author  blanchet 
Mon, 20 Jan 2014 20:21:12 +0100  
changeset 55083  0a689157e3ce 
parent 54611  31afce809794 
child 55096  916b2ac758f4 
permissions  rwrr 
10358  1 
(* 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 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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46694  44 
subsection {* Fundamental *} 
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46694  46 
subsubsection {* Relations as sets of pairs *} 
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type_synonym 'a rel = "('a * 'a) set" 

49 

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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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59 

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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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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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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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156 

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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r" 
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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A" 
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162 

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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A" 
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46694  166 
lemma reflpI: 
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"(\<And>x. r x x) \<Longrightarrow> reflp r" 

168 
by (auto intro: refl_onI simp add: reflp_def) 

169 

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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) 

174 

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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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182 

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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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189 

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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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193 

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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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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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46694  216 
lemma irrefl_distinct [code]: 
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"irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)" 

218 
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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226 

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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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230 

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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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234 

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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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239 
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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242 

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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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248 
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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252 

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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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262 

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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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266 

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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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46982  283 
lemma symp_INF: 
284 
"\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFI S r)" 

285 
by (fact sym_INTER [to_pred]) 

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286 

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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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46982  291 
lemma symp_SUP: 
292 
"\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPR S r)" 

293 
by (fact sym_UNION [to_pred]) 

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294 

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46694  296 
subsubsection {* Antisymmetry *} 
297 

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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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301 

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abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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303 
where 
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"antisymP r \<equiv> antisym {(x, y). r x y}" 
46694  305 

306 
lemma antisymI: 

307 
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" 

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by (unfold antisym_def) iprover 
46694  309 

310 
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" 

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311 
by (unfold antisym_def) iprover 
46694  312 

313 
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r" 

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314 
by (unfold antisym_def) blast 
46694  315 

316 
lemma antisym_empty [simp]: "antisym {}" 

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317 
by (unfold antisym_def) blast 
46694  318 

319 

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320 
subsubsection {* Transitivity *} 
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321 

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definition trans :: "'a rel \<Rightarrow> bool" 
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323 
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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325 

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326 
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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327 
where 
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328 
"transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)" 
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329 

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330 
lemma transp_trans_eq [pred_set_conv]: 
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331 
"transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
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332 
by (simp add: trans_def transp_def) 
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333 

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334 
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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335 
where  {* FIXME drop *} 
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336 
"transP r \<equiv> trans {(x, y). r x y}" 
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337 

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338 
lemma transI: 
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339 
"(\<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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340 
by (unfold trans_def) iprover 
46694  341 

342 
lemma transpI: 

343 
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r" 

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344 
by (fact transI [to_pred]) 
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345 

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346 
lemma transE: 
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347 
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" 
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348 
obtains "(x, z) \<in> r" 
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349 
using assms by (unfold trans_def) iprover 
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350 

46694  351 
lemma transpE: 
352 
assumes "transp r" and "r x y" and "r y z" 

353 
obtains "r x z" 

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354 
using assms by (rule transE [to_pred]) 
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355 

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356 
lemma transD: 
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357 
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" 
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358 
shows "(x, z) \<in> r" 
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359 
using assms by (rule transE) 
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360 

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361 
lemma transpD: 
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362 
assumes "transp r" and "r x y" and "r y z" 
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363 
shows "r x z" 
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364 
using assms by (rule transD [to_pred]) 
46694  365 

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366 
lemma trans_Int: 
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367 
"trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)" 
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368 
by (fast intro: transI elim: transE) 
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369 

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370 
lemma transp_inf: 
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371 
"transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)" 
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372 
by (fact trans_Int [to_pred]) 
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373 

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374 
lemma trans_INTER: 
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375 
"\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)" 
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376 
by (fast intro: transI elim: transD) 
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377 

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378 
(* FIXME thm trans_INTER [to_pred] *) 
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379 

46694  380 
lemma trans_join [code]: 
381 
"trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)" 

382 
by (auto simp add: trans_def) 

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383 

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384 
lemma transp_trans: 
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385 
"transp r \<longleftrightarrow> trans {(x, y). r x y}" 
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386 
by (simp add: trans_def transp_def) 
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387 

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388 

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389 
subsubsection {* Totality *} 
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390 

