--- a/src/HOL/Relation.thy Thu Mar 01 17:13:54 2012 +0000
+++ b/src/HOL/Relation.thy Thu Mar 01 19:34:52 2012 +0100
@@ -113,28 +113,33 @@
subsubsection {* Reflexivity *}
-definition
- refl_on :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
- "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
+definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
+where
+ "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"
-abbreviation
- refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
+abbreviation refl :: "'a rel \<Rightarrow> bool"
+where -- {* reflexivity over a type *}
"refl \<equiv> refl_on UNIV"
-definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+where
"reflp r \<longleftrightarrow> refl {(x, y). r x y}"
+lemma reflp_refl_eq [pred_set_conv]:
+ "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r"
+ by (simp add: refl_on_def reflp_def)
+
lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
-by (unfold refl_on_def) (iprover intro!: ballI)
+ by (unfold refl_on_def) (iprover intro!: ballI)
lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
-by (unfold refl_on_def) blast
+ by (unfold refl_on_def) blast
lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
-by (unfold refl_on_def) blast
+ by (unfold refl_on_def) blast
lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
-by (unfold refl_on_def) blast
+ by (unfold refl_on_def) blast
lemma reflpI:
"(\<And>x. r x x) \<Longrightarrow> reflp r"
@@ -146,32 +151,40 @@
using assms by (auto dest: refl_onD simp add: reflp_def)
lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
-by (unfold refl_on_def) blast
+ by (unfold refl_on_def) blast
+
+lemma reflp_inf:
+ "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"
+ by (auto intro: reflpI elim: reflpE)
lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
-by (unfold refl_on_def) blast
+ by (unfold refl_on_def) blast
+
+lemma reflp_sup:
+ "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"
+ by (auto intro: reflpI elim: reflpE)
lemma refl_on_INTER:
"ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
-by (unfold refl_on_def) fast
+ by (unfold refl_on_def) fast
lemma refl_on_UNION:
"ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
-by (unfold refl_on_def) blast
+ by (unfold refl_on_def) blast
-lemma refl_on_empty[simp]: "refl_on {} {}"
-by(simp add:refl_on_def)
+lemma refl_on_empty [simp]: "refl_on {} {}"
+ by (simp add:refl_on_def)
lemma refl_on_def' [nitpick_unfold, code]:
- "refl_on A r = ((\<forall>(x, y) \<in> r. x : A \<and> y : A) \<and> (\<forall>x \<in> A. (x, x) : r))"
-by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
+ "refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)"
+ by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
subsubsection {* Irreflexivity *}
-definition
- irrefl :: "('a * 'a) set => bool" where
- "irrefl r \<longleftrightarrow> (\<forall>x. (x,x) \<notin> r)"
+definition irrefl :: "'a rel \<Rightarrow> bool"
+where
+ "irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)"
lemma irrefl_distinct [code]:
"irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
@@ -180,166 +193,231 @@
subsubsection {* Symmetry *}
-definition
- sym :: "('a * 'a) set => bool" where
- "sym r \<longleftrightarrow> (ALL x y. (x,y): r --> (y,x): r)"
+definition sym :: "'a rel \<Rightarrow> bool"
+where
+ "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
+
+definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+where
+ "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
-lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
-by (unfold sym_def) iprover
+lemma symp_sym_eq [pred_set_conv]:
+ "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r"
+ by (simp add: sym_def symp_def)
-lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
-by (unfold sym_def, blast)
-
-definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
- "symp r \<longleftrightarrow> sym {(x, y). r x y}"
+lemma symI:
+ "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"
+ by (unfold sym_def) iprover
lemma sympI:
- "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
- by (auto intro: symI simp add: symp_def)
+ "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
+ by (fact symI [to_pred])
+
+lemma symE:
