author | nipkow |
Fri, 14 Mar 2008 19:57:12 +0100 | |
changeset 26271 | e324f8918c98 |
parent 24915 | fc90277c0dd7 |
child 26297 | 74012d599204 |
permissions | -rw-r--r-- |
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(* Title: HOL/Relation.thy |
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ID: $Id$ |
1983 | 3 |
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 Product_Type |
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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 :: "[('b * 'c) set, ('a * 'b) 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) : s & (y, z) : r}" |
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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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diag :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *} |
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"diag 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 :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *} |
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"refl A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)" |
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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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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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abbreviation |
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reflexive :: "('a * 'a) set => bool" where -- {* reflexivity over a type *} |
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"reflexive == refl UNIV" |
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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 reflexive_Id: "reflexive Id" |
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by (simp add: refl_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 diag_empty [simp]: "diag {} = {}" |
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by (simp add: diag_def) |
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lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A" |
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by (simp add: diag_def) |
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lemma diagI [intro!,noatp]: "a : A ==> (a, a) : diag A" |
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by (rule diag_eqI) (rule refl) |
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lemma diagE [elim!]: |
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"c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" |
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-- {* The general elimination rule. *} |
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by (unfold diag_def) (iprover elim!: UN_E singletonE) |
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lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)" |
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by blast |
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lemma diag_subset_Times: "diag 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) : s ==> (b, c) : r ==> (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) : s ==> (y, z) : r ==> 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) : s ==> (y, c) : r ==> 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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"s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C" |
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by blast |
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subsection {* Reflexivity *} |
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lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r" |
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by (unfold refl_def) (iprover intro!: ballI) |
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lemma reflD: "refl A r ==> a : A ==> (a, a) : r" |
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by (unfold refl_def) blast |
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lemma reflD1: "refl A r ==> (x, y) : r ==> x : A" |
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by (unfold refl_def) blast |
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lemma reflD2: "refl A r ==> (x, y) : r ==> y : A" |
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by (unfold refl_def) blast |
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lemma refl_Int: "refl A r ==> refl B s ==> refl (A \<inter> B) (r \<inter> s)" |
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by (unfold refl_def) blast |
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lemma refl_Un: "refl A r ==> refl B s ==> refl (A \<union> B) (r \<union> s)" |
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by (unfold refl_def) blast |
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lemma refl_INTER: |
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"ALL x:S. refl (A x) (r x) ==> refl (INTER S A) (INTER S r)" |
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by (unfold refl_def) fast |
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lemma refl_UNION: |
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"ALL x:S. refl (A x) (r x) \<Longrightarrow> refl (UNION S A) (UNION S r)" |
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by (unfold refl_def) blast |
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lemma refl_diag: "refl A (diag A)" |
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by (rule reflI [OF diag_subset_Times diagI]) |
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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_diag [simp]: "antisym (diag 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" |
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by (unfold sym_def, blast) |
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lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)" |
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by (fast intro: symI dest: symD) |
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lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)" |
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by (fast intro: symI dest: symD) |
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lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)" |
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by (fast intro: symI dest: symD) |
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lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)" |
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by (fast intro: symI dest: symD) |
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lemma sym_diag [simp]: "sym (diag A)" |
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by (rule symI) clarify |
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subsection {* Transitivity *} |
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lemma transI: |
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"(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r" |
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by (unfold trans_def) iprover |
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lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r" |
