--- a/src/HOL/Relation.thy Thu Feb 21 10:25:00 2002 +0100
+++ b/src/HOL/Relation.thy Thu Feb 21 11:05:20 2002 +0100
@@ -8,6 +8,8 @@
theory Relation = Product_Type:
+subsection {* Definitions *}
+
constdefs
converse :: "('a * 'b) set => ('b * 'a) set" ("(_^-1)" [1000] 999)
"r^-1 == {(y, x). (x, y) : r}"
@@ -16,46 +18,46 @@
constdefs
rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 60)
- "r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
+ "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
+
+ fun_rel_comp :: "['a => 'b, ('b * 'c) set] => ('a => 'c) set"
+ "fun_rel_comp f R == {g. ALL x. (f x, g x) : R}"
Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90)
- "r `` s == {y. ? x:s. (x,y):r}"
+ "r `` s == {y. EX x:s. (x,y):r}"
Id :: "('a * 'a) set" -- {* the identity relation *}
- "Id == {p. ? x. p = (x,x)}"
+ "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)}"
-
+ "diag A == UN x:A. {(x,x)}"
+
Domain :: "('a * 'b) set => 'a set"
- "Domain(r) == {x. ? y. (x,y):r}"
+ "Domain r == {x. EX y. (x,y):r}"
Range :: "('a * 'b) set => 'b set"
- "Range(r) == Domain(r^-1)"
+ "Range r == Domain(r^-1)"
Field :: "('a * 'a) set => 'a set"
- "Field r == Domain r Un Range r"
+ "Field r == Domain r Un Range r"
refl :: "['a set, ('a * 'a) set] => bool" -- {* reflexivity over a set *}
- "refl A r == r <= A <*> A & (ALL x: A. (x,x) : r)"
+ "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"
+ "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"
+ "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
trans :: "('a * 'a) set => bool" -- {* transitivity predicate *}
- "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
+ "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
single_valued :: "('a * 'b) set => bool"
- "single_valued r == !x y. (x,y):r --> (!z. (x,z):r --> y=z)"
-
- fun_rel_comp :: "['a => 'b, ('b * 'c) set] => ('a => 'c) set"
- "fun_rel_comp f R == {g. !x. (f x, g x) : R}"
+ "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}"
+ "inv_image r f == {(x, y). (f x, f y) : r}"
syntax
reflexive :: "('a * 'a) set => bool" -- {* reflexivity over a type *}
@@ -63,7 +65,7 @@
"reflexive" == "refl UNIV"
-subsection {* Identity relation *}
+subsection {* The identity relation *}
lemma IdI [intro]: "(a, a) : Id"
by (simp add: Id_def)
@@ -85,7 +87,7 @@
by (simp add: trans_def)
-subsection {* Diagonal relation: identity restricted to some set *}
+subsection {* Diagonal: identity over a set *}
lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
by (simp add: diag_def)
@@ -95,23 +97,23 @@
lemma diagE [elim!]:
"c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
- -- {* The general elimination rule *}
+ -- {* 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 <= A <*> A"
+lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
by blast
subsection {* Composition of two relations *}
-lemma rel_compI [intro]:
+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 ==>
+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)
@@ -128,25 +130,41 @@
lemma O_assoc: "(R O S) O T = R O (S O T)"
by blast
-lemma trans_O_subset: "trans r ==> r O r <= r"
+lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
by (unfold trans_def) blast
-lemma rel_comp_mono: "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)"
+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 <= A <*> B; r <= B <*> C |] ==> (r O s) <= A <*> C"
+ "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
by blast
-subsection {* Natural deduction for refl(r) *}
+
+subsection {* Composition of function and relation *}
+
+lemma fun_rel_comp_mono: "A \<subseteq> B ==> fun_rel_comp f A \<subseteq> fun_rel_comp f B"
+ by (unfold fun_rel_comp_def) fast
-lemma reflI: "r <= A <*> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
+lemma fun_rel_comp_unique:
+ "ALL x. EX! y. (f x, y) : R ==> EX! g. g : fun_rel_comp f R"
+ apply (unfold fun_rel_comp_def)
+ apply (rule_tac a = "%x. THE y. (f x, y) : R" in ex1I)
+ apply (fast dest!: theI')
+ apply (fast intro: ext the1_equality [symmetric])
+ done
+
+