+ assumes "sym r" and "(b, a) \<in> r"
+ obtains "(a, b) \<in> r"
+ using assms by (simp add: sym_def)
lemma sympE:
- assumes "symp r" and "r x y"
- obtains "r y x"
- using assms by (auto dest: symD simp add: symp_def)
+ assumes "symp r" and "r b a"
+ obtains "r a b"
+ using assms by (rule symE [to_pred])
+
+lemma symD:
+ assumes "sym r" and "(b, a) \<in> r"
+ shows "(a, b) \<in> r"
+ using assms by (rule symE)
-lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
-by (fast intro: symI dest: symD)
+lemma sympD:
+ assumes "symp r" and "r b a"
+ shows "r a b"
+ using assms by (rule symD [to_pred])
+
+lemma sym_Int:
+ "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
+ by (fast intro: symI elim: symE)
-lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
-by (fast intro: symI dest: symD)
+lemma symp_inf:
+ "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
+ by (fact sym_Int [to_pred])
+
+lemma sym_Un:
+ "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
+ by (fast intro: symI elim: symE)
+
+lemma symp_sup:
+ "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
+ by (fact sym_Un [to_pred])
-lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
-by (fast intro: symI dest: symD)
+lemma sym_INTER:
+ "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
+ by (fast intro: symI elim: symE)
+
+(* FIXME thm sym_INTER [to_pred] *)
-lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
-by (fast intro: symI dest: symD)
+lemma sym_UNION:
+ "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
+ by (fast intro: symI elim: symE)
+
+(* FIXME thm sym_UNION [to_pred] *)
subsubsection {* Antisymmetry *}
-definition
- antisym :: "('a * 'a) set => bool" where
- "antisym r \<longleftrightarrow> (ALL x y. (x,y):r --> (y,x):r --> x=y)"
+definition antisym :: "'a rel \<Rightarrow> bool"
+where
+ "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
+
+abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+where
+ "antisymP r \<equiv> antisym {(x, y). r x y}"
lemma antisymI:
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
-by (unfold antisym_def) iprover
+ by (unfold antisym_def) iprover
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
-by (unfold antisym_def) iprover
-
-abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
- "antisymP r \<equiv> antisym {(x, y). r x y}"
+ by (unfold antisym_def) iprover
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
-by (unfold antisym_def) blast
+ by (unfold antisym_def) blast
lemma antisym_empty [simp]: "antisym {}"
-by (unfold antisym_def) blast
+ by (unfold antisym_def) blast
subsubsection {* Transitivity *}
-definition
- trans :: "('a * 'a) set => bool" where
- "trans r \<longleftrightarrow> (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
+definition trans :: "'a rel \<Rightarrow> bool"
+where
+ "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
+
+definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+where
+ "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
+
+lemma transp_trans_eq [pred_set_conv]:
+ "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r"
+ by (simp add: trans_def transp_def)
+
+abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+where -- {* FIXME drop *}
+ "transP r \<equiv> trans {(x, y). r x y}"
lemma transI:
- "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
-by (unfold trans_def) iprover
-
-lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
-by (unfold trans_def) iprover
-
-abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
- "transP r \<equiv> trans {(x, y). r x y}"
-
-definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
- "transp r \<longleftrightarrow> trans {(x, y). r x y}"
+ "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
+ by (unfold trans_def) iprover
lemma transpI:
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
- by (auto intro: transI simp add: transp_def)
-
+ by (fact transI [to_pred])
+
+lemma transE:
+ assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
+ obtains "(x, z) \<in> r"
+ using assms by (unfold trans_def) iprover
+
lemma transpE:
assumes "transp r" and "r x y" and "r y z"
obtains "r x z"
- using assms by (auto dest: transD simp add: transp_def)
+ using assms by (rule transE [to_pred])
+
