author  berghofe 
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permissions  rwrr 
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(* Title: HOL/Predicate.thy 
30328  2 
Author: Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen 
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*) 
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30328  5 
header {* Predicates as relations and enumerations *} 
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theory Predicate 
23708  8 
imports Inductive Relation 
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begin 
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30328  11 
notation 
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bot ("\<bottom>") and 
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top ("\<top>") and 

30328  14 
inf (infixl "\<sqinter>" 70) and 
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sup (infixl "\<squnion>" 65) and 

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

41082  17 
Sup ("\<Squnion>_" [900] 900) 
30328  18 

41080  19 
syntax (xsymbols) 
41082  20 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 
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"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

41080  22 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 
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"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

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30328  25 

26 
subsection {* Predicates as (complete) lattices *} 

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text {* 
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Handy introduction and elimination rules for @{text "\<le>"} 
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on unary and binary predicates 
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*} 
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lemma predicate1I: 
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assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" 
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shows "P \<le> Q" 
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apply (rule le_funI) 
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apply (rule le_boolI) 
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apply (rule PQ) 
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apply assumption 
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done 
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lemma predicate1D [Pure.dest?, dest?]: 
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"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" 
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apply (erule le_funE) 
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apply (erule le_boolE) 
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apply assumption+ 
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done 
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lemma rev_predicate1D: 
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"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x" 
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by (rule predicate1D) 
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lemma predicate2I [Pure.intro!, intro!]: 
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assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" 
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shows "P \<le> Q" 
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apply (rule le_funI)+ 
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apply (rule le_boolI) 
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apply (rule PQ) 
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apply assumption 
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done 
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lemma predicate2D [Pure.dest, dest]: 
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"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" 
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apply (erule le_funE)+ 
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apply (erule le_boolE) 
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apply assumption+ 
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done 
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lemma rev_predicate2D: 
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"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y" 
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by (rule predicate2D) 
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32779  74 
subsubsection {* Equality *} 
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lemma pred_equals_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R = S)" 
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by (simp add: set_eq_iff fun_eq_iff) 
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44415  79 
lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R = S)" 
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by (simp add: set_eq_iff fun_eq_iff) 
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32779  82 

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subsubsection {* Order relation *} 

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lemma pred_subset_eq [pred_set_conv]: "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R \<subseteq> S)" 
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by (simp add: subset_iff le_fun_def) 
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44415  88 
lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R \<subseteq> S)" 
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by (simp add: subset_iff le_fun_def) 
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30328  92 
subsubsection {* Top and bottom elements *} 
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44414  94 
lemma bot1E [no_atp, elim!]: "\<bottom> x \<Longrightarrow> P" 
41550  95 
by (simp add: bot_fun_def) 
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44414  97 
lemma bot2E [elim!]: "\<bottom> x y \<Longrightarrow> P" 
41550  98 
by (simp add: bot_fun_def) 
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44414  100 
lemma bot_empty_eq: "\<bottom> = (\<lambda>x. x \<in> {})" 
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by (auto simp add: fun_eq_iff) 
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44414  103 
lemma bot_empty_eq2: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})" 
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by (auto simp add: fun_eq_iff) 
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44414  106 
lemma top1I [intro!]: "\<top> x" 
41550  107 
by (simp add: top_fun_def) 
41082  108 

44414  109 
lemma top2I [intro!]: "\<top> x y" 
41550  110 
by (simp add: top_fun_def) 
41082  111 

112 

113 
subsubsection {* Binary intersection *} 

114 

44414  115 
lemma inf1I [intro!]: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x" 
41550  116 
by (simp add: inf_fun_def) 
41082  117 

44414  118 
lemma inf2I [intro!]: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y" 
41550  119 
by (simp add: inf_fun_def) 
41082  120 

44414  121 
lemma inf1E [elim!]: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P" 
41550  122 
by (simp add: inf_fun_def) 
41082  123 

44414  124 
lemma inf2E [elim!]: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P" 
41550  125 
by (simp add: inf_fun_def) 
41082  126 

