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permissions  rwrr 
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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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by auto 
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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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319 

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lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)" 
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321 
by (simp add: inv_imagep_def) 
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322 

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30328  324 
subsubsection {* Powerset *} 
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325 

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326 
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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329 
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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334 

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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339 

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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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342 

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abbreviation single_valuedP :: "('a => 'b => bool) => bool" where 
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344 
"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 

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369 
instantiation pred :: (type) "{complete_lattice, boolean_algebra}" 
30328  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 

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396 
definition 
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397 
" P = Pred ( eval P)" 
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398 

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399 
definition 
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400 
"P  Q = Pred (eval P  eval Q)" 
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401 

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402 
instance proof 
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403 
qed (auto simp add: less_eq_pred_def less_pred_def 
30328  404 
inf_pred_def sup_pred_def bot_pred_def top_pred_def 
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405 
Inf_pred_def Sup_pred_def uminus_pred_def minus_pred_def fun_Compl_def bool_Compl_def, 
30328  406 
auto simp add: le_fun_def less_fun_def le_bool_def less_bool_def 
407 
eval_inject mem_def) 

408 

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409 
end 
30328  410 

411 
lemma bind_bind: 

412 
"(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)" 

413 
by (auto simp add: bind_def expand_fun_eq) 

414 

415 
lemma bind_single: 

416 
"P \<guillemotright>= single = P" 

417 
by (simp add: bind_def single_def) 

418 

419 
lemma single_bind: 

420 
"single x \<guillemotright>= P = P x" 

421 
by (simp add: bind_def single_def) 

422 

423 
lemma bottom_bind: 

424 
"\<bottom> \<guillemotright>= P = \<bottom>" 

425 
by (auto simp add: bot_pred_def bind_def expand_fun_eq) 

426 

427 
lemma sup_bind: 

428 
"(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R" 

429 
by (auto simp add: bind_def sup_pred_def expand_fun_eq) 

430 

431 
lemma Sup_bind: "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)" 

432 
by (auto simp add: bind_def Sup_pred_def expand_fun_eq) 

433 

434 
lemma pred_iffI: 

435 
assumes "\<And>x. eval A x \<Longrightarrow> eval B x" 

436 
and "\<And>x. eval B x \<Longrightarrow> eval A x" 

437 
shows "A = B" 

438 
proof  

439 
from assms have "\<And>x. eval A x \<longleftrightarrow> eval B x" by blast 

440 
then show ?thesis by (cases A, cases B) (simp add: expand_fun_eq) 

441 
qed 

442 

443 
lemma singleI: "eval (single x) x" 

444 
unfolding single_def by simp 

445 

446 
lemma singleI_unit: "eval (single ()) x" 

447 
by simp (rule singleI) 

448 

449 
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P" 

450 
unfolding single_def by simp 

451 

452 
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P" 

453 
by (erule singleE) simp 

454 

455 
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y" 

456 
unfolding bind_def by auto 

457 

458 
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P" 

459 
unfolding bind_def by auto 

460 

461 
lemma botE: "eval \<bottom> x \<Longrightarrow> P" 

462 
unfolding bot_pred_def by auto 

463 

464 
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x" 

465 
unfolding sup_pred_def by simp 

466 

467 
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 

468 
unfolding sup_pred_def by simp 

469 

470 
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P" 

471 
unfolding sup_pred_def by auto 

472 

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473 
lemma single_not_bot [simp]: 
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474 
"single x \<noteq> \<bottom>" 
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475 
by (auto simp add: single_def bot_pred_def expand_fun_eq) 
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476 

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477 
lemma not_bot: 
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478 
assumes "A \<noteq> \<bottom>" 
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479 
obtains x where "eval A x" 
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480 
using assms by (cases A) 
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481 
(auto simp add: bot_pred_def, auto simp add: mem_def) 
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482 

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483 

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484 
subsubsection {* Emptiness check and definite choice *} 
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485 

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486 
definition is_empty :: "'a pred \<Rightarrow> bool" where 
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487 
"is_empty A \<longleftrightarrow> A = \<bottom>" 
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488 

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489 
lemma is_empty_bot: 
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490 
"is_empty \<bottom>" 
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491 
by (simp add: is_empty_def) 
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492 

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493 
lemma not_is_empty_single: 
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494 
"\<not> is_empty (single x)" 
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495 
by (auto simp add: is_empty_def single_def bot_pred_def expand_fun_eq) 
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496 

