src/HOL/Relation.thy
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(*  Title:      HOL/Relation.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen
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*)
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header {* Relations – as sets of pairs, and binary predicates *}
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theory Relation
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imports Datatype Finite_Set
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begin
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text {* A preliminary: classical rules for reasoning on predicates *}
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declare predicate1I [Pure.intro!, intro!]
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declare predicate1D [Pure.dest, dest]
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declare predicate2I [Pure.intro!, intro!]
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declare predicate2D [Pure.dest, dest]
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declare bot1E [elim!] 
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declare bot2E [elim!]
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declare top1I [intro!]
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declare top2I [intro!]
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declare inf1I [intro!]
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declare inf2I [intro!]
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declare inf1E [elim!]
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declare inf2E [elim!]
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declare sup1I1 [intro?]
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declare sup2I1 [intro?]
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declare sup1I2 [intro?]
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declare sup2I2 [intro?]
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declare sup1E [elim!]
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declare sup2E [elim!]
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declare sup1CI [intro!]
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declare sup2CI [intro!]
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declare INF1_I [intro!]
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declare INF2_I [intro!]
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declare INF1_D [elim]
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declare INF2_D [elim]
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declare INF1_E [elim]
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declare INF2_E [elim]
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declare SUP1_I [intro]
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declare SUP2_I [intro]
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declare SUP1_E [elim!]
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declare SUP2_E [elim!]
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subsection {* Fundamental *}
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subsubsection {* Relations as sets of pairs *}
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type_synonym 'a rel = "('a * 'a) set"
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lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}
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  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
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  by auto
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lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}
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  "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
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    (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
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  using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto
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subsubsection {* Conversions between set and predicate relations *}
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lemma pred_equals_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S) \<longleftrightarrow> R = S"
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  by (simp add: set_eq_iff fun_eq_iff)
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lemma pred_equals_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R = S"
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  by (simp add: set_eq_iff fun_eq_iff)
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lemma pred_subset_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S) \<longleftrightarrow> R \<subseteq> S"
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  by (simp add: subset_iff le_fun_def)
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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"
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  by (simp add: subset_iff le_fun_def)
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lemma bot_empty_eq [pred_set_conv]: "\<bottom> = (\<lambda>x. x \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma bot_empty_eq2 [pred_set_conv]: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma top_empty_eq [pred_set_conv]: "\<top> = (\<lambda>x. x \<in> UNIV)"
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  by (auto simp add: fun_eq_iff)
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lemma top_empty_eq2 [pred_set_conv]: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"
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  by (auto simp add: fun_eq_iff)
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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)"
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  by (simp add: inf_fun_def)
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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)"
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  by (simp add: inf_fun_def)
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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)"
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  by (simp add: sup_fun_def)
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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)"
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  by (simp add: sup_fun_def)
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lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma Inf_INT_eq [pred_set_conv]: "\<Sqinter>S = (\<lambda>x. x \<in> INTER S Collect)"
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  by (simp add: fun_eq_iff)
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lemma INF_Int_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Inter>S)"
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  by (simp add: fun_eq_iff)
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lemma Inf_INT_eq2 [pred_set_conv]: "\<Sqinter>S = (\<lambda>x y. (x, y) \<in> INTER (prod_case ` S) Collect)"
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  by (simp add: fun_eq_iff)
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lemma INF_Int_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Inter>S)"
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  by (simp add: fun_eq_iff)
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lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> UNION S Collect)"
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  by (simp add: fun_eq_iff)
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lemma SUP_Sup_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union>S)"
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  by (simp add: fun_eq_iff)
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lemma Sup_SUP_eq2 [pred_set_conv]: "\<Squnion>S = (\<lambda>x y. (x, y) \<in> UNION (prod_case ` S) Collect)"
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  by (simp add: fun_eq_iff)
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lemma SUP_Sup_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Union>S)"
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  by (simp add: fun_eq_iff)
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subsection {* Properties of relations *}
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subsubsection {* Reflexivity *}
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definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
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where
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  "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"
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abbreviation refl :: "'a rel \<Rightarrow> bool"
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where -- {* reflexivity over a type *}
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  "refl \<equiv> refl_on UNIV"
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definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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where
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  "reflp r \<longleftrightarrow> (\<forall>x. r x x)"
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lemma reflp_refl_eq [pred_set_conv]:
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  "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r" 
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  by (simp add: refl_on_def reflp_def)
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lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
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  by (unfold refl_on_def) (iprover intro!: ballI)
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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
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  by (unfold refl_on_def) blast
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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
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  by (unfold refl_on_def) blast
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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
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  by (unfold refl_on_def) blast
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lemma reflpI:
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  "(\<And>x. r x x) \<Longrightarrow> reflp r"
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  by (auto intro: refl_onI simp add: reflp_def)
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lemma reflpE:
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  assumes "reflp r"
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  obtains "r x x"
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  using assms by (auto dest: refl_onD simp add: reflp_def)
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lemma reflpD:
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  assumes "reflp r"
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  shows "r x x"
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  using assms by (auto elim: reflpE)
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lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
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  by (unfold refl_on_def) blast
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lemma reflp_inf:
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  "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"
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  by (auto intro: reflpI elim: reflpE)
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lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
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  by (unfold refl_on_def) blast
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lemma reflp_sup:
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  "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"
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  by (auto intro: reflpI elim: reflpE)
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lemma refl_on_INTER:
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  "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
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  by (unfold refl_on_def) fast
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lemma refl_on_UNION:
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  "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
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  by (unfold refl_on_def) blast
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lemma refl_on_empty [simp]: "refl_on {} {}"
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  by (simp add:refl_on_def)
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lemma refl_on_def' [nitpick_unfold, code]:
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  "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)"
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  by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
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subsubsection {* Irreflexivity *}
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definition irrefl :: "'a rel \<Rightarrow> bool"
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where
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  "irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)"
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lemma irrefl_distinct [code]:
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  "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
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  by (auto simp add: irrefl_def)
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subsubsection {* Symmetry *}
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definition sym :: "'a rel \<Rightarrow> bool"
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where
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  "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
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definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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where
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  "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
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lemma symp_sym_eq [pred_set_conv]:
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  "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r" 
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  by (simp add: sym_def symp_def)
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lemma symI:
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  "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"
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  by (unfold sym_def) iprover
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lemma sympI:
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  "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
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  by (fact symI [to_pred])
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lemma symE:
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  assumes "sym r" and "(b, a) \<in> r"
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  obtains "(a, b) \<in> r"
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  using assms by (simp add: sym_def)
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lemma sympE:
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  assumes "symp r" and "r b a"
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  obtains "r a b"
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  using assms by (rule symE [to_pred])
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lemma symD:
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  assumes "sym r" and "(b, a) \<in> r"
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  shows "(a, b) \<in> r"
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  using assms by (rule symE)
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lemma sympD:
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  assumes "symp r" and "r b a"
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  shows "r a b"
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  using assms by (rule symD [to_pred])
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lemma sym_Int:
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  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
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  by (fast intro: symI elim: symE)
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lemma symp_inf:
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  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
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  by (fact sym_Int [to_pred])
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lemma sym_Un:
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  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
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  by (fast intro: symI elim: symE)
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lemma symp_sup:
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  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
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  by (fact sym_Un [to_pred])
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lemma sym_INTER:
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  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
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  by (fast intro: symI elim: symE)
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lemma symp_INF:
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  "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFI S r)"
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  by (fact sym_INTER [to_pred])
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lemma sym_UNION:
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  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
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  by (fast intro: symI elim: symE)
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lemma symp_SUP:
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  "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPR S r)"
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  by (fact sym_UNION [to_pred])
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subsubsection {* Antisymmetry *}
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definition antisym :: "'a rel \<Rightarrow> bool"
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where
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  "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
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abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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where
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  "antisymP r \<equiv> antisym {(x, y). r x y}"
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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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subsubsection {* Transitivity *}
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definition trans :: "'a rel \<Rightarrow> bool"
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where
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  "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
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definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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where
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  "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
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lemma transp_trans_eq [pred_set_conv]:
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  "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
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  by (simp add: trans_def transp_def)
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abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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where -- {* FIXME drop *}
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  "transP r \<equiv> trans {(x, y). r x y}"
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lemma transI:
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  "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
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  by (unfold trans_def) iprover
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lemma transpI:
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  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
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  by (fact transI [to_pred])
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lemma transE:
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  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
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  obtains "(x, z) \<in> r"
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  using assms by (unfold trans_def) iprover
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lemma transpE:
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  assumes "transp r" and "r x y" and "r y z"
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  obtains "r x z"
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  using assms by (rule transE [to_pred])
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lemma transD:
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  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
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  shows "(x, z) \<in> r"
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  using assms by (rule transE)
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lemma transpD:
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  assumes "transp r" and "r x y" and "r y z"
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  shows "r x z"
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  using assms by (rule transD [to_pred])
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lemma trans_Int:
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  "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
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  by (fast intro: transI elim: transE)
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lemma transp_inf:
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  "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
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   373