44414  127 
lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x" 
41550  128 
by (simp add: inf_fun_def) 
41082  129 

44414  130 
lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y" 
41550  131 
by (simp add: inf_fun_def) 
41082  132 

44414  133 
lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x" 
41550  134 
by (simp add: inf_fun_def) 
41082  135 

44414  136 
lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y" 
41550  137 
by (simp add: inf_fun_def) 
41082  138 

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lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)" 
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by (simp add: inf_fun_def) 
41082  141 

44414  142 
lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)" 
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by (simp add: inf_fun_def) 
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30328  146 
subsubsection {* Binary union *} 
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lemma sup1I1 [intro?]: "A x \<Longrightarrow> (A \<squnion> B) x" 
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by (simp add: sup_fun_def) 
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lemma sup2I1 [intro?]: "A x y \<Longrightarrow> (A \<squnion> B) x y" 
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by (simp add: sup_fun_def) 
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lemma sup1I2 [intro?]: "B x \<Longrightarrow> (A \<squnion> B) x" 
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by (simp add: sup_fun_def) 
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lemma sup2I2 [intro?]: "B x y \<Longrightarrow> (A \<squnion> B) x y" 
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by (simp add: sup_fun_def) 
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44414  160 
lemma sup1E [elim!]: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P" 
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by (simp add: sup_fun_def) iprover 
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44414  163 
lemma sup2E [elim!]: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P" 
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by (simp add: sup_fun_def) iprover 
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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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44414  171 
lemma sup1CI [intro!]: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x" 
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by (auto simp add: sup_fun_def) 
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44414  174 
lemma sup2CI [intro!]: "(\<not> B x y \<Longrightarrow> A x y) \<Longrightarrow> (A \<squnion> B) x y" 
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by (auto simp add: sup_fun_def) 
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lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)" 
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by (simp add: sup_fun_def) 
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lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)" 
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by (simp add: sup_fun_def) 
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30328  184 
subsubsection {* Intersections of families *} 
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lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)" 
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by (simp add: INF_apply) 
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lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)" 
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by (simp add: INF_apply) 
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lemma INF1_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b" 
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by (auto simp add: INF_apply) 
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lemma INF2_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c" 
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by (auto simp add: INF_apply) 
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lemma INF1_D [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b" 
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by (auto simp add: INF_apply) 
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lemma INF2_D [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c" 
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by (auto simp add: INF_apply) 
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lemma INF1_E [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" 
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by (auto simp add: INF_apply) 
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lemma INF2_E [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R" 
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by (auto simp add: INF_apply) 
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lemma INF_INT_eq: "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Sqinter>i. r i))" 
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by (simp add: INF_apply fun_eq_iff) 
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lemma INF_INT_eq2: "(\<Sqinter>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Sqinter>i. r i))" 
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by (simp add: INF_apply fun_eq_iff) 
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41082  217 
subsubsection {* Unions of families *} 
218 

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lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)" 
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by (simp add: SUP_apply) 
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lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)" 
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by (simp add: SUP_apply) 
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lemma SUP1_I [intro]: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b" 
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by (auto simp add: SUP_apply) 
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lemma SUP2_I [intro]: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c" 
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by (auto simp add: SUP_apply) 
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lemma SUP1_E [elim!]: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R" 
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by (auto simp add: SUP_apply) 
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lemma SUP2_E [elim!]: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R" 
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by (auto simp add: SUP_apply) 
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lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i. r i))" 
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by (simp add: SUP_apply fun_eq_iff) 
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lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i. r i))" 
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by (simp add: SUP_apply fun_eq_iff) 
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243 

30328  244 
subsection {* Predicates as relations *} 
245 

246 
subsubsection {* Composition *} 

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44414  248 
inductive pred_comp :: "['a \<Rightarrow> 'b \<Rightarrow> bool, 'b \<Rightarrow> 'c \<Rightarrow> bool] \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75) 
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for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where 

250 
pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (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: fun_eq_iff) 
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30328  259 
subsubsection {* Converse *} 
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inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^1)" [1000] 1000) 
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for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where 