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497 
lemma is_empty_sup: 
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498 
"is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B" 
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499 
by (auto simp add: is_empty_def intro: sup_eq_bot_eq1 sup_eq_bot_eq2) 
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500 

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501 
definition singleton :: "'a pred \<Rightarrow> 'a" where 
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502 
"singleton A = (if \<exists>!x. eval A x then THE x. eval A x else undefined)" 
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503 

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504 
lemma singleton_eqI: 
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505 
"\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton A = x" 
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506 
by (auto simp add: singleton_def) 
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507 

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508 
lemma eval_singletonI: 
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509 
"\<exists>!x. eval A x \<Longrightarrow> eval A (singleton A)" 
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510 
proof  
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511 
assume assm: "\<exists>!x. eval A x" 
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512 
then obtain x where "eval A x" .. 
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513 
moreover with assm have "singleton A = x" by (rule singleton_eqI) 
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514 
ultimately show ?thesis by simp 
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515 
qed 
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516 

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517 
lemma single_singleton: 
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518 
"\<exists>!x. eval A x \<Longrightarrow> single (singleton A) = A" 
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519 
proof  
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520 
assume assm: "\<exists>!x. eval A x" 
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521 
then have "eval A (singleton A)" 
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522 
by (rule eval_singletonI) 
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523 
moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton A = x" 
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524 
by (rule singleton_eqI) 
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525 
ultimately have "eval (single (singleton A)) = eval A" 
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526 
by (simp (no_asm_use) add: single_def expand_fun_eq) blast 
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527 
then show ?thesis by (simp add: eval_inject) 
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528 
qed 
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529 

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530 
lemma singleton_undefinedI: 
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531 
"\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton A = undefined" 
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532 
by (simp add: singleton_def) 
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533 

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534 
lemma singleton_bot: 
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535 
"singleton \<bottom> = undefined" 
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536 
by (auto simp add: bot_pred_def intro: singleton_undefinedI) 
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537 

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538 
lemma singleton_single: 
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539 
"singleton (single x) = x" 
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540 
by (auto simp add: intro: singleton_eqI singleI elim: singleE) 
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541 

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542 
lemma singleton_sup_single_single: 
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543 
"singleton (single x \<squnion> single y) = (if x = y then x else undefined)" 
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544 
proof (cases "x = y") 
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545 
case True then show ?thesis by (simp add: singleton_single) 
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546 
next 
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547 
case False 
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548 
have "eval (single x \<squnion> single y) x" 
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549 
and "eval (single x \<squnion> single y) y" 
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550 
by (auto intro: supI1 supI2 singleI) 
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551 
with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)" 
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diff
changeset

552 
by blast 
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553 
then have "singleton (single x \<squnion> single y) = undefined" 
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diff
changeset

554 
by (rule singleton_undefinedI) 
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555 
with False show ?thesis by simp 
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diff
changeset

556 
qed 
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diff
changeset

557 

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changeset

558 
lemma singleton_sup_aux: 
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559 
"singleton (A \<squnion> B) = (if A = \<bottom> then singleton B 
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changeset

560 
else if B = \<bottom> then singleton A 
22117a76f943
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diff
changeset

561 
else singleton 
22117a76f943
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diff
changeset

562 
(single (singleton A) \<squnion> single (singleton B)))" 
22117a76f943
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563 
proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)") 
22117a76f943
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diff
changeset

564 
case True then show ?thesis by (simp add: single_singleton) 
22117a76f943
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diff
changeset

565 
next 
22117a76f943
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diff
changeset

566 
case False 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
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diff
changeset

567 
from False have A_or_B: 
22117a76f943
added emptiness check predicate and singleton projection
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parents:
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diff
changeset

568 
"singleton A = undefined \<or> singleton B = undefined" 
22117a76f943
added emptiness check predicate and singleton projection
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diff
changeset

569 
by (auto intro!: singleton_undefinedI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
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diff
changeset

570 
then have rhs: "singleton 
22117a76f943
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diff
changeset

571 
(single (singleton A) \<squnion> single (singleton B)) = undefined" 
22117a76f943
added emptiness check predicate and singleton projection
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diff
changeset

572 
by (auto simp add: singleton_sup_single_single singleton_single) 
22117a76f943
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diff
changeset

573 
from False have not_unique: 
22117a76f943
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changeset

574 
"\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp 
22117a76f943
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changeset

575 
show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>") 
22117a76f943
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changeset

576 
case True 
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changeset

577 
then obtain a b where a: "eval A a" and b: "eval B b" 
22117a76f943
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diff
changeset