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391 
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" 
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392 
where 
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393 
"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)" 
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394 

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395 
abbreviation "total \<equiv> total_on UNIV" 
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396 

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397 
lemma total_on_empty [simp]: "total_on {} r" 
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398 
by (simp add: total_on_def) 
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399 

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400 

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401 
subsubsection {* Single valued relations *} 
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402 

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403 
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool" 
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404 
where 
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405 
"single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))" 
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406 

46694  407 
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
408 
"single_valuedP r \<equiv> single_valued {(x, y). r x y}" 

409 

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410 
lemma single_valuedI: 
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411 
"ALL x y. (x,y):r > (ALL z. (x,z):r > y=z) ==> single_valued r" 
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412 
by (unfold single_valued_def) 
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413 

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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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414 
lemma single_valuedD: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

415 
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" 
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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changeset

416 
by (simp add: single_valued_def) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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417 

52392  418 
lemma simgle_valued_empty[simp]: "single_valued {}" 
419 
by(simp add: single_valued_def) 

420 

46692
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tuned structure; dropped already existing syntax declarations
haftmann
parents:
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changeset

421 
lemma single_valued_subset: 
1f8b766224f6
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haftmann
parents:
46691
diff
changeset

422 
"r \<subseteq> s ==> single_valued s ==> single_valued r" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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423 
by (unfold single_valued_def) blast 
11136  424 

12905  425 

46694  426 
subsection {* Relation operations *} 
427 

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:
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diff
changeset

428 
subsubsection {* The identity relation *} 
12905  429 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

430 
definition Id :: "'a rel" 
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haftmann
parents:
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diff
changeset

431 
where 
48253
4410a709913c
a first guess to avoid the Codegenerator_Test to loop infinitely
bulwahn
parents:
47937
diff
changeset

432 
[code del]: "Id = {p. \<exists>x. p = (x, x)}" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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433 

12905  434 
lemma IdI [intro]: "(a, a) : Id" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
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435 
by (simp add: Id_def) 
12905  436 

437 
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

438 
by (unfold Id_def) (iprover elim: CollectE) 
12905  439 

440 
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" 

46752
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parents:
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changeset

441 
by (unfold Id_def) blast 
12905  442 

30198  443 
lemma refl_Id: "refl Id" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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changeset

444 
by (simp add: refl_on_def) 
12905  445 

446 
lemma antisym_Id: "antisym Id" 

447 
 {* A strange result, since @{text Id} is also symmetric. *} 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

448 
by (simp add: antisym_def) 
12905  449 

19228  450 
lemma sym_Id: "sym Id" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

451 
by (simp add: sym_def) 
19228  452 

12905  453 
lemma trans_Id: "trans Id" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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changeset

454 
by (simp add: trans_def) 
12905  455 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

456 
lemma single_valued_Id [simp]: "single_valued Id" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

457 
by (unfold single_valued_def) blast 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

458 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

459 
lemma irrefl_diff_Id [simp]: "irrefl (r  Id)" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

460 
by (simp add:irrefl_def) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

461 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

462 
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r  Id)" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

463 
unfolding antisym_def trans_def by blast 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

464 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

465 
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:
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diff
changeset

466 
by (simp add: total_on_def) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

467 

12905  468 

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

469 
subsubsection {* Diagonal: identity over a set *} 
12905  470 

46752
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parents:
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471 
definition Id_on :: "'a set \<Rightarrow> 'a rel" 
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parents:
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changeset

472 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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473 
"Id_on A = (\<Union>x\<in>A. {(x, x)})" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

474 

30198  475 
lemma Id_on_empty [simp]: "Id_on {} = {}" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

476 
by (simp add: Id_on_def) 
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

477 

30198  478 
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A" 
46752
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haftmann
parents:
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diff
changeset

479 
by (simp add: Id_on_def) 
12905  480 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53680
diff
changeset

481 
lemma Id_onI [intro!]: "a : A ==> (a, a) : Id_on A" 
46752
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haftmann
parents:
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diff
changeset

482 
by (rule Id_on_eqI) (rule refl) 
12905  483 

30198  484 
lemma Id_onE [elim!]: 
485 
"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" 

12913  486 
 {* The general elimination rule. *} 
46752
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haftmann
parents:
46696
diff
changeset

