--- a/src/HOL/Relation.thy Fri Feb 24 22:46:16 2012 +0100
+++ b/src/HOL/Relation.thy Fri Feb 24 22:46:44 2012 +0100
@@ -1,15 +1,107 @@
(* Title: HOL/Relation.thy
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1996 University of Cambridge
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen
*)
-header {* Relations *}
+header {* Relations – as sets of pairs, and binary predicates *}
theory Relation
imports Datatype Finite_Set
begin
-subsection {* Definitions *}
+notation
+ bot ("\<bottom>") and
+ top ("\<top>") and
+ inf (infixl "\<sqinter>" 70) and
+ sup (infixl "\<squnion>" 65) and
+ Inf ("\<Sqinter>_" [900] 900) and
+ Sup ("\<Squnion>_" [900] 900)
+
+syntax (xsymbols)
+ "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
+ "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
+ "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
+ "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
+
+
+subsection {* Classical rules for reasoning on predicates *}
+
+declare predicate1D [Pure.dest?, dest?]
+declare predicate2I [Pure.intro!, intro!]
+declare predicate2D [Pure.dest, dest]
+declare bot1E [elim!]
+declare bot2E [elim!]
+declare top1I [intro!]
+declare top2I [intro!]
+declare inf1I [intro!]
+declare inf2I [intro!]
+declare inf1E [elim!]
+declare inf2E [elim!]
+declare sup1I1 [intro?]
+declare sup2I1 [intro?]
+declare sup1I2 [intro?]
+declare sup2I2 [intro?]
+declare sup1E [elim!]
+declare sup2E [elim!]
+declare sup1CI [intro!]
+declare sup2CI [intro!]
+declare INF1_I [intro!]
+declare INF2_I [intro!]
+declare INF1_D [elim]
+declare INF2_D [elim]
+declare INF1_E [elim]
+declare INF2_E [elim]
+declare SUP1_I [intro]
+declare SUP2_I [intro]
+declare SUP1_E [elim!]
+declare SUP2_E [elim!]
+
+
+subsection {* Conversions between set and predicate relations *}
+
+lemma pred_equals_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R = S)"
+ by (simp add: set_eq_iff fun_eq_iff)
+
+lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R = S)"
+ by (simp add: set_eq_iff fun_eq_iff)
+
+lemma pred_subset_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"
+ by (simp add: subset_iff le_fun_def)
+
+lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"
+ by (simp add: subset_iff le_fun_def)
+
+lemma bot_empty_eq: "\<bottom> = (\<lambda>x. x \<in> {})"
+ by (auto simp add: fun_eq_iff)
+
+lemma bot_empty_eq2: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
+ by (auto simp add: fun_eq_iff)
+
+lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
+ by (simp add: inf_fun_def)
+
+lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
+ by (simp add: inf_fun_def)
+
+lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
+ by (simp add: sup_fun_def)
+
+lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
+ by (simp add: sup_fun_def)
+
+lemma INF_INT_eq: "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i. r i))"
+ by (simp add: INF_apply fun_eq_iff)
+
+lemma INF_INT_eq2: "(\<Sqinter>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i. r i))"
+ by (simp add: INF_apply fun_eq_iff)
+
+lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i. r i))"
+ by (simp add: SUP_apply fun_eq_iff)
+
+lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i. r i))"
+ by (simp add: SUP_apply fun_eq_iff)
+
+
+subsection {* Relations as sets of pairs *}
type_synonym 'a rel = "('a * 'a) set"
@@ -90,7 +182,7 @@
"inv_image r f = {(x, y). (f x, f y) : r}"
-subsection {* The identity relation *}
+subsubsection {* The identity relation *}
lemma IdI [intro]: "(a, a) : Id"
by (simp add: Id_def)
@@ -115,7 +207,7 @@
by (simp add: trans_def)
-subsection {* Diagonal: identity over a set *}
+subsubsection {* Diagonal: identity over a set *}
lemma Id_on_empty [simp]: "Id_on {} = {}"
by (simp add: Id_on_def)
@@ -142,7 +234,7 @@
by blast
-subsection {* Composition of two relations *}
+subsubsection {* Composition of two relations *}
lemma rel_compI [intro]:
"(a, b) : r ==> (b, c) : s ==> (a, c) : r O s"
@@ -194,7 +286,7 @@
by auto
-subsection {* Reflexivity *}
+subsubsection {* Reflexivity *}
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)
@@ -232,7 +324,8 @@
"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)
-subsection {* Antisymmetry *}
+
+subsubsection {* Antisymmetry *}
lemma antisymI:
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
@@ -251,7 +344,7 @@
by (unfold antisym_def) blast
-subsection {* Symmetry *}
+subsubsection {* Symmetry *}
lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
by (unfold sym_def) iprover
@@ -275,7 +368,7 @@
by (rule symI) clarify
-subsection {* Transitivity *}
+subsubsection {* Transitivity *}
lemma trans_join [code]:
"trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
@@ -300,7 +393,8 @@
lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
unfolding antisym_def trans_def by blast
-subsection {* Irreflexivity *}
+
+subsubsection {* Irreflexivity *}
lemma irrefl_distinct [code]:
"irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
@@ -310,7 +404,7 @@
by(simp add:irrefl_def)
-subsection {* Totality *}
+subsubsection {* Totality *}
lemma total_on_empty[simp]: "total_on {} r"
by(simp add:total_on_def)
@@ -318,7 +412,8 @@
lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
by(simp add: total_on_def)
-subsection {* Converse *}
+
+subsubsection {* Converse *}
lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
by (simp add: converse_def)
@@ -383,7 +478,7 @@
by (auto simp: total_on_def)
-subsection {* Domain *}