263 
conversepI: "r a b \<Longrightarrow> 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: fun_eq_iff) 
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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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44414  287 
lemma converse_meet: "(r \<sqinter> s)^1 = r^1 \<sqinter> s^1" 
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by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) 
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44414  290 
lemma converse_join: "(r \<squnion> s)^1 = r^1 \<squnion> s^1" 
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by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) 
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lemma conversep_noteq [simp]: "(op \<noteq>)^1 = op \<noteq>" 
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by (auto simp add: fun_eq_iff) 
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lemma conversep_eq [simp]: "(op =)^1 = op =" 
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by (auto simp add: fun_eq_iff) 
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30328  300 
subsubsection {* Domain *} 
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44414  302 
inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" 
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for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where 

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DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a" 

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inductive_cases DomainPE [elim!]: "DomainP r a" 
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307 

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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  312 
subsubsection {* Range *} 
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44414  314 
inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" 
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for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where 

316 
RangePI [intro]: "r a b \<Longrightarrow> 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  324 
subsubsection {* Inverse image *} 
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44414  326 
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where 
327 
"inv_imagep r f = (\<lambda>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  336 
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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340 

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lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)" 
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by (auto simp add: Powp_def fun_eq_iff) 
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lemmas Powp_mono [mono] = Pow_mono [to_pred] 
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345 

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346 

30328  347 
subsubsection {* Properties of relations *} 
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348 

44414  349 
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
350 
"antisymP r \<equiv> antisym {(x, y). r x y}" 

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351 

44414  352 
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
353 
"transP r \<equiv> trans {(x, y). r x y}" 

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354 

44414  355 
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
356 
"single_valuedP r \<equiv> single_valued {(x, y). r x y}" 

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357 

40813
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358 
(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*) 
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359 

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360 
definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
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361 
"reflp r \<longleftrightarrow> refl {(x, y). r x y}" 
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362 

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363 
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
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364 
"symp r \<longleftrightarrow> sym {(x, y). r x y}" 
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365 

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366 
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where 
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367 
"transp r \<longleftrightarrow> trans {(x, y). r x y}" 
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368 

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369 
lemma reflpI: 
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370 
"(\<And>x. r x x) \<Longrightarrow> reflp r" 
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371 
by (auto intro: refl_onI simp add: reflp_def) 
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372 

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373 
lemma reflpE: 
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374 
assumes "reflp r" 
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375 
obtains "r x x" 
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376 
using assms by (auto dest: refl_onD simp add: reflp_def) 
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377 

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378 
lemma sympI: 
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379 
"(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r" 
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380 
by (auto intro: symI simp add: symp_def) 
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381 

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382 
lemma sympE: 
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383 
assumes "symp r" and "r x y" 
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384 
obtains "r y x" 
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385 
using assms by (auto dest: symD simp add: symp_def) 
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386 

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387 
lemma transpI: 
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388 
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r" 
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389 
by (auto intro: transI simp add: transp_def) 
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390 

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391 
lemma transpE: 
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392 
assumes "transp r" and "r x y" and "r y z" 
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393 
obtains "r x z" 
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394 
using assms by (auto dest: transD simp add: transp_def) 
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395 

30328  396 

397 
subsection {* Predicates as enumerations *} 

398 

399 
subsubsection {* The type of predicate enumerations (a monad) *} 

400 

401 
datatype 'a pred = Pred "'a \<Rightarrow> bool" 

402 

403 
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where 

404 
eval_pred: "eval (Pred f) = f" 

405 

406 
lemma Pred_eval [simp]: 

407 
"Pred (eval x) = x" 

408 
by (cases x) simp 

409 

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410 
lemma pred_eqI: 
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411 
"(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q" 
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412 
by (cases P, cases Q) (auto simp add: fun_eq_iff) 
30328  413 

46038
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414 
lemma pred_eq_iff: 
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415 
"P = Q \<Longrightarrow> (\<And>w. eval P w \<longleftrightarrow> eval Q w)" 
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416 
by (simp add: pred_eqI) 
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417 

44033  418 
instantiation pred :: (type) complete_lattice 
30328  419 
begin 
420 

