P" |
|
26271 | 426 |
by (unfold Range_def) (iprover elim!: DomainE dest!: converseD) |
12905 | 427 |
|
428 |
lemma Range_empty [simp]: "Range {} = {}" |
|
26271 | 429 |
by blast |
12905 | 430 |
|
431 |
lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)" |
|
26271 | 432 |
by blast |
12905 | 433 |
|
434 |
lemma Range_Id [simp]: "Range Id = UNIV" |
|
26271 | 435 |
by blast |
12905 | 436 |
|
30198 | 437 |
lemma Range_Id_on [simp]: "Range (Id_on A) = A" |
26271 | 438 |
by auto |
12905 | 439 |
|
13830 | 440 |
lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)" |
26271 | 441 |
by blast |
12905 | 442 |
|
13830 | 443 |
lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)" |
26271 | 444 |
by blast |
12905 | 445 |
|
12913 | 446 |
lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)" |
26271 | 447 |
by blast |
12905 | 448 |
|
13830 | 449 |
lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)" |
26271 | 450 |
by blast |
451 |
||
452 |
lemma Range_converse[simp]: "Range(r^-1) = Domain r" |
|
453 |
by blast |
|
12905 | 454 |
|
22172 | 455 |
lemma snd_eq_Range: "snd ` R = Range R"; |
26271 | 456 |
by (auto intro!:image_eqI) |
457 |
||
458 |
||
459 |
subsection {* Field *} |
|
460 |
||
461 |
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s" |
|
462 |
by(auto simp:Field_def Domain_def Range_def) |
|
463 |
||
464 |
lemma Field_empty[simp]: "Field {} = {}" |
|
465 |
by(auto simp:Field_def) |
|
466 |
||
467 |
lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r" |
|
468 |
by(auto simp:Field_def) |
|
469 |
||
470 |
lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s" |
|
471 |
by(auto simp:Field_def) |
|
472 |
||
473 |
lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)" |
|
474 |
by(auto simp:Field_def) |
|
475 |
||
476 |
lemma Field_converse[simp]: "Field(r^-1) = Field r" |
|
477 |
by(auto simp:Field_def) |
|
22172 | 478 |
|
12905 | 479 |
|
480 |
subsection {* Image of a set under a relation *} |
|
481 |
||
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|
482 |
declare Image_def [noatp] |
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|
483 |
|
12913 | 484 |
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" |
26271 | 485 |
by (simp add: Image_def) |
12905 | 486 |
|
12913 | 487 |
lemma Image_singleton: "r``{a} = {b. (a, b) : r}" |
26271 | 488 |
by (simp add: Image_def) |
12905 | 489 |
|
12913 | 490 |
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" |
26271 | 491 |
by (rule Image_iff [THEN trans]) simp |
12905 | 492 |
|
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|
493 |
lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A" |
26271 | 494 |
by (unfold Image_def) blast |
12905 | 495 |
|
496 |
lemma ImageE [elim!]: |
|
12913 | 497 |
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" |
26271 | 498 |
by (unfold Image_def) (iprover elim!: CollectE bexE) |
12905 | 499 |
|
500 |
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" |
|
501 |
-- {* This version's more effective when we already have the required @{text a} *} |
|
26271 | 502 |
by blast |
12905 | 503 |
|
504 |
lemma Image_empty [simp]: "R``{} = {}" |
|
26271 | 505 |
by blast |
12905 | 506 |
|
507 |
lemma Image_Id [simp]: "Id `` A = A" |
|
26271 | 508 |
by blast |
12905 | 509 |
|
30198 | 510 |
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B" |
26271 | 511 |
by blast |
13830 | 512 |
|
513 |
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B" |
|
26271 | 514 |
by blast |
12905 | 515 |
|
13830 | 516 |
lemma Image_Int_eq: |
517 |
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B" |
|
26271 | 518 |
by (simp add: single_valued_def, blast) |
12905 | 519 |
|
13830 | 520 |
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B" |
26271 | 521 |
by blast |
12905 | 522 |
|
13812
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|
523 |
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A" |
26271 | 524 |
by blast |
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
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changeset
|
525 |
|
12913 | 526 |
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B" |
