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(* Title: HOL/Predicate.thy 
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Author: Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen 
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*) 
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header {* Predicates as relations and enumerations *} 
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theory Predicate 
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imports Inductive Relation 
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begin 
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notation 
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inf (infixl "\<sqinter>" 70) and 

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sup (infixl "\<squnion>" 65) and 

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Inf ("\<Sqinter>_" [900] 900) and 

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Sup ("\<Squnion>_" [900] 900) and 

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top ("\<top>") and 

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bot ("\<bottom>") 

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subsection {* Predicates as (complete) lattices *} 

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subsubsection {* @{const sup} on @{typ bool} *} 

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lemma sup_boolI1: 

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"P \<Longrightarrow> P \<squnion> Q" 

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by (simp add: sup_bool_eq) 

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lemma sup_boolI2: 

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"Q \<Longrightarrow> P \<squnion> Q" 

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by (simp add: sup_bool_eq) 

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lemma sup_boolE: 

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"P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R" 

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by (auto simp add: sup_bool_eq) 

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subsubsection {* Equality and Subsets *} 

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lemma pred_equals_eq: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)" 
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by (simp add: mem_def) 
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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 (simp add: expand_fun_eq mem_def) 
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lemma pred_subset_eq: "((\<lambda>x. x \<in> R) <= (\<lambda>x. x \<in> S)) = (R <= S)" 
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by (simp add: mem_def) 
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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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subsubsection {* 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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subsubsection {* 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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subsubsection {* 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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by simp 
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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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by auto 
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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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subsubsection {* 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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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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subsubsection {* 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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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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30328  186 
subsubsection {* 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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30328  219 
subsection {* Predicates as relations *} 
220 

221 
subsubsection {* Composition *} 

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inductive 
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pred_comp :: "['a => 'b => bool, 'b => 'c => bool] => 'a => 'c => bool" 
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(infixr "OO" 75) 
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for r :: "'a => 'b => bool" and s :: "'b => 'c => bool" 
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where 
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pred_compI [intro]: "r a b ==> s 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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30328  237 
subsubsection {* 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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30328  283 
subsubsection {* 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 predicate1I) 
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30328  297 
subsubsection {* 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 predicate1I) 
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30328  311 
subsubsection {* 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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30328  324 
subsubsection {* Powerset *} 
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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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328 

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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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330 
by (auto simp add: Powp_def expand_fun_eq) 
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331 

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lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq] 
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333 

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30328  335 
subsubsection {* Properties of relations *} 
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abbreviation antisymP :: "('a => 'a => bool) => bool" where 
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338 
"antisymP r == antisym {(x, y). r x y}" 
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abbreviation transP :: "('a => 'a => bool) => bool" where 
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341 
"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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30328  346 

347 
subsection {* Predicates as enumerations *} 

348 

349 
subsubsection {* The type of predicate enumerations (a monad) *} 

350 

351 
datatype 'a pred = Pred "'a \<Rightarrow> bool" 

352 

353 
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where 

354 
eval_pred: "eval (Pred f) = f" 

355 

356 
lemma Pred_eval [simp]: 

357 
"Pred (eval x) = x" 

358 
by (cases x) simp 

359 

360 
lemma eval_inject: "eval x = eval y \<longleftrightarrow> x = y" 

361 
by (cases x) auto 

362 

363 
definition single :: "'a \<Rightarrow> 'a pred" where 

364 
"single x = Pred ((op =) x)" 

365 

366 
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where 

367 
"P \<guillemotright>= f = Pred (\<lambda>x. (\<exists>y. eval P y \<and> eval (f y) x))" 

368 

369 
instantiation pred :: (type) complete_lattice 

370 
begin 

371 

372 
definition 

373 
"P \<le> Q \<longleftrightarrow> eval P \<le> eval Q" 

374 

375 
definition 

376 
"P < Q \<longleftrightarrow> eval P < eval Q" 

377 

378 
definition 

379 
"\<bottom> = Pred \<bottom>" 

380 

381 
definition 

382 
"\<top> = Pred \<top>" 

383 

384 
definition 

385 
"P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)" 

386 

387 
definition 

388 
"P \<squnion> Q = Pred (eval P \<squnion> eval Q)" 