421 
definition 

422 
"P \<le> Q \<longleftrightarrow> eval P \<le> eval Q" 

423 

424 
definition 

425 
"P < Q \<longleftrightarrow> eval P < eval Q" 

426 

427 
definition 

428 
"\<bottom> = Pred \<bottom>" 

429 

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430 
lemma eval_bot [simp]: 
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431 
"eval \<bottom> = \<bottom>" 
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432 
by (simp add: bot_pred_def) 
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433 

30328  434 
definition 
435 
"\<top> = Pred \<top>" 

436 

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437 
lemma eval_top [simp]: 
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438 
"eval \<top> = \<top>" 
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439 
by (simp add: top_pred_def) 
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440 

30328  441 
definition 
442 
"P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)" 

443 

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444 
lemma eval_inf [simp]: 
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445 
"eval (P \<sqinter> Q) = eval P \<sqinter> eval Q" 
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446 
by (simp add: inf_pred_def) 
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447 

30328  448 
definition 
449 
"P \<squnion> Q = Pred (eval P \<squnion> eval Q)" 

450 

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451 
lemma eval_sup [simp]: 
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452 
"eval (P \<squnion> Q) = eval P \<squnion> eval Q" 
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453 
by (simp add: sup_pred_def) 
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454 

30328  455 
definition 
37767  456 
"\<Sqinter>A = Pred (INFI A eval)" 
30328  457 

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458 
lemma eval_Inf [simp]: 
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459 
"eval (\<Sqinter>A) = INFI A eval" 
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460 
by (simp add: Inf_pred_def) 
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461 

30328  462 
definition 
37767  463 
"\<Squnion>A = Pred (SUPR A eval)" 
30328  464 

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465 
lemma eval_Sup [simp]: 
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466 
"eval (\<Squnion>A) = SUPR A eval" 
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467 
by (simp add: Sup_pred_def) 
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468 

44033  469 
instance proof 
44415  470 
qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def) 
44033  471 

472 
end 

473 

474 
lemma eval_INFI [simp]: 

475 
"eval (INFI A f) = INFI A (eval \<circ> f)" 

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476 
by (simp only: INF_def eval_Inf image_compose) 
44033  477 

478 
lemma eval_SUPR [simp]: 

479 
"eval (SUPR A f) = SUPR A (eval \<circ> f)" 

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480 
by (simp only: SUP_def eval_Sup image_compose) 
44033  481 

482 
instantiation pred :: (type) complete_boolean_algebra 

483 
begin 

484 

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485 
definition 
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486 
" P = Pred ( eval P)" 
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487 

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488 
lemma eval_compl [simp]: 
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489 
"eval ( P) =  eval P" 
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490 
by (simp add: uminus_pred_def) 
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491 

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492 
definition 
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493 
"P  Q = Pred (eval P  eval Q)" 
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494 

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495 
lemma eval_minus [simp]: 
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496 
"eval (P  Q) = eval P  eval Q" 
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497 
by (simp add: minus_pred_def) 
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498 

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499 
instance proof 
44415  500 
qed (auto intro!: pred_eqI simp add: uminus_apply minus_apply INF_apply SUP_apply) 
30328  501 

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502 
end 
30328  503 

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504 
definition single :: "'a \<Rightarrow> 'a pred" where 
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505 
"single x = Pred ((op =) x)" 
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506 

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507 
lemma eval_single [simp]: 
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508 
"eval (single x) = (op =) x" 
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509 
by (simp add: single_def) 
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510 

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511 
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where 
41080  512 
"P \<guillemotright>= f = (SUPR {x. eval P x} f)" 
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513 

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514 
lemma eval_bind [simp]: 
41080  515 
"eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)" 
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516 
by (simp add: bind_def) 
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517 

30328  518 
lemma bind_bind: 
519 
"(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)" 

44415  520 
by (rule pred_eqI) (auto simp add: SUP_apply) 
30328  521 

522 
lemma bind_single: 

523 
"P \<guillemotright>= single = P" 

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524 
by (rule pred_eqI) auto 
30328  525 

