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(* Title: HOL/Predicate.thy 
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ID: $Id$ 
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Author: Stefan Berghofer, TU Muenchen 
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*) 
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header {* Predicates *} 
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theory Predicate 
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imports Inductive Relation 
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begin 
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subsection {* Equality and Subsets *} 
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lemma pred_equals_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)" 
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by (auto simp add: expand_fun_eq) 
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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)) = (R = S)" 
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by (auto simp add: expand_fun_eq) 
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lemma pred_subset_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)" 
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by fast 
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lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) <= (\<lambda>x y. (x, y) \<in> S)) = (R <= S)" 
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by fast 
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subsection {* Top and bottom elements *} 
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lemma top1I [intro!]: "top x" 
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by (simp add: top_fun_eq top_bool_eq) 
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lemma top2I [intro!]: "top x y" 
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by (simp add: top_fun_eq top_bool_eq) 
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lemma bot1E [elim!]: "bot x \<Longrightarrow> P" 
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by (simp add: bot_fun_eq bot_bool_eq) 
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lemma bot2E [elim!]: "bot x y \<Longrightarrow> P" 
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by (simp add: bot_fun_eq bot_bool_eq) 
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subsection {* The empty set *} 
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lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})" 
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by (auto simp add: expand_fun_eq) 
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lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})" 
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by (auto simp add: expand_fun_eq) 
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subsection {* Binary union *} 
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lemma sup1_iff [simp]: "sup A B x \<longleftrightarrow> A x  B x" 
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by (simp add: sup_fun_eq sup_bool_eq) 
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lemma sup2_iff [simp]: "sup A B x y \<longleftrightarrow> A x y  B x y" 
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by (simp add: sup_fun_eq sup_bool_eq) 
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lemma sup_Un_eq [pred_set_conv]: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)" 
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by (simp add: expand_fun_eq) 
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lemma sup_Un_eq2 [pred_set_conv]: "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)" 
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by (simp add: expand_fun_eq) 
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lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x" 
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by simp 
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lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y" 
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lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x" 
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lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y" 
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text {* 
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\medskip Classical introduction rule: no commitment to @{text A} vs 
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@{text B}. 
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*} 
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lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x" 
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lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y" 
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lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P" 
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by simp iprover 
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lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P" 
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by simp iprover 
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subsection {* Binary intersection *} 
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lemma inf1_iff [simp]: "inf A B x \<longleftrightarrow> A x \<and> B x" 
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by (simp add: inf_fun_eq inf_bool_eq) 
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lemma inf2_iff [simp]: "inf A B x y \<longleftrightarrow> A x y \<and> B x y" 
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by (simp add: inf_fun_eq inf_bool_eq) 
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lemma inf_Int_eq [pred_set_conv]: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)" 
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by (simp add: expand_fun_eq) 
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lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)" 
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by (simp add: expand_fun_eq) 
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lemma inf1I [intro!]: "A x ==> B x ==> inf A B x" 
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lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y" 
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lemma inf1D1: "inf A B x ==> A x" 
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lemma inf2D1: "inf A B x y ==> A x y" 
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lemma inf1D2: "inf A B x ==> B x" 
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lemma inf2D2: "inf A B x y ==> B x y" 
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lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P" 
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lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P" 
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subsection {* Unions of families *} 
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lemma SUP1_iff [simp]: "(SUP x:A. B x) b = (EX x:A. B x b)" 
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by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast 
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lemma SUP2_iff [simp]: "(SUP x:A. B x) b c = (EX x:A. B x b c)" 
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by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast 
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lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b" 
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lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c" 
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by auto 
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lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R" 
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by auto 
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lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R" 
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by auto 
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lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))" 
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by (simp add: expand_fun_eq) 
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lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))" 
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by (simp add: expand_fun_eq) 
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subsection {* Intersections of families *} 
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lemma INF1_iff [simp]: "(INF x:A. B x) b = (ALL x:A. B x b)" 
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by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast 
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lemma INF2_iff [simp]: "(INF x:A. B x) b c = (ALL x:A. B x b c)" 
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by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast 
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lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b" 