578 
by (blast elim: not_bot) 
22117a76f943
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diff
changeset

579 
with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)" 
22117a76f943
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changeset

580 
by (auto simp add: sup_pred_def bot_pred_def) 
22117a76f943
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changeset

581 
then have "singleton (A \<squnion> B) = undefined" by (rule singleton_undefinedI) 
22117a76f943
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changeset

582 
with True rhs show ?thesis by simp 
22117a76f943
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haftmann
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diff
changeset

583 
next 
22117a76f943
added emptiness check predicate and singleton projection
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diff
changeset

584 
case False then show ?thesis by auto 
22117a76f943
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diff
changeset

585 
qed 
22117a76f943
added emptiness check predicate and singleton projection
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diff
changeset

586 
qed 
22117a76f943
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haftmann
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diff
changeset

587 

22117a76f943
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changeset

588 
lemma singleton_sup: 
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changeset

589 
"singleton (A \<squnion> B) = (if A = \<bottom> then singleton B 
22117a76f943
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diff
changeset

590 
else if B = \<bottom> then singleton A 
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diff
changeset

591 
else if singleton A = singleton B then singleton A else undefined)" 
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changeset

592 
using singleton_sup_aux [of A B] by (simp only: singleton_sup_single_single) 
22117a76f943
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593 

30328  594 

595 
subsubsection {* Derived operations *} 

596 

597 
definition if_pred :: "bool \<Rightarrow> unit pred" where 

598 
if_pred_eq: "if_pred b = (if b then single () else \<bottom>)" 

599 

600 
definition not_pred :: "unit pred \<Rightarrow> unit pred" where 

601 
not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())" 

602 

603 
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()" 

604 
unfolding if_pred_eq by (auto intro: singleI) 

605 

606 
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P" 

607 
unfolding if_pred_eq by (cases b) (auto elim: botE) 

608 

609 
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()" 

610 
unfolding not_pred_eq eval_pred by (auto intro: singleI) 

611 

612 
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()" 

613 
unfolding not_pred_eq by (auto intro: singleI) 

614 

615 
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

616 
unfolding not_pred_eq 

617 
by (auto split: split_if_asm elim: botE) 

618 

619 
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

620 
unfolding not_pred_eq 

621 
by (auto split: split_if_asm elim: botE) 

622 

623 

624 
subsubsection {* Implementation *} 

625 

626 
datatype 'a seq = Empty  Insert "'a" "'a pred"  Join "'a pred" "'a seq" 

627 

628 
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where 

629 
"pred_of_seq Empty = \<bottom>" 

630 
 "pred_of_seq (Insert x P) = single x \<squnion> P" 

631 
 "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq" 

632 

633 
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where 

634 
"Seq f = pred_of_seq (f ())" 

635 

636 
code_datatype Seq 

637 

638 
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool" where 

639 
"member Empty x \<longleftrightarrow> False" 

640 
 "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x" 

641 
 "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x" 

642 

643 
lemma eval_member: 

644 
"member xq = eval (pred_of_seq xq)" 

645 
proof (induct xq) 

646 
case Empty show ?case 

647 
by (auto simp add: expand_fun_eq elim: botE) 

648 
next 

649 
case Insert show ?case 

650 
by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI) 

651 
next 

652 
case Join then show ?case 

653 
by (auto simp add: expand_fun_eq elim: supE intro: supI1 supI2) 

654 
qed 

655 

656 
lemma eval_code [code]: "eval (Seq f) = member (f ())" 

657 
unfolding Seq_def by (rule sym, rule eval_member) 

658 

659 
lemma single_code [code]: 

660 
"single x = Seq (\<lambda>u. Insert x \<bottom>)" 

661 
unfolding Seq_def by simp 

662 

663 
primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where 

664 
"apply f Empty = Empty" 

665 
 "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)" 

666 
 "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)" 

667 

668 
lemma apply_bind: 

669 
"pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f" 

670 
proof (induct xq) 

671 
case Empty show ?case 

672 
by (simp add: bottom_bind) 

673 
next 

674 
case Insert show ?case 

675 
by (simp add: single_bind sup_bind) 

676 
next 

677 
case Join then show ?case 

678 
by (simp add: sup_bind) 

679 
qed 

680 

681 
lemma bind_code [code]: 

682 
"Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))" 

683 
unfolding Seq_def by (rule sym, rule apply_bind) 

684 

685 
lemma bot_set_code [code]: 

