author | mengj |
Tue, 07 Mar 2006 04:01:25 +0100 | |
changeset 19199 | b338c218cc6e |
parent 17589 | 58eeffd73be1 |
child 19228 | 30fce6da8cbe |
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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constdefs |
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converse :: "('a * 'b) set => ('b * 'a) set" ("(_^-1)" [1000] 999) |
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"r^-1 == {(y, x). (x, y) : r}" |
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syntax (xsymbols) |
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converse :: "('a * 'b) set => ('b * 'a) set" ("(_\<inverse>)" [1000] 999) |
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constdefs |
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rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 60) |
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"r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}" |
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Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90) |
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"r `` s == {y. EX x:s. (x,y):r}" |
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Id :: "('a * 'a) set" -- {* the identity relation *} |
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"Id == {p. EX x. p = (x,x)}" |
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diag :: "'a set => ('a * 'a) set" -- {* diagonal: identity over a set *} |
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"diag A == \<Union>x\<in>A. {(x,x)}" |
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Domain :: "('a * 'b) set => 'a set" |
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"Domain r == {x. EX y. (x,y):r}" |
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Range :: "('a * 'b) set => 'b set" |
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"Range r == Domain(r^-1)" |
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Field :: "('a * 'a) set => 'a set" |
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"Field r == Domain r \<union> Range r" |
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refl :: "['a set, ('a * 'a) set] => bool" -- {* 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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sym :: "('a * 'a) set => bool" -- {* symmetry predicate *} |
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"sym r == ALL x y. (x,y): r --> (y,x): r" |
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antisym:: "('a * 'a) set => bool" -- {* antisymmetry predicate *} |
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"antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y" |
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trans :: "('a * 'a) set => bool" -- {* 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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single_valued :: "('a * 'b) set => bool" |
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"single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)" |
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inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" |
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"inv_image r f == {(x, y). (f x, f y) : r}" |
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syntax |
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reflexive :: "('a * 'a) set => bool" -- {* reflexivity over a type *} |
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translations |
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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 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!]: "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 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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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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subsection {* Symmetry and Transitivity *} |
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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 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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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_Id [simp]: "Id^-1 = Id" |
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by blast |
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lemma converse_diag [simp]: "(diag A)^-1 = diag A" |
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by blast |
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lemma refl_converse: "refl A r ==> refl A (converse r)" |
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by (unfold refl_def) blast |
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lemma antisym_converse: "antisym (converse r) = antisym r" |
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by (unfold antisym_def) blast |
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lemma trans_converse: "trans (converse r) = trans r" |
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by (unfold trans_def) blast |
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subsection {* Domain *} |
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lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)" |
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by (unfold Domain_def) blast |
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lemma DomainI [intro]: "(a, b) : r ==> a : Domain r" |
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by (iprover intro!: iffD2 [OF Domain_iff]) |
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lemma DomainE [elim!]: |
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"a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P" |
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by (iprover dest!: iffD1 [OF Domain_iff]) |
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lemma Domain_empty [simp]: "Domain {} = {}" |
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by blast |
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lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)" |
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by blast |
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lemma Domain_Id [simp]: "Domain Id = UNIV" |
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by blast |
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lemma Domain_diag [simp]: "Domain (diag A) = A" |
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by blast |
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lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)" |
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by blast |
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lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)" |
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by blast |
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lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)" |
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by blast |
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lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)" |
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by blast |
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lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s" |
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by blast |
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subsection {* Range *} |
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lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)" |
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by (simp add: Domain_def Range_def) |
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lemma RangeI [intro]: "(a, b) : r ==> b : Range r" |
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by (unfold Range_def) (iprover intro!: converseI DomainI) |
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lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P" |
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by (unfold Range_def) (iprover elim!: DomainE dest!: converseD) |
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lemma Range_empty [simp]: "Range {} = {}" |
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by blast |
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lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)" |
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by blast |
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lemma Range_Id [simp]: "Range Id = UNIV" |
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by blast |
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lemma Range_diag [simp]: "Range (diag A) = A" |
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by auto |
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lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)" |
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by blast |
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lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)" |
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by blast |
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lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)" |
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by blast |
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lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)" |
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by blast |
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subsection {* Image of a set under a relation *} |
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lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" |
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by (simp add: Image_def) |
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lemma Image_singleton: "r``{a} = {b. (a, b) : r}" |
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by (simp add: Image_def) |
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lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" |
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by (rule Image_iff [THEN trans]) simp |
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lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A" |
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by (unfold Image_def) blast |
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lemma ImageE [elim!]: |
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"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" |
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by (unfold Image_def) (iprover elim!: CollectE bexE) |
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lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" |
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-- {* This version's more effective when we already have the required @{text a} *} |
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by blast |
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lemma Image_empty [simp]: "R``{} = {}" |
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by blast |
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lemma Image_Id [simp]: "Id `` A = A" |
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by blast |
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lemma Image_diag [simp]: "diag A `` B = A \<inter> B" |
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by blast |
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lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B" |
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by blast |
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lemma Image_Int_eq: |
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"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B" |
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by (simp add: single_valued_def, blast) |
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lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B" |
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by blast |
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lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A" |
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by blast |
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lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B" |
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by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) |
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lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" |
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-- {* NOT suitable for rewriting *} |
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by blast |
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lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)" |
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by blast |
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lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))" |
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by blast |
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lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))" |
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by blast |
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text{*Converse inclusion requires some assumptions*} |
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lemma Image_INT_eq: |
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"[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" |
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apply (rule equalityI) |
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apply (rule Image_INT_subset) |
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apply (simp add: single_valued_def, blast) |
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done |
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lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))" |
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by blast |
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subsection {* Single valued relations *} |
365 |
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lemma single_valuedI: |
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"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r" |
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by (unfold single_valued_def) |
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lemma single_valuedD: |
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"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" |
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by (simp add: single_valued_def) |
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subsection {* Graphs given by @{text Collect} *} |
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lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}" |
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by auto |
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lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}" |
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by auto |
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382 |
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lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}" |
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by auto |
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385 |
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386 |
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subsection {* Inverse image *} |
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lemma trans_inv_image: "trans r ==> trans (inv_image r f)" |
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apply (unfold trans_def inv_image_def) |
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apply (simp (no_asm)) |
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apply blast |
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done |
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1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
395 |
end |