+subsubsection {* Domain *}
declare Domain_def [no_atp]
@@ -444,7 +539,7 @@
by auto
-subsection {* Range *}
+subsubsection {* Range *}
lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
by (simp add: Domain_def Range_def)
@@ -493,7 +588,7 @@
by force
-subsection {* Field *}
+subsubsection {* Field *}
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
by(auto simp:Field_def Domain_def Range_def)
@@ -514,7 +609,7 @@
by(auto simp:Field_def)
-subsection {* Image of a set under a relation *}
+subsubsection {* Image of a set under a relation *}
declare Image_def [no_atp]
@@ -588,7 +683,7 @@
by blast
-subsection {* Single valued relations *}
+subsubsection {* Single valued relations *}
lemma single_valuedI:
"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
@@ -613,7 +708,7 @@
by (unfold single_valued_def) blast
-subsection {* Graphs given by @{text Collect} *}
+subsubsection {* Graphs given by @{text Collect} *}
lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
by auto
@@ -625,7 +720,7 @@
by auto
-subsection {* Inverse image *}
+subsubsection {* Inverse image *}
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
by (unfold sym_def inv_image_def) blast
@@ -643,7 +738,7 @@
unfolding inv_image_def converse_def by auto
-subsection {* Finiteness *}
+subsubsection {* Finiteness *}
lemma finite_converse [iff]: "finite (r^-1) = finite r"
apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
@@ -671,7 +766,7 @@
done
-subsection {* Miscellaneous *}
+subsubsection {* Miscellaneous *}
text {* Version of @{thm[source] lfp_induct} for binary relations *}
@@ -683,4 +778,171 @@
lemma subrelI: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
by auto
+
+subsection {* Relations as binary predicates *}
+
+subsubsection {* Composition *}
+
+inductive pred_comp :: "['a \<Rightarrow> 'b \<Rightarrow> bool, '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"
+
+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)
+
+
+subsubsection {* Converse *}
+
+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 (xsymbols)
+ conversep ("(_\<inverse>\<inverse>)" [1000] 1000)
+
+lemma conversepD:
+ assumes ab: "r^--1 a b"
+ shows "r b a" using ab
+ by cases simp
+
+lemma conversep_iff [iff]: "r^--1 a b = r b a"
+ by (iprover intro: conversepI dest: conversepD)
+
+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 conversep_conversep [simp]: "(r^--1)^--1 = r"
+ by (iprover intro: order_antisym conversepI dest: conversepD)
+
+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_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
+ by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
+
+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)
+
+
+subsubsection {* Domain *}
+
+inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"
+
+inductive_cases DomainPE [elim!]: "DomainP r a"
+
+lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
+ by (blast intro!: Orderings.order_antisym predicate1I)
+
+
+subsubsection {* Range *}
+
+inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
+ for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
+ RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"
+
+inductive_cases RangePE [elim!]: "RangeP r b"
+
+lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
+ by (blast intro!: Orderings.order_antisym predicate1I)
+
+
+subsubsection {* Inverse image *}
+
+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 in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
+ by (simp add: inv_imagep_def)
+
+
+subsubsection {* Powerset *}
+
+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)"
+ by (auto simp add: Powp_def fun_eq_iff)
+
+lemmas Powp_mono [mono] = Pow_mono [to_pred]
+
+
+subsubsection {* Properties of predicate relations *}
+
+abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "antisymP r \<equiv> antisym {(x, y). r x y}"
+
+abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "transP r \<equiv> trans {(x, y). r x y}"
+
+abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
+ "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
+
+(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
+
+definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
+
+definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "symp r \<longleftrightarrow> sym {(x, y). r x y}"
+
+definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
+ "transp r \<longleftrightarrow> trans {(x, y). r x y}"
+
+lemma reflpI:
+ "(\<And>x. r x x) \<Longrightarrow> reflp r"
+ by (auto intro: refl_onI simp add: reflp_def)
+
+lemma reflpE:
+ assumes "reflp r"
+ obtains "r x x"
+ using assms by (auto dest: refl_onD simp add: reflp_def)
+
+lemma sympI:
+ "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
+ by (auto intro: symI simp add: symp_def)
+
+lemma sympE:
+ assumes "symp r" and "r x y"
+ obtains "r y x"
+ using assms by (auto dest: symD simp add: symp_def)
+
+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)
+
+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)
+
+no_notation
+ bot ("\<bottom>") and
+ top ("\<top>") and
+ inf (infixl "\<sqinter>" 70) and
+ sup (infixl "\<squnion>" 65) and
+ Inf ("\<Sqinter>_" [900] 900) and
+ Sup ("\<Squnion>_" [900] 900)
+
+no_syntax (xsymbols)
+ "_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10)
+ "_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
+ "_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10)
+ "_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
+
end