686 
"\<bottom> = Seq (\<lambda>u. Empty)" 

687 
unfolding Seq_def by simp 

688 

30376  689 
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where 
690 
"adjunct P Empty = Join P Empty" 

691 
 "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)" 

692 
 "adjunct P (Join Q xq) = Join Q (adjunct P xq)" 

693 

694 
lemma adjunct_sup: 

695 
"pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq" 

696 
by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute) 

697 

30328  698 
lemma sup_code [code]: 
699 
"Seq f \<squnion> Seq g = Seq (\<lambda>u. case f () 

700 
of Empty \<Rightarrow> g () 

701 
 Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g) 

30376  702 
 Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))" 
30328  703 
proof (cases "f ()") 
704 
case Empty 

705 
thus ?thesis 

30376  706 
unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"] sup_bot) 
30328  707 
next 
708 
case Insert 

709 
thus ?thesis 

710 
unfolding Seq_def by (simp add: sup_assoc) 

711 
next 

712 
case Join 

713 
thus ?thesis 

30376  714 
unfolding Seq_def 
715 
by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute) 

30328  716 
qed 
717 

30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
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diff
changeset

718 
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
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diff
changeset

719 
"contained Empty Q \<longleftrightarrow> True" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
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diff
changeset

720 
 "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
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diff
changeset

721 
 "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

722 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
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diff
changeset

723 
lemma single_less_eq_eval: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
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diff
changeset

724 
"single x \<le> P \<longleftrightarrow> eval P x" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

725 
by (auto simp add: single_def less_eq_pred_def mem_def) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

726 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
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diff
changeset

727 
lemma contained_less_eq: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

728 
"contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

729 
by (induct xq) (simp_all add: single_less_eq_eval) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
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diff
changeset

730 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

731 
lemma less_eq_pred_code [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

732 
"Seq f \<le> Q = (case f () 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

733 
of Empty \<Rightarrow> True 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

734 
 Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

735 
 Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

736 
by (cases "f ()") 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

737 
(simp_all add: Seq_def single_less_eq_eval contained_less_eq) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

738 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

739 
lemma eq_pred_code [code]: 
31133  740 
fixes P Q :: "'a pred" 
30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

741 
shows "eq_class.eq P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

742 
unfolding eq by auto 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

743 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

744 
lemma [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

745 
"pred_case f P = f (eval P)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

746 
by (cases P) simp 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

747 

42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

748 
lemma [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

749 
"pred_rec f P = f (eval P)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

750 
by (cases P) simp 
30328  751 

31105
95f66b234086
added general preprocessing of equality in predicates for code generation
bulwahn
parents:
30430
diff
changeset

752 
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x" 
95f66b234086
added general preprocessing of equality in predicates for code generation
bulwahn
parents:
30430
diff
changeset

753 

95f66b234086
added general preprocessing of equality in predicates for code generation
bulwahn
parents:
30430
diff
changeset

754 
lemma eq_is_eq: "eq x y \<equiv> (x = y)" 
31108  755 
by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases) 
30948  756 

31216  757 
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where 
758 
"map f P = P \<guillemotright>= (single o f)" 

759 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
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diff
changeset

760 
primrec null :: "'a seq \<Rightarrow> bool" where 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

761 
"null Empty \<longleftrightarrow> True" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

762 
 "null (Insert x P) \<longleftrightarrow> False" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

763 
 "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

764 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

765 
lemma null_is_empty: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

766 
"null xq \<longleftrightarrow> is_empty (pred_of_seq xq)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
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diff
changeset

767 
by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

768 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

769 
lemma is_empty_code [code]: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

770 
"is_empty (Seq f) \<longleftrightarrow> null (f ())" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

771 
by (simp add: null_is_empty Seq_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

772 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

773 
primrec the_only :: "'a seq \<Rightarrow> 'a" where 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

774 
[code del]: "the_only Empty = undefined" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

775 
 "the_only (Insert x P) = (if is_empty P then x else let y = singleton P in if x = y then x else undefined)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

776 
 "the_only (Join P xq) = (if is_empty P then the_only xq else if null xq then singleton P 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

777 
else let x = singleton P; y = the_only xq in 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

778 
if x = y then x else undefined)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

779 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

780 
lemma the_only_singleton: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

781 
"the_only xq = singleton (pred_of_seq xq)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

782 
by (induct xq) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

783 
(auto simp add: singleton_bot singleton_single is_empty_def 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

784 
null_is_empty Let_def singleton_sup) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