526 
lemma single_bind: 

527 
"single x \<guillemotright>= P = P x" 

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528 
by (rule pred_eqI) auto 
30328  529 

530 
lemma bottom_bind: 

531 
"\<bottom> \<guillemotright>= P = \<bottom>" 

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532 
by (rule pred_eqI) auto 
30328  533 

534 
lemma sup_bind: 

535 
"(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R" 

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536 
by (rule pred_eqI) auto 
30328  537 

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538 
lemma Sup_bind: 
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539 
"(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)" 
44415  540 
by (rule pred_eqI) (auto simp add: SUP_apply) 
30328  541 

542 
lemma pred_iffI: 

543 
assumes "\<And>x. eval A x \<Longrightarrow> eval B x" 

544 
and "\<And>x. eval B x \<Longrightarrow> eval A x" 

545 
shows "A = B" 

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546 
using assms by (auto intro: pred_eqI) 
30328  547 

548 
lemma singleI: "eval (single x) x" 

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549 
by simp 
30328  550 

551 
lemma singleI_unit: "eval (single ()) x" 

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552 
by simp 
30328  553 

554 
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

555 
by simp 
30328  556 

557 
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

558 
by simp 
30328  559 

560 
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

561 
by auto 
30328  562 

563 
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

564 
by auto 
30328  565 

566 
lemma botE: "eval \<bottom> x \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

567 
by auto 
30328  568 

569 
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

570 
by auto 
30328  571 

572 
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

573 
by auto 
30328  574 

575 
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P" 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

576 
by auto 
30328  577 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

578 
lemma single_not_bot [simp]: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

579 
"single x \<noteq> \<bottom>" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

580 
by (auto simp add: single_def bot_pred_def fun_eq_iff) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

581 

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

582 
lemma not_bot: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

583 
assumes "A \<noteq> \<bottom>" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

584 
obtains x where "eval A x" 
45970
b6d0cff57d96
adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents:
45630
diff
changeset

585 
using assms by (cases A) (auto simp add: bot_pred_def) 
b6d0cff57d96
adjusted to set/pred distinction by means of type constructor `set`
haftmann
parents:
45630
diff
changeset

586 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

587 

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

588 
subsubsection {* Emptiness check and definite choice *} 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

589 

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

590 
definition is_empty :: "'a pred \<Rightarrow> bool" where 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

591 
"is_empty A \<longleftrightarrow> A = \<bottom>" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

592 

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

593 
lemma is_empty_bot: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

594 
"is_empty \<bottom>" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

595 
by (simp add: is_empty_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

596 

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

597 
lemma not_is_empty_single: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

598 
"\<not> is_empty (single x)" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

599 
by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

600 

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

601 
lemma is_empty_sup: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

602 
"is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B" 
36008  603 
by (auto simp add: is_empty_def) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

604 

40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

605 
definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where 
33111  606 
"singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

607 

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

608 
lemma singleton_eqI: 
33110  609 
"\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

610 
by (auto simp add: singleton_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

611 

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

612 
lemma eval_singletonI: 
33110  613 
"\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

614 
proof  
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

615 
assume assm: "\<exists>!x. eval A x" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

616 
then obtain x where "eval A x" .. 
33110  617 
moreover with assm have "singleton dfault A = x" by (rule singleton_eqI) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

618 
ultimately show ?thesis by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

619 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

620 

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

621 
lemma single_singleton: 
33110  622 
"\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

623 
proof  
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

624 
assume assm: "\<exists>!x. eval A x" 
33110  625 
then have "eval A (singleton dfault A)" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

626 
by (rule eval_singletonI) 
33110  627 
moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

628 
by (rule singleton_eqI) 
33110  629 
ultimately have "eval (single (singleton dfault A)) = eval A" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

630 
by (simp (no_asm_use) add: single_def fun_eq_iff) blast 
40616
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

631 
then have "\<And>x. eval (single (singleton dfault A)) x = eval A x" 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

632 
by simp 
c5ee1e06d795
eval simp rules for predicate type, simplify primitive proofs
haftmann
parents:
39302
diff
changeset