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by auto 
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lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c" 
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by auto 
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lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b" 
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by auto 
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lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c" 
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by auto 
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lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R" 
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by auto 
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lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R" 
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by auto 
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lemma INF_INT_eq: "(INF i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (INT i. r i))" 
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by (simp add: expand_fun_eq) 
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lemma INF_INT_eq2: "(INF i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (INT i. r i))" 
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by (simp add: expand_fun_eq) 
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subsection {* Composition of two relations *} 
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inductive 
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pred_comp :: "['b => 'c => bool, 'a => 'b => bool] => 'a => 'c => bool" 
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(infixr "OO" 75) 
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for r :: "'b => 'c => bool" and s :: "'a => 'b => bool" 
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where 
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pred_compI [intro]: "s a b ==> r b c ==> (r OO s) a c" 
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inductive_cases pred_compE [elim!]: "(r OO s) a c" 
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lemma pred_comp_rel_comp_eq [pred_set_conv]: 
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"((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)" 
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by (auto simp add: expand_fun_eq elim: pred_compE) 
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subsection {* Converse *} 
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inductive 
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conversep :: "('a => 'b => bool) => 'b => 'a => bool" 
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("(_^1)" [1000] 1000) 
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for r :: "'a => 'b => bool" 
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where 
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conversepI: "r a b ==> r^1 b a" 
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notation (xsymbols) 
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conversep ("(_\<inverse>\<inverse>)" [1000] 1000) 
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lemma conversepD: 
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assumes ab: "r^1 a b" 
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shows "r b a" using ab 
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by cases simp 
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lemma conversep_iff [iff]: "r^1 a b = r b a" 
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by (iprover intro: conversepI dest: conversepD) 
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lemma conversep_converse_eq [pred_set_conv]: 
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"(\<lambda>x y. (x, y) \<in> r)^1 = (\<lambda>x y. (x, y) \<in> r^1)" 
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by (auto simp add: expand_fun_eq) 
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lemma conversep_conversep [simp]: "(r^1)^1 = r" 
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by (iprover intro: order_antisym conversepI dest: conversepD) 
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lemma converse_pred_comp: "(r OO s)^1 = s^1 OO r^1" 
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by (iprover intro: order_antisym conversepI pred_compI 
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elim: pred_compE dest: conversepD) 
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lemma converse_meet: "(inf r s)^1 = inf r^1 s^1" 
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by (simp add: inf_fun_eq inf_bool_eq) 
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(iprover intro: conversepI ext dest: conversepD) 
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lemma converse_join: "(sup r s)^1 = sup r^1 s^1" 
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by (simp add: sup_fun_eq sup_bool_eq) 
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(iprover intro: conversepI ext dest: conversepD) 
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lemma conversep_noteq [simp]: "(op ~=)^1 = op ~=" 
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by (auto simp add: expand_fun_eq) 
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lemma conversep_eq [simp]: "(op =)^1 = op =" 
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by (auto simp add: expand_fun_eq) 
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subsection {* Domain *} 
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inductive 
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DomainP :: "('a => 'b => bool) => 'a => bool" 
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for r :: "'a => 'b => bool" 
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where 
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DomainPI [intro]: "r a b ==> DomainP r a" 
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inductive_cases DomainPE [elim!]: "DomainP r a" 
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lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)" 
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by (blast intro!: Orderings.order_antisym) 
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subsection {* Range *} 
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inductive 
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RangeP :: "('a => 'b => bool) => 'b => bool" 
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for r :: "'a => 'b => bool" 
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where 
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RangePI [intro]: "r a b ==> RangeP r b" 
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inductive_cases RangePE [elim!]: "RangeP r b" 
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lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)" 
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by (blast intro!: Orderings.order_antisym) 
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subsection {* Inverse image *} 
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definition 
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inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where 
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"inv_imagep r f == %x y. r (f x) (f y)" 
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lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)" 
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by (simp add: inv_image_def inv_imagep_def) 
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lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)" 
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by (simp add: inv_imagep_def) 
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subsection {* The Powerset operator *} 
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definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where 
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"Powp A == \<lambda>B. \<forall>x \<in> B. A x" 
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lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)" 
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by (auto simp add: Powp_def expand_fun_eq) 
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subsection {* Properties of relations  predicate versions *} 
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abbreviation antisymP :: "('a => 'a => bool) => bool" where 
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"antisymP r == antisym {(x, y). r x y}" 
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abbreviation transP :: "('a => 'a => bool) => bool" where 
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"transP r == trans {(x, y). r x y}" 
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abbreviation single_valuedP :: "('a => 'b => bool) => bool" where 
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"single_valuedP r == single_valued {(x, y). r x y}" 
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end 