785 

22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

786 
lemma singleton_code [code]: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

787 
"singleton (Seq f) = (case f () 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

788 
of Empty \<Rightarrow> undefined 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

789 
 Insert x P \<Rightarrow> if is_empty P then x 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
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790 
else let y = singleton P in 
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791 
if x = y then x else undefined 
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792 
 Join P xq \<Rightarrow> if is_empty P then the_only xq 
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793 
else if null xq then singleton P 
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794 
else let x = singleton P; y = the_only xq in 
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795 
if x = y then x else undefined)" 
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796 
by (cases "f ()") 
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797 
(auto simp add: Seq_def the_only_singleton is_empty_def 
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798 
null_is_empty singleton_bot singleton_single singleton_sup Let_def) 
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799 

30948  800 
ML {* 
801 
signature PREDICATE = 

802 
sig 

803 
datatype 'a pred = Seq of (unit > 'a seq) 

804 
and 'a seq = Empty  Insert of 'a * 'a pred  Join of 'a pred * 'a seq 

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805 
val yield: 'a pred > ('a * 'a pred) option 
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806 
val yieldn: int > 'a pred > 'a list * 'a pred 
31222  807 
val map: ('a > 'b) > 'a pred > 'b pred 
30948  808 
end; 
809 

810 
structure Predicate : PREDICATE = 

811 
struct 

812 

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813 
@{code_datatype pred = Seq}; 
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814 
@{code_datatype seq = Empty  Insert  Join}; 
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815 

32372  816 
fun yield (@{code Seq} f) = next (f ()) 
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817 
and next @{code Empty} = NONE 
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818 
 next (@{code Insert} (x, P)) = SOME (x, P) 
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819 
 next (@{code Join} (P, xq)) = (case yield P 
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820 
of NONE => next xq 
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821 
 SOME (x, Q) => SOME (x, @{code Seq} (fn _ => @{code Join} (Q, xq)))) 
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822 

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823 
fun anamorph f k x = (if k = 0 then ([], x) 
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824 
else case f x 
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825 
of NONE => ([], x) 
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826 
 SOME (v, y) => let 
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827 
val (vs, z) = anamorph f (k  1) y 
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828 
in (v :: vs, z) end) 
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829 

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830 
fun yieldn P = anamorph yield P; 
30948  831 

31222  832 
fun map f = @{code map} f; 
833 

30948  834 
end; 
835 
*} 

836 

837 
code_reserved Eval Predicate 

838 

839 
code_type pred and seq 

840 
(Eval "_/ Predicate.pred" and "_/ Predicate.seq") 

841 

842 
code_const Seq and Empty and Insert and Join 

843 
(Eval "Predicate.Seq" and "Predicate.Empty" and "Predicate.Insert/ (_,/ _)" and "Predicate.Join/ (_,/ _)") 

844 

31122  845 
text {* dummy setup for @{text code_pred} and @{text values} keywords *} 
31108  846 

847 
ML {* 

31122  848 
local 
849 

850 
structure P = OuterParse; 

851 

852 
val opt_modes = Scan.optional (P.$$$ "("  P.!!! (Scan.repeat1 P.xname  P.$$$ ")")) []; 

853 

854 
in 

855 

856 
val _ = OuterSyntax.local_theory_to_proof "code_pred" "sets up goal for cases rule from given introduction rules and compiles predicate" 

857 
OuterKeyword.thy_goal (P.term_group >> (K (Proof.theorem_i NONE (K I) [[]]))); 

858 

31216  859 
val _ = OuterSyntax.improper_command "values" "enumerate and print comprehensions" 
31122  860 
OuterKeyword.diag ((opt_modes  P.term) 
861 
>> (fn (modes, t) => Toplevel.no_timing o Toplevel.keep 

862 
(K ()))); 

863 

864 
end 

31108  865 
*} 
30959
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866 

30328  867 
no_notation 
868 
inf (infixl "\<sqinter>" 70) and 

869 
sup (infixl "\<squnion>" 65) and 

870 
Inf ("\<Sqinter>_" [900] 900) and 

871 
Sup ("\<Squnion>_" [900] 900) and 

872 
top ("\<top>") and 

873 
bot ("\<bottom>") and 

874 
bind (infixl "\<guillemotright>=" 70) 

875 

876 
hide (open) type pred seq 

30378  877 
hide (open) const Pred eval single bind if_pred not_pred 
31216  878 
Empty Insert Join Seq member pred_of_seq "apply" adjunct eq map 
30328  879 

880 
end 