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by (unfold trans_def) iprover |
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lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)" |
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by (fast intro: transI elim: transD) |
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lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)" |
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by (fast intro: transI elim: transD) |
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lemma trans_diag [simp]: "trans (diag A)" |
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by (fast intro: transI elim: transD) |
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subsection {* Converse *} |
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lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)" |
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by (simp add: converse_def) |
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lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1" |
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by (simp add: converse_def) |
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lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r" |
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by (simp add: converse_def) |
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lemma converseE [elim!]: |
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"yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P" |
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-- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} |
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by (unfold converse_def) (iprover elim!: CollectE splitE bexE) |
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lemma converse_converse [simp]: "(r^-1)^-1 = r" |
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by (unfold converse_def) blast |
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lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1" |
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by blast |
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lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1" |
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by blast |
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lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1" |
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by blast |
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292 |
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)" |
|
26271 | 293 |
by fast |
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|
295 |
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)" |
|
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by blast |
19228 | 297 |
|
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lemma converse_Id [simp]: "Id^-1 = Id" |
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by blast |
12905 | 300 |
|
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lemma converse_diag [simp]: "(diag A)^-1 = diag A" |
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by blast |
12905 | 303 |
|
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lemma refl_converse [simp]: "refl A (converse r) = refl A r" |
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by (unfold refl_def) auto |
12905 | 306 |
|
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lemma sym_converse [simp]: "sym (converse r) = sym r" |
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by (unfold sym_def) blast |
19228 | 309 |
|
310 |
lemma antisym_converse [simp]: "antisym (converse r) = antisym r" |
|
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by (unfold antisym_def) blast |
12905 | 312 |
|
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lemma trans_converse [simp]: "trans (converse r) = trans r" |
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by (unfold trans_def) blast |
12905 | 315 |
|
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lemma sym_conv_converse_eq: "sym r = (r^-1 = r)" |
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by (unfold sym_def) fast |
19228 | 318 |
|
319 |
lemma sym_Un_converse: "sym (r \<union> r^-1)" |
|
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by (unfold sym_def) blast |
19228 | 321 |
|
322 |
lemma sym_Int_converse: "sym (r \<inter> r^-1)" |
|
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by (unfold sym_def) blast |
19228 | 324 |
|
12913 | 325 |
|
12905 | 326 |
subsection {* Domain *} |
327 |
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declare Domain_def [noatp] |
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329 |
|
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lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)" |
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by (unfold Domain_def) blast |
12905 | 332 |
|
333 |
lemma DomainI [intro]: "(a, b) : r ==> a : Domain r" |
|
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by (iprover intro!: iffD2 [OF Domain_iff]) |
12905 | 335 |
|
336 |
lemma DomainE [elim!]: |
|
337 |
"a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P" |
|
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by (iprover dest!: iffD1 [OF Domain_iff]) |
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|
340 |
lemma Domain_empty [simp]: "Domain {} = {}" |
|
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by blast |
12905 | 342 |
|
343 |
lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)" |
|
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by blast |
12905 | 345 |
|
346 |
lemma Domain_Id [simp]: "Domain Id = UNIV" |
|
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by blast |
12905 | 348 |
|
349 |
lemma Domain_diag [simp]: "Domain (diag A) = A" |
|
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by blast |
12905 | 351 |
|
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lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)" |
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by blast |
12905 | 354 |
|
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lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)" |
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by blast |
12905 | 357 |
|
12913 | 358 |
lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)" |
26271 | 359 |
by blast |
12905 | 360 |
|
13830 | 361 |
lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)" |
26271 | 362 |
by blast |
363 |
||
364 |
lemma Domain_converse[simp]: "Domain(r^-1) = Range r" |
|
365 |
by(auto simp:Range_def) |
|
12905 | 366 |
|
12913 | 367 |
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s" |
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by blast |
12905 | 369 |
|
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lemma fst_eq_Domain: "fst ` R = Domain R"; |
26271 | 371 |
by (auto intro!:image_eqI) |
22172 | 372 |
|
12905 | 373 |
|
374 |
subsection {* Range *} |
|
375 |
||
376 |
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)" |
|
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by (simp add: Domain_def Range_def) |
12905 | 378 |
|
379 |
lemma RangeI [intro]: "(a, b) : r ==> b : Range r" |
|
26271 | 380 |
by (unfold Range_def) (iprover intro!: converseI DomainI) |
