src/HOL/Relation.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 24286 7619080e49f0
child 24915 fc90277c0dd7
permissions -rw-r--r--
moved Finite_Set before Datatype
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(*  Title:      HOL/Relation.thy
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    ID:         $Id$
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    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 FixedPoint
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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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lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
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  by fast
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lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-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 [simp]: "refl A (converse r) = refl A r"
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  by (unfold refl_def) auto
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lemma sym_converse [simp]: "sym (converse r) = sym r"
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  by (unfold sym_def) blast
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lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
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  by (unfold antisym_def) blast
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lemma trans_converse [simp]: "trans (converse r) = trans r"
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  by (unfold trans_def) blast
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lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
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  by (unfold sym_def) fast
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lemma sym_Un_converse: "sym (r \<union> r^-1)"
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  by (unfold sym_def) blast
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lemma sym_Int_converse: "sym (r \<inter> r^-1)"
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  by (unfold sym_def) blast
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subsection {* Domain *}
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declare Domain_def [noatp]
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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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   341
  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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   347
  by blast
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   348
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lemma Domain_diag [simp]: "Domain (diag A) = A"
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   350
  by blast
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   352
lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
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   353
  by blast
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   354
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   355
lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
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   356
  by blast
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   357
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   358
lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
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   359
  by blast
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   360
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   361
lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
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   362
  by blast
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   363
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   364
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
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   365
  by blast
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   366
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   367
lemma fst_eq_Domain: "fst ` R = Domain R";
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   368
  apply auto
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   369
  apply (rule image_eqI, auto) 
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   370
  done
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   371
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   372
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subsection {* Range *}
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   375
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
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   376
  by (simp add: Domain_def Range_def)
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   377
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   378
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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   380
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   381
lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
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   382
  by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
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   383
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   384
lemma Range_empty [simp]: "Range {} = {}"
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   385
  by blast
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   386
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   387
lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
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   388
  by blast
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   389
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   390
lemma Range_Id [simp]: "Range Id = UNIV"
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   391
  by blast
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   392
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   393
lemma Range_diag [simp]: "Range (diag A) = A"
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   394
  by auto
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   395
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   396
lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
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   397
  by blast
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   398
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   399
lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
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   400
  by blast
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   401
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   402
lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
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   403
  by blast
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   404
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   405
lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
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   406
  by blast
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   407
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   408
lemma snd_eq_Range: "snd ` R = Range R";
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   409
  apply auto
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   410