26271 | 527 |
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) |
12905 | 528 |
|
13830 | 529 |
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" |
12905 | 530 |
-- {* NOT suitable for rewriting *} |
26271 | 531 |
by blast |
12905 | 532 |
|
12913 | 533 |
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)" |
26271 | 534 |
by blast |
12905 | 535 |
|
13830 | 536 |
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))" |
26271 | 537 |
by blast |
13830 | 538 |
|
539 |
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))" |
|
26271 | 540 |
by blast |
12905 | 541 |
|
13830 | 542 |
text{*Converse inclusion requires some assumptions*} |
543 |
lemma Image_INT_eq: |
|
544 |
"[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" |
|
545 |
apply (rule equalityI) |
|
546 |
apply (rule Image_INT_subset) |
|
547 |
apply (simp add: single_valued_def, blast) |
|
548 |
done |
|
12905 | 549 |
|
12913 | 550 |
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))" |
26271 | 551 |
by blast |
12905 | 552 |
|
553 |
||
12913 | 554 |
subsection {* Single valued relations *} |
555 |
||
556 |
lemma single_valuedI: |
|
12905 | 557 |
"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r" |
26271 | 558 |
by (unfold single_valued_def) |
12905 | 559 |
|
560 |
lemma single_valuedD: |
|
561 |
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" |
|
26271 | 562 |
by (simp add: single_valued_def) |
12905 | 563 |
|
19228 | 564 |
lemma single_valued_rel_comp: |
565 |
"single_valued r ==> single_valued s ==> single_valued (r O s)" |
|
26271 | 566 |
by (unfold single_valued_def) blast |
19228 | 567 |
|
568 |
lemma single_valued_subset: |
|
569 |
"r \<subseteq> s ==> single_valued s ==> single_valued r" |
|
26271 | 570 |
by (unfold single_valued_def) blast |
19228 | 571 |
|
572 |
lemma single_valued_Id [simp]: "single_valued Id" |
|
26271 | 573 |
by (unfold single_valued_def) blast |
19228 | 574 |
|
30198 | 575 |
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)" |
26271 | 576 |
by (unfold single_valued_def) blast |
19228 | 577 |
|
12905 | 578 |
|
579 |
subsection {* Graphs given by @{text Collect} *} |
|
580 |
||
581 |
lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}" |
|
26271 | 582 |
by auto |
12905 | 583 |
|
584 |
lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}" |
|
26271 | 585 |
by auto |
12905 | 586 |
|
587 |
lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}" |
|
26271 | 588 |
by auto |
12905 | 589 |
|
590 |
||
12913 | 591 |
subsection {* Inverse image *} |
12905 | 592 |
|
19228 | 593 |
lemma sym_inv_image: "sym r ==> sym (inv_image r f)" |
26271 | 594 |
by (unfold sym_def inv_image_def) blast |
19228 | 595 |
|
12913 | 596 |
lemma trans_inv_image: "trans r ==> trans (inv_image r f)" |
12905 | 597 |
apply (unfold trans_def inv_image_def) |
598 |
apply (simp (no_asm)) |
|
599 |
apply blast |
|
600 |
done |
|
601 |
||
23709 | 602 |
|
29609 | 603 |
subsection {* Finiteness *} |
604 |
||
605 |
lemma finite_converse [iff]: "finite (r^-1) = finite r" |
|
606 |
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r") |
|
607 |
apply simp |
|
608 |
apply (rule iffI) |
|
609 |
apply (erule finite_imageD [unfolded inj_on_def]) |
|
610 |
apply (simp split add: split_split) |
|
611 |
apply (erule finite_imageI) |
|
612 |
apply (simp add: converse_def image_def, auto) |
|
613 |
apply (rule bexI) |
|
614 |
prefer 2 apply assumption |
|
615 |
apply simp |
|
616 |
done |
|
617 |
||
618 |
text {* \paragraph{Finiteness of transitive closure} (Thanks to Sidi |
|
619 |
Ehmety) *} |
|
620 |
||
621 |
lemma finite_Field: "finite r ==> finite (Field r)" |
|
622 |
-- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *} |
|
623 |
apply (induct set: finite) |
|
624 |
apply (auto simp add: Field_def Domain_insert Range_insert) |
|
625 |
done |
|
626 |
||
627 |
||
23709 | 628 |
subsection {* Version of @{text lfp_induct} for binary relations *} |
629 |
||
630 |
lemmas lfp_induct2 = |
|
631 |
lfp_induct_set [of "(a, b)", split_format (complete)] |
|
632 |
||
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
633 |
end |