633 
then show ?thesis by (rule pred_eqI) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

634 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

635 

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

636 
lemma singleton_undefinedI: 
33111  637 
"\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

638 
by (simp add: singleton_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

639 

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

640 
lemma singleton_bot: 
33111  641 
"singleton dfault \<bottom> = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

642 
by (auto simp add: bot_pred_def intro: singleton_undefinedI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

643 

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

644 
lemma singleton_single: 
33110  645 
"singleton dfault (single x) = x" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

646 
by (auto simp add: intro: singleton_eqI singleI elim: singleE) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

647 

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

648 
lemma singleton_sup_single_single: 
33111  649 
"singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

650 
proof (cases "x = y") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

651 
case True then show ?thesis by (simp add: singleton_single) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

652 
next 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

653 
case False 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

654 
have "eval (single x \<squnion> single y) x" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

655 
and "eval (single x \<squnion> single y) y" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

656 
by (auto intro: supI1 supI2 singleI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

657 
with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

658 
by blast 
33111  659 
then have "singleton dfault (single x \<squnion> single y) = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

660 
by (rule singleton_undefinedI) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

661 
with False show ?thesis by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

662 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

663 

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

664 
lemma singleton_sup_aux: 
33110  665 
"singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B 
666 
else if B = \<bottom> then singleton dfault A 

667 
else singleton dfault 

668 
(single (singleton dfault A) \<squnion> single (singleton dfault B)))" 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

669 
proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

670 
case True then show ?thesis by (simp add: single_singleton) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

671 
next 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

672 
case False 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

673 
from False have A_or_B: 
33111  674 
"singleton dfault A = dfault () \<or> singleton dfault B = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

675 
by (auto intro!: singleton_undefinedI) 
33110  676 
then have rhs: "singleton dfault 
33111  677 
(single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

678 
by (auto simp add: singleton_sup_single_single singleton_single) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

679 
from False have not_unique: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

680 
"\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

681 
show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

682 
case True 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

683 
then obtain a b where a: "eval A a" and b: "eval B b" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

684 
by (blast elim: not_bot) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

685 
with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

686 
by (auto simp add: sup_pred_def bot_pred_def) 
33111  687 
then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

688 
with True rhs show ?thesis by simp 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

689 
next 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

690 
case False then show ?thesis by auto 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

691 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

692 
qed 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

693 

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

694 
lemma singleton_sup: 
33110  695 
"singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B 
696 
else if B = \<bottom> then singleton dfault A 

33111  697 
else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())" 
33110  698 
using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single) 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

699 

30328  700 

701 
subsubsection {* Derived operations *} 

702 

703 
definition if_pred :: "bool \<Rightarrow> unit pred" where 

704 
if_pred_eq: "if_pred b = (if b then single () else \<bottom>)" 

705 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

706 
definition holds :: "unit pred \<Rightarrow> bool" where 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

707 
holds_eq: "holds P = eval P ()" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

708 

30328  709 
definition not_pred :: "unit pred \<Rightarrow> unit pred" where 
710 
not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())" 

711 

712 
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()" 

713 
unfolding if_pred_eq by (auto intro: singleI) 

714 

715 
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P" 

716 
unfolding if_pred_eq by (cases b) (auto elim: botE) 

717 

718 
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()" 

719 
unfolding not_pred_eq eval_pred by (auto intro: singleI) 

720 

721 
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()" 

722 
unfolding not_pred_eq by (auto intro: singleI) 

723 

724 
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

725 
unfolding not_pred_eq 

726 
by (auto split: split_if_asm elim: botE) 

727 

728 
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" 

729 
unfolding not_pred_eq 

730 
by (auto split: split_if_asm elim: botE) 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

731 
lemma "f () = False \<or> f () = True" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

732 
by simp 
30328  733 

37549  734 
lemma closure_of_bool_cases [no_atp]: 
44007  735 
fixes f :: "unit \<Rightarrow> bool" 
736 
assumes "f = (\<lambda>u. False) \<Longrightarrow> P f" 