+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 {* Natural deduction for antisym(r) *}
+
+subsection {* Antisymmetry *}
lemma antisymI:
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
@@ -155,7 +173,8 @@
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
by (unfold antisym_def) rules
-subsection {* Natural deduction for trans(r) *}
+
+subsection {* Transitivity *}
lemma transI:
"(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
@@ -164,21 +183,21 @@
lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
by (unfold trans_def) rules
-subsection {* Natural deduction for r^-1 *}
-lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a):r)"
+subsection {* Converse *}
+
+lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
by (simp add: converse_def)
-lemma converseI: "(a,b):r ==> (b,a): r^-1"
+lemma converseI: "(a, b) : r ==> (b, a) : r^-1"
by (simp add: converse_def)
-lemma converseD: "(a,b) : r^-1 ==> (b,a) : r"
+lemma converseD: "(a,b) : r^-1 ==> (b, a) : r"
by (simp add: converse_def)
-(*More general than converseD, as it "splits" the member of the relation*)
-
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"
@@ -190,7 +209,7 @@
lemma converse_Id [simp]: "Id^-1 = Id"
by blast
-lemma converse_diag [simp]: "(diag A) ^-1 = diag A"
+lemma converse_diag [simp]: "(diag A)^-1 = diag A"
by blast
lemma refl_converse: "refl A r ==> refl A (converse r)"
@@ -202,6 +221,7 @@
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)"
@@ -229,16 +249,16 @@
lemma Domain_Un_eq: "Domain(A Un B) = Domain(A) Un Domain(B)"
by blast
-lemma Domain_Int_subset: "Domain(A Int B) <= Domain(A) Int Domain(B)"
+lemma Domain_Int_subset: "Domain(A Int B) \<subseteq> Domain(A) Int Domain(B)"
by blast
-lemma Domain_Diff_subset: "Domain(A) - Domain(B) <= Domain(A - B)"
+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 <= s ==> Domain r <= Domain s"
+lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
by blast
@@ -268,10 +288,10 @@
lemma Range_Un_eq: "Range(A Un B) = Range(A) Un Range(B)"
by blast
-lemma Range_Int_subset: "Range(A Int B) <= Range(A) Int Range(B)"
+lemma Range_Int_subset: "Range(A Int B) \<subseteq> Range(A) Int Range(B)"
by blast
-lemma Range_Diff_subset: "Range(A) - Range(B) <= Range(A - B)"
+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)"
@@ -282,20 +302,20 @@
ML {* overload_1st_set "Relation.Image" *}
-lemma Image_iff: "(b : r``A) = (EX x:A. (x,b):r)"
+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}"
+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)"
+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"
+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"
+ "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"
@@ -311,35 +331,36 @@
lemma Image_diag [simp]: "diag A `` B = A Int B"
by blast
-lemma Image_Int_subset: "R `` (A Int B) <= R `` A Int R `` B"
+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 <= A <*> B ==> r``C <= B"
+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'<=r; A'<=A |] ==> (r' `` A') <= (r `` A)"
+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)) <= (INT x:A.(r `` (B x)))"
+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 <= B) = (A <= - ((r^-1) `` (-B)))"
+lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
by blast
-subsection "single_valued"
-lemma single_valuedI:
+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)
@@ -360,24 +381,9 @@
by auto
-subsection {* Composition of function and relation *}
-
-lemma fun_rel_comp_mono: "A <= B ==> fun_rel_comp f A <= fun_rel_comp f B"
- by (unfold fun_rel_comp_def) fast
+subsection {* Inverse image *}
-lemma fun_rel_comp_unique:
- "ALL x. EX! y. (f x, y) : R ==> EX! g. g : fun_rel_comp f R"
- apply (unfold fun_rel_comp_def)
- apply (rule_tac a = "%x. THE y. (f x, y) : R" in ex1I)
- apply (fast dest!: theI')
- apply (fast intro: ext the1_equality [symmetric])
- done
-
-
-subsection "inverse image"
-
-lemma trans_inv_image:
- "trans r ==> trans (inv_image r f)"
+lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
apply (unfold trans_def inv_image_def)
apply (simp (no_asm))
apply blast