+lemma transD:
+ assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
+ shows "(x, z) \<in> r"
+ using assms by (rule transE)
+
+lemma transpD:
+ assumes "transp r" and "r x y" and "r y z"
+ shows "r x z"
+ using assms by (rule transD [to_pred])
-lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
-by (fast intro: transI elim: transD)
+lemma trans_Int:
+ "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
+ by (fast intro: transI elim: transE)
-lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
-by (fast intro: transI elim: transD)
+lemma transp_inf:
+ "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
+ by (fact trans_Int [to_pred])
+
+lemma trans_INTER:
+ "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
+ by (fast intro: transI elim: transD)
+
+(* FIXME thm trans_INTER [to_pred] *)
lemma trans_join [code]:
"trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
by (auto simp add: trans_def)
+lemma transp_trans:
+ "transp r \<longleftrightarrow> trans {(x, y). r x y}"
+ by (simp add: trans_def transp_def)
+
subsubsection {* Totality *}
-definition
- total_on :: "'a set => ('a * 'a) set => bool" where
- "total_on A r \<longleftrightarrow> (\<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r)"
+definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
+where
+ "total_on A r \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
abbreviation "total \<equiv> total_on UNIV"
-lemma total_on_empty[simp]: "total_on {} r"
-by(simp add:total_on_def)
+lemma total_on_empty [simp]: "total_on {} r"
+ by (simp add: total_on_def)
subsubsection {* Single valued relations *}
-definition
- single_valued :: "('a * 'b) set => bool" where
- "single_valued r \<longleftrightarrow> (ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z))"
-
-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)
+definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
+where
+ "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
"single_valuedP r \<equiv> single_valued {(x, y). r x y}"
+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)
+
lemma single_valued_subset:
"r \<subseteq> s ==> single_valued s ==> single_valued r"
-by (unfold single_valued_def) blast
+ by (unfold single_valued_def) blast
subsection {* Relation operations *}
subsubsection {* The identity relation *}
-definition
- Id :: "('a * 'a) set" where
- "Id = {p. EX x. p = (x,x)}"
+definition Id :: "'a rel"
+where
+ "Id = {p. \<exists>x. p = (x, x)}"
lemma IdI [intro]: "(a, a) : Id"
-by (simp add: Id_def)
+ by (simp add: Id_def)
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
-by (unfold Id_def) (iprover elim: CollectE)
+ by (unfold Id_def) (iprover elim: CollectE)
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
-by (unfold Id_def) blast
+ by (unfold Id_def) blast
lemma refl_Id: "refl Id"
-by (simp add: refl_on_def)
+ by (simp add: refl_on_def)
lemma antisym_Id: "antisym Id"
-- {* A strange result, since @{text Id} is also symmetric. *}
-by (simp add: antisym_def)
+ by (simp add: antisym_def)
lemma sym_Id: "sym Id"
-by (simp add: sym_def)
+ by (simp add: sym_def)
lemma trans_Id: "trans Id"
-by (simp add: trans_def)
+ by (simp add: trans_def)
lemma single_valued_Id [simp]: "single_valued Id"
by (unfold single_valued_def) blast
@@ -356,45 +434,45 @@
subsubsection {* Diagonal: identity over a set *}
-definition
- Id_on :: "'a set => ('a * 'a) set" where
- "Id_on A = (\<Union>x\<in>A. {(x,x)})"
+definition Id_on :: "'a set \<Rightarrow> 'a rel"
+where
+ "Id_on A = (\<Union>x\<in>A. {(x, x)})"
lemma Id_on_empty [simp]: "Id_on {} = {}"
-by (simp add: Id_on_def)
+ by (simp add: Id_on_def)
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
-by (simp add: Id_on_def)
+ by (simp add: Id_on_def)
lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A"
-by (rule Id_on_eqI) (rule refl)
+ by (rule Id_on_eqI) (rule refl)
lemma Id_onE [elim!]:
"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
-- {* The general elimination rule. *}
-by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
+ by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
-by blast
+ by blast
lemma Id_on_def' [nitpick_unfold]:
"Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
-by auto
+ by auto