487 
by (unfold Id_on_def) (iprover elim!: UN_E singletonE) 
12905  488 

30198  489 
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)" 
46752
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haftmann
parents:
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diff
changeset

490 
by blast 
12905  491 

45967  492 
lemma Id_on_def' [nitpick_unfold]: 
44278
1220ecb81e8f
observe distinction between sets and predicates more properly
haftmann
parents:
41792
diff
changeset

493 
"Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

494 
by auto 
40923
be80c93ac0a2
adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents:
36772
diff
changeset

495 

30198  496 
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A" 
46752
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haftmann
parents:
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diff
changeset

497 
by blast 
12905  498 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

499 
lemma refl_on_Id_on: "refl_on A (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

500 
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

501 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

502 
lemma antisym_Id_on [simp]: "antisym (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

503 
by (unfold antisym_def) blast 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

504 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

505 
lemma sym_Id_on [simp]: "sym (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

506 
by (rule symI) clarify 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

507 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

508 
lemma trans_Id_on [simp]: "trans (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

509 
by (fast intro: transI elim: transD) 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

510 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

511 
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

512 
by (unfold single_valued_def) blast 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

513 

12905  514 

46694  515 
subsubsection {* Composition *} 
12905  516 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

517 
inductive_set relcomp :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

518 
for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set" 
46694  519 
where 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

520 
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

521 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

522 
notation relcompp (infixr "OO" 75) 
12905  523 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

524 
lemmas relcomppI = relcompp.intros 
12905  525 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

526 
text {* 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

527 
For historic reasons, the elimination rules are not wholly corresponding. 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

528 
Feel free to consolidate this. 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

529 
*} 
46694  530 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

531 
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

532 
inductive_cases relcomppE [elim!]: "(r OO s) a c" 
46694  533 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

534 
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow> 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

535 
(\<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

536 
by (cases xz) (simp, erule relcompEpair, iprover) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

537 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

538 
lemma R_O_Id [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

539 
"R O Id = R" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

540 
by fast 
46694  541 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

542 
lemma Id_O_R [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

543 
"Id O R = R" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

544 
by fast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

545 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

546 
lemma relcomp_empty1 [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

547 
"{} O R = {}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

548 
by blast 
12905  549 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

550 
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

551 
"\<bottom> OO R = \<bottom>" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

552 
by (fact relcomp_empty1 [to_pred]) 
12905  553 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

554 
lemma relcomp_empty2 [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

555 
"R O {} = {}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

556 
by blast 
12905  557 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

558 
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

559 
"R OO \<bottom> = \<bottom>" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

560 
by (fact relcomp_empty2 [to_pred]) 
23185  561 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

562 
lemma O_assoc: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

563 
"(R O S) O T = R O (S O T)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

564 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

565 

46883
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

566 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

567 
lemma relcompp_assoc: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

568 
"(r OO s) OO t = r OO (s OO t)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

569 
by (fact O_assoc [to_pred]) 
23185  570 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

571 
lemma trans_O_subset: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

572 
"trans r \<Longrightarrow> r O r \<subseteq> r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

573 
by (unfold trans_def) blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

574 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

575 
lemma transp_relcompp_less_eq: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

576 
"transp r \<Longrightarrow> r OO r \<le> r " 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

577 
by (fact trans_O_subset [to_pred]) 
12905  578 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

579 
lemma relcomp_mono: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

580 
"r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

581 
by blast 
12905  582 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

583 
lemma relcompp_mono: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

584 
"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

585 
by (fact relcomp_mono [to_pred]) 
12905  586 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

587 
lemma relcomp_subset_Sigma: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

588 
"r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

589 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

590 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

591 
lemma relcomp_distrib [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

592 
"R O (S \<union> T) = (R O S) \<union> (R O T)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

593 
by auto 
12905  594 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

595 
lemma relcompp_distrib [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

596 
"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

597 
by (fact relcomp_distrib [to_pred]) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

598 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

599 
lemma relcomp_distrib2 [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

600 
"(S \<union> T) O R = (S O R) \<union> (T O R)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

601 
by auto 
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset

602 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

603 
lemma relcompp_distrib2 [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

604 
"(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

605 
by (fact relcomp_distrib2 [to_pred]) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

606 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

607 
lemma relcomp_UNION_distrib: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