  by (fact trans_Int [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   374
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   375
lemma trans_INTER:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   376
  "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   377
  by (fast intro: transI elim: transD)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   378
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   379
(* FIXME thm trans_INTER [to_pred] *)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   380
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   381
lemma trans_join [code]:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   382
  "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   383
  by (auto simp add: trans_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   384
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   385
lemma transp_trans:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   386
  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   387
  by (simp add: trans_def transp_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   388
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   389
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   390
subsubsection {* Totality *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   391
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   392
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   393
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   394
  "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)"
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   395
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   396
abbreviation "total \<equiv> total_on UNIV"
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   397
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   398
lemma total_on_empty [simp]: "total_on {} r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   399
  by (simp add: total_on_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   400
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   401
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   402
subsubsection {* Single valued relations *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   403
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   404
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   405
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   406
  "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   407
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   408
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   409
  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   410
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   411
lemma single_valuedI:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   412
  "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   413
  by (unfold single_valued_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   414
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   415
lemma single_valuedD:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   416
  "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   417
  by (simp add: single_valued_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   418
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   419
lemma single_valued_subset:
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   420
  "r \<subseteq> s ==> single_valued s ==> single_valued r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   421
  by (unfold single_valued_def) blast
11136
e34e7f6d9b57 moved inv_image to Relation
oheimb
parents: 10832
diff changeset
   422
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   423
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   424
subsection {* Relation operations *}
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   425
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   426
subsubsection {* The identity relation *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   427
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   428
definition Id :: "'a rel"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   429
where
48253
4410a709913c a first guess to avoid the Codegenerator_Test to loop infinitely
bulwahn
parents: 47937
diff changeset
   430
  [code del]: "Id = {p. \<exists>x. p = (x, x)}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   431
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   432
lemma IdI [intro]: "(a, a) : Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   433
  by (simp add: Id_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   434
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   435
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   436
  by (unfold Id_def) (iprover elim: CollectE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   437
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   438
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   439
  by (unfold Id_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   440
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   441
lemma refl_Id: "refl Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   442
  by (simp add: refl_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   443
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   444
lemma antisym_Id: "antisym Id"
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   445
  -- {* A strange result, since @{text Id} is also symmetric. *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   446
  by (simp add: antisym_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   447
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   448
lemma sym_Id: "sym Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   449
  by (simp add: sym_def)
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   450
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   451
lemma trans_Id: "trans Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   452
  by (simp add: trans_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   453
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   454
lemma single_valued_Id [simp]: "single_valued Id"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   455
  by (unfold single_valued_def) blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   456
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   457
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   458
  by (simp add:irrefl_def)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   459
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   460
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   461
  unfolding antisym_def trans_def by blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   462
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   463
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   464
  by (simp add: total_on_def)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   465
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   466
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   467
subsubsection {* Diagonal: identity over a set *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   468
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   469
definition Id_on  :: "'a set \<Rightarrow> 'a rel"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   470
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   471
  "Id_on A = (\<Union>x\<in>A. {(x, x)})"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   472
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   473
lemma Id_on_empty [simp]: "Id_on {} = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   474
  by (simp add: Id_on_def) 
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   475
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   476
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   477
  by (simp add: Id_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   478
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 33218
diff changeset
   479
lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   480
  by (rule Id_on_eqI) (rule refl)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   481
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   482
lemma Id_onE [elim!]:
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   483
  "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   484
  -- {* The general elimination rule. *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   485
  by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   486
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   487
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   488
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   489
45967
76cf71ed15c7 dropped obsolete code equation for Id
haftmann
parents: 45139
diff changeset
   490
lemma Id_on_def' [nitpick_unfold]:
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 41792
diff changeset
   491
  "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   492
  by auto
40923
be80c93ac0a2 adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents: 36772
diff changeset
   493
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   494
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   495
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   496
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   497
lemma refl_on_Id_on: "refl_on A (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   498
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   499
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   500
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   501
  by (unfold antisym_def) blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   502
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   503
lemma sym_Id_on [simp]: "sym (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   504
  by (rule symI) clarify
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   505
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   506
lemma trans_Id_on [simp]: "trans (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   507
  by (fast intro: transI elim: transD)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   508
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   509
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   510
  by (unfold single_valued_def) blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   511
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   512
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   513
subsubsection {* Composition *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   514
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   515
inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   516
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   517
where
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   518
  relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   519
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   520
notation relcompp (infixr "OO" 75)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   521
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   522
lemmas relcomppI = relcompp.intros
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   523
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   524
text {*
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   525
  For historic reasons, the elimination rules are not wholly corresponding.