  apply (rule image_eqI, auto) 
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   411
  done
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   412
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   413
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   414
subsection {* Image of a set under a relation *}
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   415
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   416
declare Image_def [noatp]
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   417
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   418
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
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   419
  by (simp add: Image_def)
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   420
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   421
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
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   422
  by (simp add: Image_def)
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   423
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   424
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
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   425
  by (rule Image_iff [THEN trans]) simp
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   426
paulson@24286
   427
lemma ImageI [intro,noatp]: "(a, b) : r ==> a : A ==> b : r``A"
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   428
  by (unfold Image_def) blast
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   429
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   430
lemma ImageE [elim!]:
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   431
    "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
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   432
  by (unfold Image_def) (iprover elim!: CollectE bexE)
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   433
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   434
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
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   435
  -- {* This version's more effective when we already have the required @{text a} *}
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   436
  by blast
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   437
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   438
lemma Image_empty [simp]: "R``{} = {}"
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   439
  by blast
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   440
berghofe@12905
   441
lemma Image_Id [simp]: "Id `` A = A"
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   442
  by blast
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   443
paulson@13830
   444
lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
paulson@13830
   445
  by blast
paulson@13830
   446
paulson@13830
   447
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
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   448
  by blast
berghofe@12905
   449
paulson@13830
   450
lemma Image_Int_eq:
paulson@13830
   451
     "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
paulson@13830
   452
  by (simp add: single_valued_def, blast) 
berghofe@12905
   453
paulson@13830
   454
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
berghofe@12905
   455
  by blast
berghofe@12905
   456
paulson@13812
   457
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
paulson@13812
   458
  by blast
paulson@13812
   459
wenzelm@12913
   460
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
nipkow@17589
   461
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
berghofe@12905
   462
paulson@13830
   463
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
berghofe@12905
   464
  -- {* NOT suitable for rewriting *}
berghofe@12905
   465
  by blast
berghofe@12905
   466
wenzelm@12913
   467
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
berghofe@12905
   468
  by blast
berghofe@12905
   469
paulson@13830
   470
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
paulson@13830
   471
  by blast
paulson@13830
   472
paulson@13830
   473
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
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   474
  by blast
berghofe@12905
   475
paulson@13830
   476
text{*Converse inclusion requires some assumptions*}
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   477
lemma Image_INT_eq:
paulson@13830
   478
     "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
paulson@13830
   479
apply (rule equalityI)
paulson@13830
   480
 apply (rule Image_INT_subset) 
paulson@13830
   481
apply  (simp add: single_valued_def, blast)
paulson@13830
   482
done
berghofe@12905
   483
wenzelm@12913
   484
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
berghofe@12905
   485
  by blast
berghofe@12905
   486
berghofe@12905
   487
wenzelm@12913
   488
subsection {* Single valued relations *}
wenzelm@12913
   489
wenzelm@12913
   490
lemma single_valuedI:
berghofe@12905
   491
  "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
berghofe@12905
   492
  by (unfold single_valued_def)
berghofe@12905
   493
berghofe@12905
   494
lemma single_valuedD:
berghofe@12905
   495
  "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
berghofe@12905
   496
  by (simp add: single_valued_def)
berghofe@12905
   497
huffman@19228
   498
lemma single_valued_rel_comp:
huffman@19228
   499
  "single_valued r ==> single_valued s ==> single_valued (r O s)"
huffman@19228
   500
  by (unfold single_valued_def) blast
huffman@19228
   501
huffman@19228
   502
lemma single_valued_subset:
huffman@19228
   503
  "r \<subseteq> s ==> single_valued s ==> single_valued r"
huffman@19228
   504
  by (unfold single_valued_def) blast
huffman@19228
   505
huffman@19228
   506
lemma single_valued_Id [simp]: "single_valued Id"
huffman@19228
   507
  by (unfold single_valued_def) blast
huffman@19228
   508
huffman@19228
   509
lemma single_valued_diag [simp]: "single_valued (diag A)"
huffman@19228
   510
  by (unfold single_valued_def) blast
huffman@19228
   511
berghofe@12905
   512
berghofe@12905
   513
subsection {* Graphs given by @{text Collect} *}
berghofe@12905
   514
berghofe@12905
   515
lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
berghofe@12905
   516
  by auto
berghofe@12905
   517
berghofe@12905
   518
lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
berghofe@12905
   519
  by auto
berghofe@12905
   520
berghofe@12905
   521
lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
berghofe@12905
   522
  by auto
berghofe@12905
   523
berghofe@12905
   524
wenzelm@12913
   525
subsection {* Inverse image *}
berghofe@12905
   526
huffman@19228
   527
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
huffman@19228
   528
  by (unfold sym_def inv_image_def) blast
huffman@19228
   529
wenzelm@12913
   530
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
berghofe@12905
   531
  apply (unfold trans_def inv_image_def)
berghofe@12905
   532
  apply (simp (no_asm))
berghofe@12905
   533
  apply blast
berghofe@12905
   534
  done
berghofe@12905
   535
haftmann@23709
   536
haftmann@23709
   537
subsection {* Version of @{text lfp_induct} for binary relations *}
haftmann@23709
   538
haftmann@23709
   539
lemmas lfp_induct2 = 
haftmann@23709
   540
  lfp_induct_set [of "(a, b)", split_format (complete)]
haftmann@23709
   541
nipkow@1128
   542
end