608 
"s O UNION I r = (\<Union>i\<in>I. s O r i) " 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

609 
by auto 
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset

610 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

611 
(* FIXME thm relcomp_UNION_distrib [to_pred] *) 
36772  612 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

613 
lemma relcomp_UNION_distrib2: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

614 
"UNION I r O s = (\<Union>i\<in>I. r i O s) " 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

615 
by auto 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

616 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

617 
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *) 
36772  618 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

619 
lemma single_valued_relcomp: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

620 
"single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

621 
by (unfold single_valued_def) blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

622 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

623 
lemma relcomp_unfold: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

624 
"r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

625 
by (auto simp add: set_eq_iff) 
12905  626 

55083  627 
lemma eq_OO: "op= OO R = R" 
628 
by blast 

629 

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

630 

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

631 
subsubsection {* Converse *} 
12913  632 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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changeset

633 
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^1)" [1000] 999) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

634 
for r :: "('a \<times> 'b) set" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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changeset

635 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

636 
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^1" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

637 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

638 
notation (xsymbols) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

639 
converse ("(_\<inverse>)" [1000] 999) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

640 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

641 
notation 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

642 
conversep ("(_^1)" [1000] 1000) 
46694  643 

644 
notation (xsymbols) 

645 
conversep ("(_\<inverse>\<inverse>)" [1000] 1000) 

646 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

647 
lemma converseI [sym]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

648 
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

649 
by (fact converse.intros) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

650 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

651 
lemma conversepI (* CANDIDATE [sym] *): 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

652 
"r a b \<Longrightarrow> r\<inverse>\<inverse> b a" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

653 
by (fact conversep.intros) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

654 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

655 
lemma converseD [sym]: 
e9e7209eb375
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parents:
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diff
changeset

656 
"(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

657 
by (erule converse.cases) iprover 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

658 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

659 
lemma conversepD (* CANDIDATE [sym] *): 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

660 
"r\<inverse>\<inverse> b a \<Longrightarrow> r a b" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

661 
by (fact converseD [to_pred]) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

662 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

663 
lemma converseE [elim!]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

664 
 {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

665 
"yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

666 
by (cases yx) (simp, erule converse.cases, iprover) 
46694  667 

46882  668 
lemmas conversepE [elim!] = conversep.cases 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

669 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

670 
lemma converse_iff [iff]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

671 
"(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

672 
by (auto intro: converseI) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

673 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

674 
lemma conversep_iff [iff]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

675 
"r\<inverse>\<inverse> a b = r b a" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

676 
by (fact converse_iff [to_pred]) 
46694  677 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

678 
lemma converse_converse [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

679 
"(r\<inverse>)\<inverse> = r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

680 
by (simp add: set_eq_iff) 
46694  681 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

682 
lemma conversep_conversep [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

683 
"(r\<inverse>\<inverse>)\<inverse>\<inverse> = r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

684 
by (fact converse_converse [to_pred]) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

685 

53680  686 
lemma converse_empty[simp]: "{}\<inverse> = {}" 
687 
by auto 

688 

689 
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV" 

690 
by auto 

691 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

692 
lemma converse_relcomp: "(r O s)^1 = s^1 O r^1" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

693 
by blast 
46694  694 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

695 
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

696 
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

697 
elim: relcomppE dest: conversepD) 
46694  698 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

699 
lemma converse_Int: "(r \<inter> s)^1 = r^1 \<inter> s^1" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

700 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

701 

46694  702 
lemma converse_meet: "(r \<sqinter> s)^1 = r^1 \<sqinter> s^1" 
703 
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) 

704 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

705 
lemma converse_Un: "(r \<union> s)^1 = r^1 \<union> s^1" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

706 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

707 

46694  708 
lemma converse_join: "(r \<squnion> s)^1 = r^1 \<squnion> s^1" 
709 
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) 

710 

19228  711 
lemma converse_INTER: "(INTER S r)^1 = (INT x:S. (r x)^1)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

712 
by fast 
19228  713 

714 
lemma converse_UNION: "(UNION S r)^1 = (UN x:S. (r x)^1)" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

715 
by blast 
19228  716 

52749  717 
lemma converse_mono[simp]: "r^1 \<subseteq> s ^1 \<longleftrightarrow> r \<subseteq> s" 
718 
by auto 