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   526
  Feel free to consolidate this.
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   527
*}
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   528
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   529
inductive_cases relcompEpair: "(a, c) \<in> r O s"
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   530
inductive_cases relcomppE [elim!]: "(r OO s) a c"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   531
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   532
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   533
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   534
  by (cases xz) (simp, erule relcompEpair, iprover)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   535
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   536
lemma R_O_Id [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   537
  "R O Id = R"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   538
  by fast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   539
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   540
lemma Id_O_R [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   541
  "Id O R = R"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   542
  by fast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   543
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   544
lemma relcomp_empty1 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   545
  "{} O R = {}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   546
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   547
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   548
lemma relcompp_bot1 [simp]:
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   549
  "\<bottom> OO R = \<bottom>"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   550
  by (fact relcomp_empty1 [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   551
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   552
lemma relcomp_empty2 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   553
  "R O {} = {}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   554
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   555
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   556
lemma relcompp_bot2 [simp]:
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   557
  "R OO \<bottom> = \<bottom>"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   558
  by (fact relcomp_empty2 [to_pred])
23185
1fa87978cf27 Added simp-rules: "R O {} = {}" and "{} O R = {}"
krauss
parents: 22172
diff changeset
   559
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   560
lemma O_assoc:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   561
  "(R O S) O T = R O (S O T)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   562
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   563
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   564
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   565
lemma relcompp_assoc:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   566
  "(r OO s) OO t = r OO (s OO t)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   567
  by (fact O_assoc [to_pred])
23185
1fa87978cf27 Added simp-rules: "R O {} = {}" and "{} O R = {}"
krauss
parents: 22172
diff changeset
   568
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   569
lemma trans_O_subset:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   570
  "trans r \<Longrightarrow> r O r \<subseteq> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   571
  by (unfold trans_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   572
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   573
lemma transp_relcompp_less_eq:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   574
  "transp r \<Longrightarrow> r OO r \<le> r "
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   575
  by (fact trans_O_subset [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   576
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   577
lemma relcomp_mono:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   578
  "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   579
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   580
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   581
lemma relcompp_mono:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   582
  "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   583
  by (fact relcomp_mono [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   584
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   585
lemma relcomp_subset_Sigma:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   586
  "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   587
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   588
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   589
lemma relcomp_distrib [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   590
  "R O (S \<union> T) = (R O S) \<union> (R O T)" 
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   591
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   592
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   593
lemma relcompp_distrib [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   594
  "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   595
  by (fact relcomp_distrib [to_pred])
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   596
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   597
lemma relcomp_distrib2 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   598
  "(S \<union> T) O R = (S O R) \<union> (T O R)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   599
  by auto
28008
f945f8d9ad4d added distributivity of relation composition over union [simp]
krauss
parents: 26297
diff changeset
   600
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   601
lemma relcompp_distrib2 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   602
  "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   603
  by (fact relcomp_distrib2 [to_pred])
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   604
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   605
lemma relcomp_UNION_distrib:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   606
  "s O UNION I r = (\<Union>i\<in>I. s O r i) "
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   607
  by auto
28008
f945f8d9ad4d added distributivity of relation composition over union [simp]
krauss
parents: 26297
diff changeset
   608
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   609
(* FIXME thm relcomp_UNION_distrib [to_pred] *)
36772
ef97c5006840 added lemmas rel_comp_UNION_distrib(2)
krauss
parents: 36729
diff changeset
   610
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   611
lemma relcomp_UNION_distrib2:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   612
  "UNION I r O s = (\<Union>i\<in>I. r i O s) "
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   613
  by auto
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   614
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   615
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *)
36772
ef97c5006840 added lemmas rel_comp_UNION_distrib(2)
krauss
parents: 36729
diff changeset
   616
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   617
lemma single_valued_relcomp:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   618
  "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   619
  by (unfold single_valued_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   620
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   621
lemma relcomp_unfold:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   622
  "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   623
  by (auto simp add: set_eq_iff)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   624
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   625
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   626
subsubsection {* Converse *}
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   627
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   628
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   629
  for r :: "('a \<times> 'b) set"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   630
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   631
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   632
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   633
notation (xsymbols)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   634
  converse  ("(_\<inverse>)" [1000] 999)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   635
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   636
notation
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   637
  conversep ("(_^--1)" [1000] 1000)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   638
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   639
notation (xsymbols)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   640
  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   641
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   642
lemma converseI [sym]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   643
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   644
  by (fact converse.intros)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   645
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   646
lemma conversepI (* CANDIDATE [sym] *):
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   647
  "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   648
  by (fact conversep.intros)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   649
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   650
lemma converseD [sym]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   651
  "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   652
  by (erule converse.cases) iprover
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   653
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   654
lemma conversepD (* CANDIDATE [sym] *):
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   655
  "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   656
  by (fact converseD [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   657
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   658
lemma converseE [elim!]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   659
  -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   660
  "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   661
  by (cases yx) (simp, erule converse.cases, iprover)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   662
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   663
lemmas conversepE [elim!] = conversep.cases
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   664
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   665