737 
assumes "f = (\<lambda>u. True) \<Longrightarrow> P f" 

738 
shows "P f" 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

739 
proof  
44007  740 
have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)" 
33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

741 
apply (cases "f ()") 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

742 
apply (rule disjI2) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

743 
apply (rule ext) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

744 
apply (simp add: unit_eq) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

745 
apply (rule disjI1) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

746 
apply (rule ext) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

747 
apply (simp add: unit_eq) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

748 
done 
41550  749 
from this assms show ?thesis by blast 
33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

750 
qed 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

751 

f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

752 
lemma unit_pred_cases: 
44007  753 
assumes "P \<bottom>" 
754 
assumes "P (single ())" 

755 
shows "P Q" 

44415  756 
using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q) 
44007  757 
fix f 
758 
assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))" 

759 
then have "P (Pred f)" 

760 
by (cases _ f rule: closure_of_bool_cases) simp_all 

761 
moreover assume "Q = Pred f" 

762 
ultimately show "P Q" by simp 

763 
qed 

764 

33754
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

765 
lemma holds_if_pred: 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

766 
"holds (if_pred b) = b" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

767 
unfolding if_pred_eq holds_eq 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

768 
by (cases b) (auto intro: singleI elim: botE) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

769 

f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

770 
lemma if_pred_holds: 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

771 
"if_pred (holds P) = P" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

772 
unfolding if_pred_eq holds_eq 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

773 
by (rule unit_pred_cases) (auto intro: singleI elim: botE) 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

774 

f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

775 
lemma is_empty_holds: 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

776 
"is_empty P \<longleftrightarrow> \<not> holds P" 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

777 
unfolding is_empty_def holds_eq 
f2957bd46faf
adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents:
33622
diff
changeset

778 
by (rule unit_pred_cases) (auto elim: botE intro: singleI) 
30328  779 

41311  780 
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where 
781 
"map f P = P \<guillemotright>= (single o f)" 

782 

783 
lemma eval_map [simp]: 

44363  784 
"eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))" 
44415  785 
by (auto simp add: map_def comp_def) 
41311  786 

41505
6d19301074cf
"enriched_type" replaces less specific "type_lifting"
haftmann
parents:
41372
diff
changeset

787 
enriched_type map: map 
44363  788 
by (rule ext, rule pred_eqI, auto)+ 
41311  789 

790 

30328  791 
subsubsection {* Implementation *} 
792 

793 
datatype 'a seq = Empty  Insert "'a" "'a pred"  Join "'a pred" "'a seq" 

794 

795 
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where 

44414  796 
"pred_of_seq Empty = \<bottom>" 
797 
 "pred_of_seq (Insert x P) = single x \<squnion> P" 

798 
 "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq" 

30328  799 

800 
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where 

801 
"Seq f = pred_of_seq (f ())" 

802 

803 
code_datatype Seq 

804 

805 
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool" where 

806 
"member Empty x \<longleftrightarrow> False" 

44414  807 
 "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x" 
808 
 "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x" 

30328  809 

810 
lemma eval_member: 

811 
"member xq = eval (pred_of_seq xq)" 

812 
proof (induct xq) 

813 
case Empty show ?case 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

814 
by (auto simp add: fun_eq_iff elim: botE) 
30328  815 
next 
816 
case Insert show ?case 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

817 
by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI) 
30328  818 
next 
819 
case Join then show ?case 

39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

820 
by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2) 
30328  821 
qed 
822 

46038
bb2f7488a0f1
conversions from sets to predicates and vice versa; extensionality on predicates
haftmann
parents:
45970
diff
changeset

823 
lemma eval_code [(* FIXME declare simp *)code]: "eval (Seq f) = member (f ())" 
30328  824 
unfolding Seq_def by (rule sym, rule eval_member) 
825 

826 
lemma single_code [code]: 

827 
"single x = Seq (\<lambda>u. Insert x \<bottom>)" 

828 
unfolding Seq_def by simp 

829 

41080  830 
primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where 
44415  831 
"apply f Empty = Empty" 
832 
 "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)" 

833 
 "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)" 