719 

720 
lemma conversep_mono[simp]: "r^1 \<le> s ^1 \<longleftrightarrow> r \<le> s" 

721 
by (fact converse_mono[to_pred]) 

722 

723 
lemma converse_inject[simp]: "r^1 = s ^1 \<longleftrightarrow> r = s" 

52730  724 
by auto 
725 

52749  726 
lemma conversep_inject[simp]: "r^1 = s ^1 \<longleftrightarrow> r = s" 
727 
by (fact converse_inject[to_pred]) 

728 

729 
lemma converse_subset_swap: "r \<subseteq> s ^1 = (r ^1 \<subseteq> s)" 

730 
by auto 

731 

732 
lemma conversep_le_swap: "r \<le> s ^1 = (r ^1 \<le> s)" 

733 
by (fact converse_subset_swap[to_pred]) 

52730  734 

12905  735 
lemma converse_Id [simp]: "Id^1 = Id" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

736 
by blast 
12905  737 

30198  738 
lemma converse_Id_on [simp]: "(Id_on A)^1 = Id_on A" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

739 
by blast 
12905  740 

30198  741 
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

742 
by (unfold refl_on_def) auto 
12905  743 

19228  744 
lemma sym_converse [simp]: "sym (converse r) = sym r" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

745 
by (unfold sym_def) blast 
19228  746 

747 
lemma antisym_converse [simp]: "antisym (converse r) = antisym r" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

748 
by (unfold antisym_def) blast 
12905  749 

19228  750 
lemma trans_converse [simp]: "trans (converse r) = trans r" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

751 
by (unfold trans_def) blast 
12905  752 

19228  753 
lemma sym_conv_converse_eq: "sym r = (r^1 = r)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

754 
by (unfold sym_def) fast 
19228  755 

756 
lemma sym_Un_converse: "sym (r \<union> r^1)" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

757 
by (unfold sym_def) blast 
19228  758 

759 
lemma sym_Int_converse: "sym (r \<inter> r^1)" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

760 
by (unfold sym_def) blast 
19228  761 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

762 
lemma total_on_converse [simp]: "total_on A (r^1) = total_on A r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

763 
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

764 

52749  765 
lemma finite_converse [iff]: "finite (r^1) = finite r" 
54611
31afce809794
set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents:
54555
diff
changeset

766 
unfolding converse_def conversep_iff using [[simproc add: finite_Collect]] 
31afce809794
set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents:
54555
diff
changeset

767 
by (auto elim: finite_imageD simp: inj_on_def) 
12913  768 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

769 
lemma conversep_noteq [simp]: "(op \<noteq>)^1 = op \<noteq>" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

770 
by (auto simp add: fun_eq_iff) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

771 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

772 
lemma conversep_eq [simp]: "(op =)^1 = op =" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

773 
by (auto simp add: fun_eq_iff) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

774 

53680  775 
lemma converse_unfold [code]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

776 
"r\<inverse> = {(y, x). (x, y) \<in> r}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

777 
by (simp add: set_eq_iff) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

778 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

779 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

780 
subsubsection {* Domain, range and field *} 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

781 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

782 
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

783 
for r :: "('a \<times> 'b) set" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

784 
where 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

785 
DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

786 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

787 
abbreviation (input) "DomainP \<equiv> Domainp" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

788 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

789 
lemmas DomainPI = Domainp.DomainI 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

790 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

791 
inductive_cases DomainE [elim!]: "a \<in> Domain r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

792 
inductive_cases DomainpE [elim!]: "Domainp r a" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

793 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

794 
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

795 
for r :: "('a \<times> 'b) set" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

796 
where 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

797 
RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

798 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

799 
abbreviation (input) "RangeP \<equiv> Rangep" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

800 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

801 
lemmas RangePI = Rangep.RangeI 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

802 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

803 
inductive_cases RangeE [elim!]: "b \<in> Range r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

804 
inductive_cases RangepE [elim!]: "Rangep r b" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

805 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

806 
definition Field :: "'a rel \<Rightarrow> 'a set" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

807 
where 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

808 
"Field r = Domain r \<union> Range r" 
12905  809 

46694  810 
lemma Domain_fst [code]: 
811 
"Domain r = fst ` r" 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