389 

390 
definition 

31932  391 
[code del]: "\<Sqinter>A = Pred (INFI A eval)" 
30328  392 

393 
definition 

31932  394 
[code del]: "\<Squnion>A = Pred (SUPR A eval)" 
30328  395 

396 
instance by default 

397 
(auto simp add: less_eq_pred_def less_pred_def 

398 
inf_pred_def sup_pred_def bot_pred_def top_pred_def 

399 
Inf_pred_def Sup_pred_def, 

400 
auto simp add: le_fun_def less_fun_def le_bool_def less_bool_def 

401 
eval_inject mem_def) 

402 

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403 
end 
30328  404 

405 
lemma bind_bind: 

406 
"(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)" 

407 
by (auto simp add: bind_def expand_fun_eq) 

408 

409 
lemma bind_single: 

410 
"P \<guillemotright>= single = P" 

411 
by (simp add: bind_def single_def) 

412 

413 
lemma single_bind: 

414 
"single x \<guillemotright>= P = P x" 

415 
by (simp add: bind_def single_def) 

416 

417 
lemma bottom_bind: 

418 
"\<bottom> \<guillemotright>= P = \<bottom>" 

419 
by (auto simp add: bot_pred_def bind_def expand_fun_eq) 

420 

421 
lemma sup_bind: 

422 
"(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R" 

423 
by (auto simp add: bind_def sup_pred_def expand_fun_eq) 

424 

425 
lemma Sup_bind: "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)" 

426 
by (auto simp add: bind_def Sup_pred_def expand_fun_eq) 

427 

428 
lemma pred_iffI: 

429 
assumes "\<And>x. eval A x \<Longrightarrow> eval B x" 

430 
and "\<And>x. eval B x \<Longrightarrow> eval A x" 

431 
shows "A = B" 

432 
proof  

433 
from assms have "\<And>x. eval A x \<longleftrightarrow> eval B x" by blast 

434 
then show ?thesis by (cases A, cases B) (simp add: expand_fun_eq) 

435 
qed 

436 

437 
lemma singleI: "eval (single x) x" 

438 
unfolding single_def by simp 

439 

440 
lemma singleI_unit: "eval (single ()) x" 

441 
by simp (rule singleI) 

442 

443 
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P" 

444 
unfolding single_def by simp 

445 

446 
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P" 

447 
by (erule singleE) simp 

448 

449 
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y" 

450 
unfolding bind_def by auto 

451 

452 
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P" 

453 
unfolding bind_def by auto 

454 

455 
lemma botE: "eval \<bottom> x \<Longrightarrow> P" 

456 
unfolding bot_pred_def by auto 

457 

458 
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x" 

459 
unfolding sup_pred_def by simp 

460 

461 
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 

462 
unfolding sup_pred_def by simp 

463 

464 
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P" 

465 
unfolding sup_pred_def by auto 

466 

467 

468 
subsubsection {* Derived operations *} 

469 

470 
definition if_pred :: "bool \<Rightarrow> unit pred" where 

471 
if_pred_eq: "if_pred b = (if b then single () else \<bottom>)" 

472 

473 
definition not_pred :: "unit pred \<Rightarrow> unit pred" where 

474 
not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())" 

475 

476 
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()" 

477 
unfolding if_pred_eq by (auto intro: singleI) 

478 

479 
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P" 

480 
unfolding if_pred_eq by (cases b) (auto elim: botE) 

481 

482 
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()" 

483 
unfolding not_pred_eq eval_pred by (auto intro: singleI) 

484 

485 
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()" 

486 
unfolding not_pred_eq by (auto intro: singleI) 

487 

488 
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

489 
unfolding not_pred_eq 

490 
by (auto split: split_if_asm elim: botE) 

491 

492 
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

493 
unfolding not_pred_eq 

494 
by (auto split: split_if_asm elim: botE) 

495 

496 

497 
subsubsection {* Implementation *} 

498 

499 
datatype 'a seq = Empty  Insert "'a" "'a pred"  Join "'a pred" "'a seq" 

500 

501 
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where 

502 
"pred_of_seq Empty = \<bottom>" 