lemma converse_iff [iff]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   666
  "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   667
  by (auto intro: converseI)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   668
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   669
lemma conversep_iff [iff]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   670
  "r\<inverse>\<inverse> a b = r b a"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   671
  by (fact converse_iff [to_pred])
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   672
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   673
lemma converse_converse [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   674
  "(r\<inverse>)\<inverse> = r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   675
  by (simp add: set_eq_iff)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   676
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   677
lemma conversep_conversep [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   678
  "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   679
  by (fact converse_converse [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   680
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   681
lemma converse_relcomp: "(r O s)^-1 = s^-1 O r^-1"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   682
  by blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   683
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   684
lemma converse_relcompp: "(r OO s)^--1 = s^--1 OO r^--1"
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   685
  by (iprover intro: order_antisym conversepI relcomppI
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   686
    elim: relcomppE dest: conversepD)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   687
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   688
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   689
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   690
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   691
lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   692
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   693
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   694
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   695
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   696
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   697
lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   698
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   699
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   700
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   701
  by fast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   702
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   703
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   704
  by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   705
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   706
lemma converse_Id [simp]: "Id^-1 = Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   707
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   708
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   709
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   710
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   711
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   712
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   713
  by (unfold refl_on_def) auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   714
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   715
lemma sym_converse [simp]: "sym (converse r) = sym r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   716
  by (unfold sym_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   717
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   718
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   719
  by (unfold antisym_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   720
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   721
lemma trans_converse [simp]: "trans (converse r) = trans r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   722
  by (unfold trans_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   723
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   724
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   725
  by (unfold sym_def) fast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   726
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   727
lemma sym_Un_converse: "sym (r \<union> r^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   728
  by (unfold sym_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   729
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   730
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   731
  by (unfold sym_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   732
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   733
lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   734
  by (auto simp: total_on_def)
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   735
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   736
lemma finite_converse [iff]: "finite (r^-1) = finite r"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   737
  apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   738
   apply simp
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   739
   apply (rule iffI)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   740
    apply (erule finite_imageD [unfolded inj_on_def])
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   741
    apply (simp split add: split_split)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   742
   apply (erule finite_imageI)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   743
  apply (simp add: set_eq_iff image_def, auto)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   744
  apply (rule bexI)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   745
   prefer 2 apply assumption
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   746
  apply simp
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   747
  done
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   748
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   749
lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   750
  by (auto simp add: fun_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   751
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   752
lemma conversep_eq [simp]: "(op =)^--1 = op ="
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   753
  by (auto simp add: fun_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   754
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   755
lemma converse_unfold:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   756
  "r\<inverse> = {(y, x). (x, y) \<in> r}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   757
  by (simp add: set_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   758
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   759
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   760
subsubsection {* Domain, range and field *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   761
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   762
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   763
  for r :: "('a \<times> 'b) set"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   764
where
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   765
  DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   766
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   767
abbreviation (input) "DomainP \<equiv> Domainp"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   768
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   769
lemmas DomainPI = Domainp.DomainI
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   770
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   771
inductive_cases DomainE [elim!]: "a \<in> Domain r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   772
inductive_cases DomainpE [elim!]: "Domainp r a"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   773
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   774
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   775
  for r :: "('a \<times> 'b) set"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   776
where
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   777
  RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   778
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   779
abbreviation (input) "RangeP \<equiv> Rangep"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   780
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   781
lemmas RangePI = Rangep.RangeI
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   782
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   783
inductive_cases RangeE [elim!]: "b \<in> Range r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   784
inductive_cases RangepE [elim!]: "Rangep r b"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   785
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   786
definition Field :: "'a rel \<Rightarrow> 'a set"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   787
where
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   788
  "Field r = Domain r \<union> Range r"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   789
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   790
lemma Domain_fst [code]:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   791
  "Domain r = fst ` r"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   792
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   793
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   794
lemma Range_snd [code]:
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   795
  "Range r = snd ` r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   796
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   797
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   798
lemma fst_eq_Domain: "fst ` R = Domain R"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   799
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   800
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   801
lemma snd_eq_Range: "snd ` R = Range R"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   802
  by force
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   803
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   804
lemma Domain_empty [simp]: "Domain {} = {}"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   805
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   806
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   807
lemma Range_empty [simp]: "Range {} = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   808
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   809
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   810