812 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

813 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

814 
lemma Range_snd [code]: 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

815 
"Range r = snd ` r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

816 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

817 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

818 
lemma fst_eq_Domain: "fst ` R = Domain R" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

819 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

820 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

821 
lemma snd_eq_Range: "snd ` R = Range R" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

822 
by force 
46694  823 

824 
lemma Domain_empty [simp]: "Domain {} = {}" 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

825 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

826 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

827 
lemma Range_empty [simp]: "Range {} = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

828 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

829 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

830 
lemma Field_empty [simp]: "Field {} = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

831 
by (simp add: Field_def) 
46694  832 

833 
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}" 

834 
by auto 

835 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

836 
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

837 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

838 

46882  839 
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)" 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

840 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

841 

46882  842 
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)" 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

843 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

844 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

845 
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r" 
46884  846 
by (auto simp add: Field_def) 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

847 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

848 
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

849 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

850 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

851 
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)" 
46694  852 
by blast 
853 

854 
lemma Domain_Id [simp]: "Domain Id = UNIV" 

855 
by blast 

856 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

857 
lemma Range_Id [simp]: "Range Id = UNIV" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

858 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

859 

46694  860 
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A" 
861 
by blast 

862 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

863 
lemma Range_Id_on [simp]: "Range (Id_on A) = A" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

864 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

865 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

866 
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B" 
46694  867 
by blast 
868 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

869 
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

870 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

871 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

872 
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

873 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

874 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

875 
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B" 
46694  876 
by blast 
877 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

878 
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

879 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

880 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

881 
lemma Domain_Diff_subset: "Domain A  Domain B \<subseteq> Domain (A  B)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

882 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

883 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

884 
lemma Range_Diff_subset: "Range A  Range B \<subseteq> Range (A  B)" 
46694  885 
by blast 
886 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

887 
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)" 
46694  888 
by blast 
889 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

890 
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

891 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

892 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

893 
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

894 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

895 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

896 
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

897 
by auto 
46694  898 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

899 
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r" 
46694  900 
by blast 
901 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

902 
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

903 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

904 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

905 
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

906 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

907 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

908 
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

909 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

910 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

911 
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)" 
46884  912 
by (induct set: finite) auto 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

913 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

914 
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)" 
46884  915 
by (induct set: finite) auto 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

916 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

917 
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

918 
by (simp add: Field_def finite_Domain finite_Range) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

919 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

920 
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

921 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

922 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

923 
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

924 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

925 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

926 
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

927 
by (auto simp: Field_def Domain_def Range_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

928 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

929 
lemma Domain_unfold: 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

930 
"Domain r = {x. \<exists>y. (x, y) \<in> r}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

931 
by blast 
46694  932 

933 
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)" 

934 
by auto 

935 

936 
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)" 

937 
by auto 

938 

12905  939 

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

940 
subsubsection {* Image of a set under a relation *} 
12905  941 

50420  942 
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "``" 90) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

943 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

944 
"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

945 

12913  946 
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

947 
by (simp add: Image_def) 
12905  948 

12913  949 
lemma Image_singleton: "r``{a} = {b. (a, b) : r}" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

950 
by (simp add: Image_def) 
12905  951 

12913  952 
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

953 
by (rule Image_iff [THEN trans]) simp 
12905  954 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53680
diff
changeset

955 
lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

956 
by (unfold Image_def) blast 
12905  957 

958 
lemma ImageE [elim!]: 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

959 
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

960 
by (unfold Image_def) (iprover elim!: CollectE bexE) 
12905  961 

962 
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" 

963 
 {* This version's more effective when we already have the required @{text a} *} 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

964 
by blast 
12905  965 

966 
lemma Image_empty [simp]: "R``{} = {}" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

967 
by blast 
12905  968 

969 
lemma Image_Id [simp]: "Id `` A = A" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

970 
by blast 
12905  971 

30198  972 
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

973 
by blast 
13830  974 

975 
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

976 
by blast 
12905  977 

13830  978 
lemma Image_Int_eq: 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

979 
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

980 
by (simp add: single_valued_def, blast) 
12905  981 

13830  982 
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

983 
by blast 
12905  984 

13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

985 
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

986 
by blast 
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

987 

12913  988 
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

989 
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) 
12905  990 

13830  991 
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" 
12905  992 
 {* NOT suitable for rewriting *} 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