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
-by blast
+ by blast
lemma refl_on_Id_on: "refl_on A (Id_on A)"
-by (rule refl_onI [OF Id_on_subset_Times Id_onI])
+ by (rule refl_onI [OF Id_on_subset_Times Id_onI])
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
-by (unfold antisym_def) blast
+ by (unfold antisym_def) blast
lemma sym_Id_on [simp]: "sym (Id_on A)"
-by (rule symI) clarify
+ by (rule symI) clarify
lemma trans_Id_on [simp]: "trans (Id_on A)"
-by (fast intro: transI elim: transD)
+ by (fast intro: transI elim: transD)
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
by (unfold single_valued_def) blast
@@ -402,184 +480,228 @@
subsubsection {* Composition *}
-definition rel_comp :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a * 'c) set" (infixr "O" 75)
+inductive_set rel_comp :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75)
+ for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
where
- "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
+ rel_compI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
-lemma rel_compI [intro]:
- "(a, b) : r ==> (b, c) : s ==> (a, c) : r O s"
-by (unfold rel_comp_def) blast
+abbreviation pred_comp (infixr "OO" 75) where
+ "pred_comp \<equiv> rel_compp"
-lemma rel_compE [elim!]: "xz : r O s ==>
- (!!x y z. xz = (x, z) ==> (x, y) : r ==> (y, z) : s ==> P) ==> P"
-by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
+lemmas pred_compI = rel_compp.intros
-inductive pred_comp :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75)
-for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
-where
- pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (r OO s) a c"
+text {*
+ For historic reasons, the elimination rules are not wholly corresponding.
+ Feel free to consolidate this.
+*}
+inductive_cases rel_compEpair: "(a, c) \<in> r O s"
inductive_cases pred_compE [elim!]: "(r OO s) a c"
-lemma pred_comp_rel_comp_eq [pred_set_conv]:
- "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
- by (auto simp add: fun_eq_iff)
+lemma rel_compE [elim!]: "xz \<in> r O s \<Longrightarrow>
+ (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s \<Longrightarrow> P) \<Longrightarrow> P"
+ by (cases xz) (simp, erule rel_compEpair, iprover)
+
+lemmas pred_comp_rel_comp_eq = rel_compp_rel_comp_eq
+
+lemma R_O_Id [simp]:
+ "R O Id = R"
+ by fast
-lemma rel_compEpair:
- "(a, c) : r O s ==> (!!y. (a, y) : r ==> (y, c) : s ==> P) ==> P"
-by (iprover elim: rel_compE Pair_inject ssubst)
+lemma Id_O_R [simp]:
+ "Id O R = R"
+ by fast
+
+lemma rel_comp_empty1 [simp]:
+ "{} O R = {}"
+ by blast
-lemma R_O_Id [simp]: "R O Id = R"
-by fast
+(* CANDIDATE lemma pred_comp_bot1 [simp]:
+ ""
+ by (fact rel_comp_empty1 [to_pred]) *)
-lemma Id_O_R [simp]: "Id O R = R"
-by fast
+lemma rel_comp_empty2 [simp]:
+ "R O {} = {}"
+ by blast
-lemma rel_comp_empty1[simp]: "{} O R = {}"
-by blast
+(* CANDIDATE lemma pred_comp_bot2 [simp]:
+ ""
+ by (fact rel_comp_empty2 [to_pred]) *)
-lemma rel_comp_empty2[simp]: "R O {} = {}"
-by blast
+lemma O_assoc:
+ "(R O S) O T = R O (S O T)"
+ by blast
+
+lemma pred_comp_assoc:
+ "(r OO s) OO t = r OO (s OO t)"
+ by (fact O_assoc [to_pred])
-lemma O_assoc: "(R O S) O T = R O (S O T)"
-by blast
+lemma trans_O_subset:
+ "trans r \<Longrightarrow> r O r \<subseteq> r"
+ by (unfold trans_def) blast
+
+lemma transp_pred_comp_less_eq:
+ "transp r \<Longrightarrow> r OO r \<le> r "
+ by (fact trans_O_subset [to_pred])
-lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
-by (unfold trans_def) blast
+lemma rel_comp_mono:
+ "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
+ by blast
-lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
-by blast
+lemma pred_comp_mono:
+ "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
+ by (fact rel_comp_mono [to_pred])
lemma rel_comp_subset_Sigma:
- "r \<subseteq> A \<times> B ==> s \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
-by blast
+ "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
+ by blast
+
+lemma rel_comp_distrib [simp]:
+ "R O (S \<union> T) = (R O S) \<union> (R O T)"
+ by auto
-lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
-by auto
+lemma pred_comp_distrib (* CANDIDATE [simp] *):
+ "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
