(* Title: HOL/Relation.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
header {* Relations *}
theory Relation = Product_Type:
subsection {* Definitions *}
constdefs
converse :: "('a * 'b) set => ('b * 'a) set" ("(_^-1)" [1000] 999)
"r^-1 == {(y, x). (x, y) : r}"
syntax (xsymbols)
converse :: "('a * 'b) set => ('b * 'a) set" ("(_\<inverse>)" [1000] 999)
constdefs
rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 60)
"r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90)
"r `` s == {y. EX x:s. (x,y):r}"
Id :: "('a * 'a) set" -- {* the identity relation *}
"Id == {p. EX x. p = (x,x)}"
diag :: "'a set => ('a * 'a) set" -- {* diagonal: identity over a set *}
"diag A == UN x:A. {(x,x)}"
Domain :: "('a * 'b) set => 'a set"
"Domain r == {x. EX y. (x,y):r}"
Range :: "('a * 'b) set => 'b set"
"Range r == Domain(r^-1)"
Field :: "('a * 'a) set => 'a set"
"Field r == Domain r Un Range r"
refl :: "['a set, ('a * 'a) set] => bool" -- {* reflexivity over a set *}
"refl A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
sym :: "('a * 'a) set => bool" -- {* symmetry predicate *}
"sym r == ALL x y. (x,y): r --> (y,x): r"
antisym:: "('a * 'a) set => bool" -- {* antisymmetry predicate *}
"antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
trans :: "('a * 'a) set => bool" -- {* transitivity predicate *}
"trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
single_valued :: "('a * 'b) set => bool"
"single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set"
"inv_image r f == {(x, y). (f x, f y) : r}"
syntax
reflexive :: "('a * 'a) set => bool" -- {* reflexivity over a type *}
translations
"reflexive" == "refl UNIV"
subsection {* The identity relation *}
lemma IdI [intro]: "(a, a) : Id"
by (simp add: Id_def)
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
by (unfold Id_def) (rules elim: CollectE)
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
by (unfold Id_def) blast
lemma reflexive_Id: "reflexive Id"
by (simp add: refl_def)
lemma antisym_Id: "antisym Id"
-- {* A strange result, since @{text Id} is also symmetric. *}
by (simp add: antisym_def)
lemma trans_Id: "trans Id"
by (simp add: trans_def)
subsection {* Diagonal: identity over a set *}
lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
by (simp add: diag_def)
lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
by (rule diag_eqI) (rule refl)
lemma diagE [elim!]:
"c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
-- {* The general elimination rule. *}
by (unfold diag_def) (rules elim!: UN_E singletonE)
lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
by blast
lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
by blast
subsection {* Composition of two relations *}
lemma rel_compI [intro]:
"(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
by (unfold rel_comp_def) blast
lemma rel_compE [elim!]: "xz : r O s ==>
(!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r ==> P) ==> P"
by (unfold rel_comp_def) (rules elim!: CollectE splitE exE conjE)
lemma rel_compEpair:
"(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
by (rules elim: rel_compE Pair_inject ssubst)
lemma R_O_Id [simp]: "R O Id = R"
by fast
lemma Id_O_R [simp]: "Id O R = R"
by fast
lemma O_assoc: "(R O S) O T = R O (S O T)"
by blast
lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
by (unfold trans_def) blast
lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
by blast
lemma rel_comp_subset_Sigma:
"s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
by blast
subsection {* Reflexivity *}
lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
by (unfold refl_def) (rules intro!: ballI)
lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
by (unfold refl_def) blast
subsection {* Antisymmetry *}
lemma antisymI:
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
by (unfold antisym_def) rules
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
by (unfold antisym_def) rules
subsection {* Transitivity *}
lemma transI:
"(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
by (unfold trans_def) rules
lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
by (unfold trans_def) rules
subsection {* Converse *}
lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
by (simp add: converse_def)
lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
by (simp add: converse_def)
lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
by (simp add: converse_def)
lemma converseE [elim!]:
"yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
-- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
by (unfold converse_def) (rules elim!: CollectE splitE bexE)
lemma converse_converse [simp]: "(r^-1)^-1 = r"
by (unfold converse_def) blast
lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
by blast
lemma converse_Id [simp]: "Id^-1 = Id"
by blast
lemma converse_diag [simp]: "(diag A)^-1 = diag A"
by blast
lemma refl_converse: "refl A r ==> refl A (converse r)"
by (unfold refl_def) blast
lemma antisym_converse: "antisym (converse r) = antisym r"
by (unfold antisym_def) blast
lemma trans_converse: "trans (converse r) = trans r"
by (unfold trans_def) blast
subsection {* Domain *}
lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
by (unfold Domain_def) blast
lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
by (rules intro!: iffD2 [OF Domain_iff])
lemma DomainE [elim!]:
"a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
by (rules dest!: iffD1 [OF Domain_iff])
lemma Domain_empty [simp]: "Domain {} = {}"
by blast
lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
by blast
lemma Domain_Id [simp]: "Domain Id = UNIV"
by blast
lemma Domain_diag [simp]: "Domain (diag A) = A"
by blast
lemma Domain_Un_eq: "Domain(A Un B) = Domain(A) Un Domain(B)"
by blast
lemma Domain_Int_subset: "Domain(A Int B) \<subseteq> Domain(A) Int Domain(B)"
by blast
lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
by blast
lemma Domain_Union: "Domain (Union S) = (UN A:S. Domain A)"
by blast
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
by blast
subsection {* Range *}
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
by (simp add: Domain_def Range_def)
lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
by (unfold Range_def) (rules intro!: converseI DomainI)
lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
by (unfold Range_def) (rules elim!: DomainE dest!: converseD)
lemma Range_empty [simp]: "Range {} = {}"
by blast
lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
by blast
lemma Range_Id [simp]: "Range Id = UNIV"
by blast
lemma Range_diag [simp]: "Range (diag A) = A"
by auto
lemma Range_Un_eq: "Range(A Un B) = Range(A) Un Range(B)"
by blast
lemma Range_Int_subset: "Range(A Int B) \<subseteq> Range(A) Int Range(B)"
by blast
lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
by blast
lemma Range_Union: "Range (Union S) = (UN A:S. Range A)"
by blast
subsection {* Image of a set under a relation *}
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
by (simp add: Image_def)
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
by (simp add: Image_def)
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
by (rule Image_iff [THEN trans]) simp
lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
by (unfold Image_def) blast
lemma ImageE [elim!]:
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
by (unfold Image_def) (rules elim!: CollectE bexE)
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
-- {* This version's more effective when we already have the required @{text a} *}
by blast
lemma Image_empty [simp]: "R``{} = {}"
by blast
lemma Image_Id [simp]: "Id `` A = A"
by blast
lemma Image_diag [simp]: "diag A `` B = A Int B"
by blast
lemma Image_Int_subset: "R `` (A Int B) \<subseteq> R `` A Int R `` B"
by blast
lemma Image_Un: "R `` (A Un B) = R `` A Un R `` B"
by blast
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
by (rules intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
lemma Image_eq_UN: "r``B = (UN y: B. r``{y})"
-- {* NOT suitable for rewriting *}
by blast
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
by blast
lemma Image_UN: "(r `` (UNION A B)) = (UN x:A.(r `` (B x)))"
by blast
lemma Image_INT_subset: "(r `` (INTER A B)) \<subseteq> (INT x:A.(r `` (B x)))"
-- {* Converse inclusion fails *}
by blast
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
by blast
subsection {* Single valued relations *}
lemma single_valuedI:
"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
by (unfold single_valued_def)
lemma single_valuedD:
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
by (simp add: single_valued_def)
subsection {* Graphs given by @{text Collect} *}
lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
by auto
lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
by auto
lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
by auto
subsection {* Inverse image *}
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
apply (unfold trans_def inv_image_def)
apply (simp (no_asm))
apply blast
done
end