993 
by blast 
12905  994 

12913  995 
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

996 
by blast 
12905  997 

13830  998 
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

999 
by blast 
13830  1000 

54410
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1001 
lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)" 
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1002 
by auto 
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1003 

13830  1004 
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1005 
by blast 
12905  1006 

13830  1007 
text{*Converse inclusion requires some assumptions*} 
1008 
lemma Image_INT_eq: 

1009 
"[single_valued (r\<inverse>); A\<noteq>{}] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" 

1010 
apply (rule equalityI) 

1011 
apply (rule Image_INT_subset) 

1012 
apply (simp add: single_valued_def, blast) 

1013 
done 

12905  1014 

12913  1015 
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq>  ((r^1) `` (B)))" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1016 
by blast 
12905  1017 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

1018 
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1019 
by auto 
12905  1020 

54410
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1021 
lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)" 
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1022 
by auto 
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1023 

0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1024 
lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)" 
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset

1025 
by auto 
12905  1026 

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

1027 
subsubsection {* Inverse image *} 
12905  1028 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1029 
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1030 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1031 
"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

1032 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1033 
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1034 
where 
46694  1035 
"inv_imagep r f = (\<lambda>x y. r (f x) (f y))" 
1036 

1037 
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)" 

1038 
by (simp add: inv_image_def inv_imagep_def) 

1039 

19228  1040 
lemma sym_inv_image: "sym r ==> sym (inv_image r f)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1041 
by (unfold sym_def inv_image_def) blast 
19228  1042 

12913  1043 
lemma trans_inv_image: "trans r ==> trans (inv_image r f)" 
12905  1044 
apply (unfold trans_def inv_image_def) 
1045 
apply (simp (no_asm)) 

1046 
apply blast 

1047 
done 

1048 

32463
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset

1049 
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

1050 
by (auto simp:inv_image_def) 
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset

1051 

33218  1052 
lemma converse_inv_image[simp]: "(inv_image R f)^1 = inv_image (R^1) f" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1053 
unfolding inv_image_def converse_unfold by auto 
33218  1054 

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

1055 
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

1056 
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

1057 

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

1058 

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

1059 
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

1060 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1061 
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1062 
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

1063 
"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

1064 

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

1065 
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

1066 
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

1067 

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

1068 
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

1069 

48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1070 
subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *} 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1071 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1072 
lemma Id_on_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1073 
assumes "finite A" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1074 
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

1075 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1076 
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

1077 
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

1078 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1079 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1080 
lemma comp_fun_commute_Image_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1081 
"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

1082 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1083 
interpret comp_fun_idem Set.insert 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1084 
by (fact comp_fun_idem_insert) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1085 
show ?thesis 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1086 
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

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 Image_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1090 
assumes "finite R" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1091 
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

1092 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1093 
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

1094 
by (rule comp_fun_commute_Image_fold) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1095 
have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))" 
52749  1096 
by (force intro: rev_ImageI) 
48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1097 
show ?thesis using assms by (induct R) (auto simp: *) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1098 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1099 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1100 
lemma insert_relcomp_union_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1101 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1102 
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

1103 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1104 
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

1105 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1106 
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

1107 
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

1108 
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

1109 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1110 
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

1111 
show ?thesis unfolding * 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1112 
using `finite S` by (induct S) (auto split: prod.split) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1113 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1114 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1115 
lemma insert_relcomp_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1116 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1117 
shows "Set.insert x R O S = 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1118 
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

1119 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1120 
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

1121 
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

1122 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1123 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1124 
lemma comp_fun_commute_relcomp_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1125 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1126 
shows "comp_fun_commute (\<lambda>(x,y) A. 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1127 
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

1128 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1129 
have *: "\<And>a b A. 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1130 
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

1131 
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

1132 
show ?thesis by default (auto simp: *) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1133 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1134 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1135 
lemma relcomp_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1136 
assumes "finite R" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1137 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1138 
shows "R O S = Finite_Set.fold 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1139 
(\<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" 
52749  1140 
using assms by (induct R) 
1141 
(auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold 

48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1142 
cong: if_cong) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1143 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1144 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1145 

1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset

1146 
end 
46689  1147 