+ by (fact rel_comp_distrib [to_pred])
+
+lemma rel_comp_distrib2 [simp]:
+ "(S \<union> T) O R = (S O R) \<union> (T O R)"
+ by auto
-lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
-by auto
+lemma pred_comp_distrib2 (* CANDIDATE [simp] *):
+ "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
+ by (fact rel_comp_distrib2 [to_pred])
+
+lemma rel_comp_UNION_distrib:
+ "s O UNION I r = (\<Union>i\<in>I. s O r i) "
+ by auto
-lemma rel_comp_UNION_distrib: "s O UNION I r = UNION I (%i. s O r i)"
-by auto
+(* FIXME thm rel_comp_UNION_distrib [to_pred] *)
-lemma rel_comp_UNION_distrib2: "UNION I r O s = UNION I (%i. r i O s)"
-by auto
+lemma rel_comp_UNION_distrib2:
+ "UNION I r O s = (\<Union>i\<in>I. r i O s) "
+ by auto
+
+(* FIXME thm rel_comp_UNION_distrib2 [to_pred] *)
lemma single_valued_rel_comp:
- "single_valued r ==> single_valued s ==> single_valued (r O s)"
-by (unfold single_valued_def) blast
+ "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
+ by (unfold single_valued_def) blast
+
+lemma rel_comp_unfold:
+ "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
+ by (auto simp add: set_eq_iff)
subsubsection {* Converse *}
-definition
- converse :: "('a * 'b) set => ('b * 'a) set"
- ("(_^-1)" [1000] 999) where
- "r^-1 = {(y, x). (x, y) : r}"
+inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999)
+ for r :: "('a \<times> 'b) set"
+where
+ "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1"
notation (xsymbols)
converse ("(_\<inverse>)" [1000] 999)
-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) (iprover elim!: CollectE splitE bexE)
-
-lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
- by (simp add: converse_def)
-
-inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^--1)" [1000] 1000)
- for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
- conversepI: "r a b \<Longrightarrow> r^--1 b a"
+notation
+ conversep ("(_^--1)" [1000] 1000)
notation (xsymbols)
conversep ("(_\<inverse>\<inverse>)" [1000] 1000)
-lemma conversepD:
- assumes ab: "r^--1 a b"
- shows "r b a" using ab
- by cases simp
+lemma converseI [sym]:
+ "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
+ by (fact converse.intros)
+
+lemma conversepI (* CANDIDATE [sym] *):
+ "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
+ by (fact conversep.intros)
+
+lemma converseD [sym]:
+ "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
+ by (erule converse.cases) iprover
+
+lemma conversepD (* CANDIDATE [sym] *):
+ "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
+ by (fact converseD [to_pred])
+
+lemma converseE [elim!]:
+ -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
+ "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
+ by (cases yx) (simp, erule converse.cases, iprover)
-lemma conversep_iff [iff]: "r^--1 a b = r b a"
- by (iprover intro: conversepI dest: conversepD)
+lemmas conversepE (* CANDIDATE [elim!] *) = conversep.cases
+
+lemma converse_iff [iff]:
+ "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
+ by (auto intro: converseI)
+
+lemma conversep_iff [iff]:
+ "r\<inverse>\<inverse> a b = r b a"
+ by (fact converse_iff [to_pred])
-lemma conversep_converse_eq [pred_set_conv]:
- "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"
- by (auto simp add: fun_eq_iff)
+lemma converse_converse [simp]:
+ "(r\<inverse>)\<inverse> = r"
+ by (simp add: set_eq_iff)
-lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
- by (iprover intro: order_antisym conversepI dest: conversepD)
+lemma conversep_conversep [simp]:
+ "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
+ by (fact converse_converse [to_pred])
+
+lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
+ by blast
lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
by (iprover intro: order_antisym conversepI pred_compI
elim: pred_compE dest: conversepD)
+lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
+ by blast
+
lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
+lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
+ by blast
+
lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
-lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
- by (auto simp add: fun_eq_iff)
-
-lemma conversep_eq [simp]: "(op =)^--1 = op ="