lemma Field_empty [simp]: "Field {} = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   811
  by (simp add: Field_def)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   812
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   813
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   814
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   815
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   816
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   817
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   818
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   819
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   820
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   821
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   822
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   823
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   824
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   825
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   826
  by (auto simp add: Field_def)
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   827
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   828
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   829
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   830
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   831
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   832
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   833
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   834
lemma Domain_Id [simp]: "Domain Id = UNIV"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   835
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   836
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   837
lemma Range_Id [simp]: "Range Id = UNIV"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   838
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   839
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   840
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   841
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   842
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   843
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   844
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   845
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   846
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   847
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   848
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   849
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   850
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   851
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   852
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   853
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   854
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   855
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   856
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   857
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   858
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   859
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   860
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   861
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   862
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   863
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   864
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   865
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   866
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   867
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   868
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   869
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   870
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   871
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   872
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   873
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   874
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   875
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   876
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   877
  by auto
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   878
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   879
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   880
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   881
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   882
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   883
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   884
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   885
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   886
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   887
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   888
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   889
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   890
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   891
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   892
  by (induct set: finite) auto
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   893
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   894
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   895
  by (induct set: finite) auto
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   896
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   897
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   898
  by (simp add: Field_def finite_Domain finite_Range)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   899
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   900
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   901
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   902
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   903
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   904
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   905
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   906
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   907
  by (auto simp: Field_def Domain_def Range_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   908
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   909
lemma Domain_unfold:
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   910
  "Domain r = {x. \<exists>y. (x, y) \<in> r}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   911
  by blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   912
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   913
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   914
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   915
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   916
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   917
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   918
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   919
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   920
subsubsection {* Image of a set under a relation *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   921
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   922
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixl "``" 90)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   923
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   924
  "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   925
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 33218
diff changeset
   926
declare Image_def [no_atp]
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23709
diff changeset
   927
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   928
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   929
  by (simp add: Image_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   930
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   931
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   932
  by (simp add: Image_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   933
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   934
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   935
  by (rule Image_iff [THEN trans]) simp
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   936
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 33218
diff changeset
   937
lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   938
  by (unfold Image_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   939
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   940
lemma ImageE [elim!]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   941
  "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   942
  by (unfold Image_def) (iprover elim!: CollectE bexE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   943
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   944
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   945
  -- {* This version's more effective when we already have the required @{text a} *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   946
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   947
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   948
lemma Image_empty [simp]: "R``{} = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   949
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   950
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   951
lemma Image_Id [simp]: "Id `` A = A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   952
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   953
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   954
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   955
  by blast
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   956
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   957
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   958
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   959
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   960
lemma Image_Int_eq:
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   961
  "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   962
  by (simp add: single_valued_def, blast) 
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   963
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   964
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   965
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   966
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   967
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   968
  by blast
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   969
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   970
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   971
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   972
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   973
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   974