503 
 "pred_of_seq (Insert x P) = single x \<squnion> P" 

504 
 "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq" 

505 

506 
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where 

507 
"Seq f = pred_of_seq (f ())" 

508 

509 
code_datatype Seq 

510 

511 
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool" where 

512 
"member Empty x \<longleftrightarrow> False" 

513 
 "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x" 

514 
 "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x" 

515 

516 
lemma eval_member: 

517 
"member xq = eval (pred_of_seq xq)" 

518 
proof (induct xq) 

519 
case Empty show ?case 

520 
by (auto simp add: expand_fun_eq elim: botE) 

521 
next 

522 
case Insert show ?case 

523 
by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI) 

524 
next 

525 
case Join then show ?case 

526 
by (auto simp add: expand_fun_eq elim: supE intro: supI1 supI2) 

527 
qed 

528 

529 
lemma eval_code [code]: "eval (Seq f) = member (f ())" 

530 
unfolding Seq_def by (rule sym, rule eval_member) 

531 

532 
lemma single_code [code]: 

533 
"single x = Seq (\<lambda>u. Insert x \<bottom>)" 

534 
unfolding Seq_def by simp 

535 

536 
primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where 

537 
"apply f Empty = Empty" 

538 
 "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)" 

539 
 "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)" 

540 

541 
lemma apply_bind: 

542 
"pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f" 

543 
proof (induct xq) 

544 
case Empty show ?case 

545 
by (simp add: bottom_bind) 

546 
next 

547 
case Insert show ?case 

548 
by (simp add: single_bind sup_bind) 

549 
next 

550 
case Join then show ?case 

551 
by (simp add: sup_bind) 

552 
qed 

553 

554 
lemma bind_code [code]: 

555 
"Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))" 

556 
unfolding Seq_def by (rule sym, rule apply_bind) 

557 

558 
lemma bot_set_code [code]: 

559 
"\<bottom> = Seq (\<lambda>u. Empty)" 

560 
unfolding Seq_def by simp 

561 

30376  562 
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where 
563 
"adjunct P Empty = Join P Empty" 

564 
 "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)" 

565 
 "adjunct P (Join Q xq) = Join Q (adjunct P xq)" 

566 

567 
lemma adjunct_sup: 

568 
"pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq" 

569 
by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute) 

570 

30328  571 
lemma sup_code [code]: 
572 
"Seq f \<squnion> Seq g = Seq (\<lambda>u. case f () 

573 
of Empty \<Rightarrow> g () 

574 
 Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g) 

30376  575 
 Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))" 
30328  576 
proof (cases "f ()") 
577 
case Empty 

578 
thus ?thesis 

30376  579 
unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"] sup_bot) 
30328  580 
next 
581 
case Insert 

582 
thus ?thesis 

583 
unfolding Seq_def by (simp add: sup_assoc) 

584 
next 

585 
case Join 

586 
thus ?thesis 

30376  587 
unfolding Seq_def 
588 
by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute) 

30328  589 
qed 
590 

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591 
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where 
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592 
"contained Empty Q \<longleftrightarrow> True" 
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593 
 "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q" 
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594 
 "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q" 
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595 

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596 
lemma single_less_eq_eval: 
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597 
"single x \<le> P \<longleftrightarrow> eval P x" 
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598 
by (auto simp add: single_def less_eq_pred_def mem_def) 
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599 

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600 
lemma contained_less_eq: 
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601 
"contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q" 
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602 
by (induct xq) (simp_all add: single_less_eq_eval) 
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603 

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604 
lemma less_eq_pred_code [code]: 
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605 
"Seq f \<le> Q = (case f () 
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606 
of Empty \<Rightarrow> True 
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607 
 Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q 
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608 
 Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)" 
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609 
by (cases "f ()") 
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610 
(simp_all add: Seq_def single_less_eq_eval contained_less_eq) 
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611 

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612 
lemma eq_pred_code [code]: 
31133  613 
fixes P Q :: "'a pred" 
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614 
shows "eq_class.eq P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P" 
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615 
unfolding eq by auto 
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616 

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617 
lemma [code]: 
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618 
"pred_case f P = f (eval P)" 
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619 
by (cases P) simp 
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620 