- by (auto simp add: fun_eq_iff)
-
-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_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
-by blast
-
-lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
-by blast
-
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
-by fast
+ by fast
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
-by blast
+ by blast
lemma converse_Id [simp]: "Id^-1 = Id"
-by blast
+ by blast
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
-by blast
+ by blast
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
-by (unfold refl_on_def) auto
+ by (unfold refl_on_def) auto
lemma sym_converse [simp]: "sym (converse r) = sym r"
-by (unfold sym_def) blast
+ by (unfold sym_def) blast
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
-by (unfold antisym_def) blast
+ by (unfold antisym_def) blast
lemma trans_converse [simp]: "trans (converse r) = trans r"
-by (unfold trans_def) blast
+ by (unfold trans_def) blast
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
-by (unfold sym_def) fast
+ by (unfold sym_def) fast
lemma sym_Un_converse: "sym (r \<union> r^-1)"
-by (unfold sym_def) blast
+ by (unfold sym_def) blast
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
-by (unfold sym_def) blast
+ by (unfold sym_def) blast
-lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
-by (auto simp: total_on_def)
+lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r"
+ by (auto simp: total_on_def)
lemma finite_converse [iff]: "finite (r^-1) = finite r"
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
@@ -588,50 +710,61 @@
apply (erule finite_imageD [unfolded inj_on_def])
apply (simp split add: split_split)
apply (erule finite_imageI)
- apply (simp add: converse_def image_def, auto)
+ apply (simp add: set_eq_iff image_def, auto)
apply (rule bexI)
prefer 2 apply assumption
apply simp
done
+lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
+ by (auto simp add: fun_eq_iff)
+
+lemma conversep_eq [simp]: "(op =)^--1 = op ="
+ by (auto simp add: fun_eq_iff)
+
+lemma converse_unfold:
+ "r\<inverse> = {(y, x). (x, y) \<in> r}"
+ by (simp add: set_eq_iff)
+
subsubsection {* Domain, range and field *}
-definition
- Domain :: "('a * 'b) set => 'a set" where
- "Domain r = {x. EX y. (x,y):r}"
+definition Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set"
+where
+ "Domain r = {x. \<exists>y. (x, y) \<in> r}"
-definition
- Range :: "('a * 'b) set => 'b set" where
- "Range r = Domain(r^-1)"
+definition Range :: "('a \<times> 'b) set \<Rightarrow> 'b set"
+where
+ "Range r = Domain (r\<inverse>)"
-definition
- Field :: "('a * 'a) set => 'a set" where
+definition Field :: "'a rel \<Rightarrow> 'a set"
+where
"Field r = Domain r \<union> Range r"
declare Domain_def [no_atp]
lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
-by (unfold Domain_def) blast
+ by (unfold Domain_def) blast
lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
-by (iprover intro!: iffD2 [OF Domain_iff])
+ by (iprover intro!: iffD2 [OF Domain_iff])
lemma DomainE [elim!]:
"a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
-by (iprover dest!: iffD1 [OF Domain_iff])
+ by (iprover dest!: iffD1 [OF Domain_iff])
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
-by (simp add: Domain_def Range_def)
+ by (simp add: Domain_def Range_def)
lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
-by (unfold Range_def) (iprover intro!: converseI DomainI)
+ by (unfold Range_def) (iprover intro!: converseI DomainI)
lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
-by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
+ by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
- for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
+where
DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"
inductive_cases DomainPE [elim!]: "DomainP r a"
@@ -640,7 +773,8 @@
by (blast intro!: Orderings.order_antisym predicate1I)
inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
- for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"
+where
RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"
inductive_cases RangePE [elim!]: "RangeP r b"
@@ -679,8 +813,8 @@
lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
by blast
-lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
- by(auto simp: Range_def)
+lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
+ by auto
lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
by blast
@@ -731,7 +865,7 @@
lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
by blast
-lemma Range_converse [simp]: "Range(r^-1) = Domain r"
+lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
by blast
lemma snd_eq_Range: "snd ` R = Range R"
@@ -767,73 +901,76 @@
apply (auto simp add: Field_def Domain_insert Range_insert)
done
+lemma Domain_unfold:
+ "Domain r = {x. \<exists>y. (x, y) \<in> r}"
+ by (fact Domain_def)
+
subsubsection {* Image of a set under a relation *}
-definition
- Image :: "[('a * 'b) set, 'a set] => 'b set"
- (infixl "``" 90) where
- "r `` s = {y. EX x:s. (x,y):r}"
+definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixl "``" 90)
+where
+ "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
declare Image_def [no_atp]
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
-by (simp add: Image_def)
+ by (simp add: Image_def)
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
-by (simp add: Image_def)
+ by (simp add: Image_def)
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
-by (rule Image_iff [THEN trans]) simp
+ by (rule Image_iff [THEN trans]) simp
lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"
-by (unfold Image_def) blast
+ by (unfold Image_def) blast
lemma ImageE [elim!]:
- "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
-by (unfold Image_def) (iprover elim!: CollectE bexE)
+ "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
+ by (unfold Image_def) (iprover 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
+ by blast
lemma Image_empty [simp]: "R``{} = {}"
-by blast
+ by blast
lemma Image_Id [simp]: "Id `` A = A"
-by blast
+ by blast
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
-by blast
+ by blast
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
-by blast
+ by blast
lemma Image_Int_eq:
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
-by (simp add: single_valued_def, blast)
+ by (simp add: single_valued_def, blast)
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
-by blast
+ by blast
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
-by blast
+ by blast
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
-by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
+ by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
-- {* NOT suitable for rewriting *}
-by blast
+ by blast
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
-by blast
+ by blast
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
-by blast
+ by blast
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
-by blast
+ by blast
text{*Converse inclusion requires some assumptions*}
lemma Image_INT_eq:
@@ -844,26 +981,27 @@
done
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
-by blast
+ by blast
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}"
-by auto
+ by auto
subsubsection {* Inverse image *}
-definition
- inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
- "inv_image r f = {(x, y). (f x, f y) : r}"
+definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
+where
+ "inv_image r f = {(x, y). (f x, f y) \<in> r}"
-definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
+definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
+where
"inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
by (simp add: inv_image_def inv_imagep_def)
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
-by (unfold sym_def inv_image_def) blast
+ by (unfold sym_def inv_image_def) blast
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
apply (unfold trans_def inv_image_def)
@@ -875,7 +1013,7 @@
by (auto simp:inv_image_def)
lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
-unfolding inv_image_def converse_def by auto
+ unfolding inv_image_def converse_unfold by auto
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
by (simp add: inv_imagep_def)
@@ -883,7 +1021,8 @@
subsubsection {* Powerset *}
-definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
+definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+where
"Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"