  -- {* NOT suitable for rewriting *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   975
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   976
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   977
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   978
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   979
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   980
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   981
  by blast
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   982
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   983
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   984
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   985
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   986
text{*Converse inclusion requires some assumptions*}
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   987
lemma Image_INT_eq:
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   988
     "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   989
apply (rule equalityI)
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   990
 apply (rule Image_INT_subset) 
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   991
apply  (simp add: single_valued_def, blast)
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   992
done
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   993
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   994
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   995
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   996
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   997
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   998
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   999
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1000
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1001
subsubsection {* Inverse image *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1002
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1003
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1004
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1005
  "inv_image r f = {(x, y). (f x, f y) \<in> r}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
  1006
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1007
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1008
where
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1009
  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1010
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1011
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1012
  by (simp add: inv_image_def inv_imagep_def)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1013
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
  1014
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1015
  by (unfold sym_def inv_image_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
  1016
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
  1017
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1018
  apply (unfold trans_def inv_image_def)
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1019
  apply (simp (no_asm))
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1020
  apply blast
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1021
  done
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1022
32463
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1023
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1024
  by (auto simp:inv_image_def)
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1025
33218
ecb5cd453ef2 lemma converse_inv_image
krauss
parents: 32876
diff changeset
  1026
lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1027
  unfolding inv_image_def converse_unfold by auto
33218
ecb5cd453ef2 lemma converse_inv_image
krauss
parents: 32876
diff changeset
  1028
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1029
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1030
  by (simp add: inv_imagep_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1031
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1032
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1033
subsubsection {* Powerset *}
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1034
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1035
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1036
where
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1037
  "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1038
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1039
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1040
  by (auto simp add: Powp_def fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1041
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1042
lemmas Powp_mono [mono] = Pow_mono [to_pred]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1043
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1044
subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *}
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1045
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1046
lemma Id_on_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1047
  assumes "finite A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1048
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1049
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1050
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)" by default auto
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1051
  show ?thesis using assms unfolding Id_on_def by (induct A) simp_all
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1052
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1053
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1054
lemma comp_fun_commute_Image_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1055
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1056
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1057
  interpret comp_fun_idem Set.insert
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1058
      by (fact comp_fun_idem_insert)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1059
  show ?thesis 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1060
  by default (auto simp add: fun_eq_iff comp_fun_commute split:prod.split)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1061
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1062
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1063
lemma Image_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1064
  assumes "finite R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1065
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1066
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1067
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1068
    by (rule comp_fun_commute_Image_fold)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1069
  have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1070
    by (auto intro: rev_ImageI)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1071
  show ?thesis using assms by (induct R) (auto simp: *)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1072
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1073
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1074
lemma insert_relcomp_union_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1075
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1076
  shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1077
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1078
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1079
  proof - 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1080
    interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1081
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1082
    by default (auto simp add: fun_eq_iff split:prod.split)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1083
  qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1084
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x,z) \<in> S}" by (auto simp: relcomp_unfold intro!: exI)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1085
  show ?thesis unfolding *
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1086
  using `finite S` by (induct S) (auto split: prod.split)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1087
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1088
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1089
lemma insert_relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1090
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1091
  shows "Set.insert x R O S = 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1092
    Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1093
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1094
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)" by auto
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1095
  then show ?thesis by (auto simp: insert_relcomp_union_fold[OF assms])
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1096
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1097
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1098
lemma comp_fun_commute_relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1099
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1100
  shows "comp_fun_commute (\<lambda>(x,y) A. 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1101
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1102
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1103
  have *: "\<And>a b A. 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1104
    Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1105
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1106
  show ?thesis by default (auto simp: *)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1107
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1108
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1109
lemma relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1110
  assumes "finite R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1111
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1112
  shows "R O S = Finite_Set.fold 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1113
    (\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1114
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1115
  show ?thesis using assms by (induct R) 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1116
    (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1117
      cong: if_cong)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1118
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1119
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1120
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1121
1128
64b30e3cc6d4 Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff changeset
  1122
end
46689
f559866a7aa2 marked candidates for rule declarations
haftmann
parents: 46664
diff changeset
  1123