12905 | 381 |
|
382 |
lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P" |
|
26271 | 383 |
by (unfold Range_def) (iprover elim!: DomainE dest!: converseD) |
12905 | 384 |
|
385 |
lemma Range_empty [simp]: "Range {} = {}" |
|
26271 | 386 |
by blast |
12905 | 387 |
|
388 |
lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)" |
|
26271 | 389 |
by blast |
12905 | 390 |
|
391 |
lemma Range_Id [simp]: "Range Id = UNIV" |
|
26271 | 392 |
by blast |
12905 | 393 |
|
394 |
lemma Range_diag [simp]: "Range (diag A) = A" |
|
26271 | 395 |
by auto |
12905 | 396 |
|
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lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)" |
26271 | 398 |
by blast |
12905 | 399 |
|
13830 | 400 |
lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)" |
26271 | 401 |
by blast |
12905 | 402 |
|
12913 | 403 |
lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)" |
26271 | 404 |
by blast |
12905 | 405 |
|
13830 | 406 |
lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)" |
26271 | 407 |
by blast |
408 |
||
409 |
lemma Range_converse[simp]: "Range(r^-1) = Domain r" |
|
410 |
by blast |
|
12905 | 411 |
|
22172 | 412 |
lemma snd_eq_Range: "snd ` R = Range R"; |
26271 | 413 |
by (auto intro!:image_eqI) |
414 |
||
415 |
||
416 |
subsection {* Field *} |
|
417 |
||
418 |
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s" |
|
419 |
by(auto simp:Field_def Domain_def Range_def) |
|
420 |
||
421 |
lemma Field_empty[simp]: "Field {} = {}" |
|
422 |
by(auto simp:Field_def) |
|
423 |
||
424 |
lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r" |
|
425 |
by(auto simp:Field_def) |
|
426 |
||
427 |
lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s" |
|
428 |
by(auto simp:Field_def) |
|
429 |
||
430 |
lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)" |
|
431 |
by(auto simp:Field_def) |
|
432 |
||
433 |
lemma Field_converse[simp]: "Field(r^-1) = Field r" |
|
434 |
by(auto simp:Field_def) |
|
22172 | 435 |
|
12905 | 436 |
|
437 |
subsection {* Image of a set under a relation *} |
|
438 |
||
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declare Image_def [noatp] |
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440 |
|
12913 | 441 |
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" |
26271 | 442 |
by (simp add: Image_def) |
12905 | 443 |
|
12913 | 444 |
lemma Image_singleton: "r``{a} = {b. (a, b) : r}" |
26271 | 445 |
by (simp add: Image_def) |
12905 | 446 |
|
12913 | 447 |
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" |
26271 | 448 |
by (rule Image_iff [THEN trans]) simp |
12905 | 449 |
|
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450 |
lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A" |
26271 | 451 |
by (unfold Image_def) blast |
12905 | 452 |
|
453 |
lemma ImageE [elim!]: |
|
12913 | 454 |
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" |
26271 | 455 |
by (unfold Image_def) (iprover elim!: CollectE bexE) |
12905 | 456 |
|
457 |
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" |
|
458 |
-- {* This version's more effective when we already have the required @{text a} *} |
|
26271 | 459 |
by blast |
12905 | 460 |
|
461 |
lemma Image_empty [simp]: "R``{} = {}" |
|
26271 | 462 |
by blast |
12905 | 463 |
|
464 |
lemma Image_Id [simp]: "Id `` A = A" |
|
26271 | 465 |
by blast |
12905 | 466 |
|
13830 | 467 |
lemma Image_diag [simp]: "diag A `` B = A \<inter> B" |
26271 | 468 |
by blast |
13830 | 469 |
|
470 |
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B" |
|
26271 | 471 |
by blast |
12905 | 472 |
|
13830 | 473 |
lemma Image_Int_eq: |
474 |
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B" |
|
26271 | 475 |
by (simp add: single_valued_def, blast) |
12905 | 476 |
|
13830 | 477 |
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B" |
26271 | 478 |
by blast |
12905 | 479 |
|
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480 |
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A" |
26271 | 481 |
by blast |
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482 |
|
12913 | 483 |
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B" |
26271 | 484 |
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) |
12905 | 485 |
|
13830 | 486 |
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" |
12905 | 487 |
-- {* NOT suitable for rewriting *} |
26271 | 488 |
by blast |
12905 | 489 |
|
12913 | 490 |
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)" |
26271 | 491 |
by blast |
12905 | 492 |
|
13830 | 493 |
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))" |
26271 | 494 |
by blast |
13830 | 495 |
|
496 |
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))" |
|
26271 | 497 |
by blast |
12905 | 498 |
|
13830 | 499 |
text{*Converse inclusion requires some assumptions*} |
500 |
lemma Image_INT_eq: |
|
501 |
"[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" |
|
502 |
apply (rule equalityI) |
|
503 |
apply (rule Image_INT_subset) |
|
504 |
apply (simp add: single_valued_def, blast) |
|
505 |
done |
|
12905 | 506 |
|
12913 | 507 |
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))" |
26271 | 508 |
by blast |
12905 | 509 |
|
510 |
||
12913 | 511 |
subsection {* Single valued relations *} |
512 |
||
513 |
lemma single_valuedI: |
|
12905 | 514 |
"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r" |
26271 | 515 |
by (unfold single_valued_def) |
12905 | 516 |
|
517 |
lemma single_valuedD: |
|
518 |
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" |
|
26271 | 519 |
by (simp add: single_valued_def) |
12905 | 520 |
|
19228 | 521 |
lemma single_valued_rel_comp: |
522 |
"single_valued r ==> single_valued s ==> single_valued (r O s)" |
|
26271 | 523 |
by (unfold single_valued_def) blast |
19228 | 524 |
|
525 |
lemma single_valued_subset: |
|
526 |
"r \<subseteq> s ==> single_valued s ==> single_valued r" |
|
26271 | 527 |
by (unfold single_valued_def) blast |
19228 | 528 |
|
529 |
lemma single_valued_Id [simp]: "single_valued Id" |
|
26271 | 530 |
by (unfold single_valued_def) blast |
19228 | 531 |
|
532 |
lemma single_valued_diag [simp]: "single_valued (diag A)" |
|
26271 | 533 |
by (unfold single_valued_def) blast |
19228 | 534 |
|
12905 | 535 |
|
536 |
subsection {* Graphs given by @{text Collect} *} |
|
537 |
||
538 |
lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}" |
|
26271 | 539 |
by auto |
12905 | 540 |
|
541 |
lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}" |
|
26271 | 542 |
by auto |
12905 | 543 |
|
544 |
lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}" |
|
26271 | 545 |
by auto |
12905 | 546 |
|
547 |
||
12913 | 548 |
subsection {* Inverse image *} |
12905 | 549 |
|
19228 | 550 |
lemma sym_inv_image: "sym r ==> sym (inv_image r f)" |
26271 | 551 |
by (unfold sym_def inv_image_def) blast |
19228 | 552 |
|
12913 | 553 |
lemma trans_inv_image: "trans r ==> trans (inv_image r f)" |
12905 | 554 |
apply (unfold trans_def inv_image_def) |
555 |
apply (simp (no_asm)) |
|
556 |
apply blast |
|
557 |
done |
|
558 |
||
23709 | 559 |
|
560 |
subsection {* Version of @{text lfp_induct} for binary relations *} |
|
561 |
||
562 |
lemmas lfp_induct2 = |
|
563 |
lfp_induct_set [of "(a, b)", split_format (complete)] |
|
564 |
||
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|
565 |
end |