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621 
lemma [code]: 
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622 
"pred_rec f P = f (eval P)" 
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623 
by (cases P) simp 
30328  624 

31105
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625 
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x" 
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626 

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627 
lemma eq_is_eq: "eq x y \<equiv> (x = y)" 
31108  628 
by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases) 
30948  629 

31216  630 
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where 
631 
"map f P = P \<guillemotright>= (single o f)" 

632 

30948  633 
ML {* 
634 
signature PREDICATE = 

635 
sig 

636 
datatype 'a pred = Seq of (unit > 'a seq) 

637 
and 'a seq = Empty  Insert of 'a * 'a pred  Join of 'a pred * 'a seq 

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638 
val yield: 'a pred > ('a * 'a pred) option 
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639 
val yieldn: int > 'a pred > 'a list * 'a pred 
31222  640 
val map: ('a > 'b) > 'a pred > 'b pred 
30948  641 
end; 
642 

643 
structure Predicate : PREDICATE = 

644 
struct 

645 

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646 
@{code_datatype pred = Seq}; 
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647 
@{code_datatype seq = Empty  Insert  Join}; 
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648 

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649 
fun yield (Seq f) = next (f ()) 
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650 
and next @{code Empty} = NONE 
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651 
 next (@{code Insert} (x, P)) = SOME (x, P) 
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652 
 next (@{code Join} (P, xq)) = (case yield P 
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653 
of NONE => next xq 
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654 
 SOME (x, Q) => SOME (x, @{code Seq} (fn _ => @{code Join} (Q, xq)))) 
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655 

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656 
fun anamorph f k x = (if k = 0 then ([], x) 
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657 
else case f x 
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658 
of NONE => ([], x) 
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659 
 SOME (v, y) => let 
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660 
val (vs, z) = anamorph f (k  1) y 
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661 
in (v :: vs, z) end) 
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662 

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663 
fun yieldn P = anamorph yield P; 
30948  664 

31222  665 
fun map f = @{code map} f; 
666 

30948  667 
end; 
668 
*} 

669 

670 
code_reserved Eval Predicate 

671 

672 
code_type pred and seq 

673 
(Eval "_/ Predicate.pred" and "_/ Predicate.seq") 

674 

675 
code_const Seq and Empty and Insert and Join 

676 
(Eval "Predicate.Seq" and "Predicate.Empty" and "Predicate.Insert/ (_,/ _)" and "Predicate.Join/ (_,/ _)") 

677 

31122  678 
text {* dummy setup for @{text code_pred} and @{text values} keywords *} 
31108  679 

680 
ML {* 

31122  681 
local 
682 

683 
structure P = OuterParse; 

684 

685 
val opt_modes = Scan.optional (P.$$$ "("  P.!!! (Scan.repeat1 P.xname  P.$$$ ")")) []; 

686 

687 
in 

688 

689 
val _ = OuterSyntax.local_theory_to_proof "code_pred" "sets up goal for cases rule from given introduction rules and compiles predicate" 

690 
OuterKeyword.thy_goal (P.term_group >> (K (Proof.theorem_i NONE (K I) [[]]))); 

691 

31216  692 
val _ = OuterSyntax.improper_command "values" "enumerate and print comprehensions" 
31122  693 
OuterKeyword.diag ((opt_modes  P.term) 
694 
>> (fn (modes, t) => Toplevel.no_timing o Toplevel.keep 

695 
(K ()))); 

696 

697 
end 

31108  698 
*} 
30959
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699 

30328  700 
no_notation 
701 
inf (infixl "\<sqinter>" 70) and 

702 
sup (infixl "\<squnion>" 65) and 

703 
Inf ("\<Sqinter>_" [900] 900) and 

704 
Sup ("\<Squnion>_" [900] 900) and 

705 
top ("\<top>") and 

706 
bot ("\<bottom>") and 

707 
bind (infixl "\<guillemotright>=" 70) 

708 

709 
hide (open) type pred seq 

30378  710 
hide (open) const Pred eval single bind if_pred not_pred 
31216  711 
Empty Insert Join Seq member pred_of_seq "apply" adjunct eq map 
30328  712 

713 
end 