30328  834 

835 
lemma apply_bind: 

836 
"pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f" 

837 
proof (induct xq) 

838 
case Empty show ?case 

839 
by (simp add: bottom_bind) 

840 
next 

841 
case Insert show ?case 

842 
by (simp add: single_bind sup_bind) 

843 
next 

844 
case Join then show ?case 

845 
by (simp add: sup_bind) 

846 
qed 

847 

848 
lemma bind_code [code]: 

849 
"Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))" 

850 
unfolding Seq_def by (rule sym, rule apply_bind) 

851 

852 
lemma bot_set_code [code]: 

853 
"\<bottom> = Seq (\<lambda>u. Empty)" 

854 
unfolding Seq_def by simp 

855 

30376  856 
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where 
44415  857 
"adjunct P Empty = Join P Empty" 
858 
 "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)" 

859 
 "adjunct P (Join Q xq) = Join Q (adjunct P xq)" 

30376  860 

861 
lemma adjunct_sup: 

862 
"pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq" 

863 
by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute) 

864 

30328  865 
lemma sup_code [code]: 
866 
"Seq f \<squnion> Seq g = Seq (\<lambda>u. case f () 

867 
of Empty \<Rightarrow> g () 

868 
 Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g) 

30376  869 
 Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))" 
30328  870 
proof (cases "f ()") 
871 
case Empty 

872 
thus ?thesis 

34007
aea892559fc5
tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents:
33988
diff
changeset

873 
unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"]) 
30328  874 
next 
875 
case Insert 

876 
thus ?thesis 

877 
unfolding Seq_def by (simp add: sup_assoc) 

878 
next 

879 
case Join 

880 
thus ?thesis 

30376  881 
unfolding Seq_def 
882 
by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute) 

30328  883 
qed 
884 

30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

885 
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where 
44415  886 
"contained Empty Q \<longleftrightarrow> True" 
887 
 "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q" 

888 
 "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q" 

30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

889 

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

890 
lemma single_less_eq_eval: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

891 
"single x \<le> P \<longleftrightarrow> eval P x" 
44415  892 
by (auto simp add: less_eq_pred_def le_fun_def) 
30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

893 

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

894 
lemma contained_less_eq: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

895 
"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

896 
by (induct xq) (simp_all add: single_less_eq_eval) 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

897 

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

898 
lemma less_eq_pred_code [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

899 
"Seq f \<le> Q = (case f () 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

900 
of Empty \<Rightarrow> True 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

901 
 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

902 
 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

903 
by (cases "f ()") 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

904 
(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

905 

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

906 
lemma eq_pred_code [code]: 
31133  907 
fixes P Q :: "'a pred" 
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

908 
shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P" 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

909 
by (auto simp add: equal) 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

910 

97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

911 
lemma [code nbe]: 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

912 
"HOL.equal (x :: 'a pred) x \<longleftrightarrow> True" 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38651
diff
changeset

913 
by (fact equal_refl) 
30430
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

914 

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

915 
lemma [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

916 
"pred_case f P = f (eval P)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

917 
by (cases P) simp 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

918 

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

919 
lemma [code]: 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

920 
"pred_rec f P = f (eval P)" 
42ea5d85edcc
explicit code equations for some rarely used pred operations
haftmann
parents:
30378
diff
changeset

921 
by (cases P) simp 
30328  922 

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

923 
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

924 

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

925 
lemma eq_is_eq: "eq x y \<equiv> (x = y)" 
31108  926 
by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases) 
30948  927 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

928 
primrec null :: "'a seq \<Rightarrow> bool" where 
44415  929 
"null Empty \<longleftrightarrow> True" 
930 
 "null (Insert x P) \<longleftrightarrow> False" 

931 
 "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq" 

32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

932 

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

933 
lemma null_is_empty: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

934 
"null xq \<longleftrightarrow> is_empty (pred_of_seq xq)" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

935 
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

936 

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

937 
lemma is_empty_code [code]: 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

938 
"is_empty (Seq f) \<longleftrightarrow> null (f ())" 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

939 
by (simp add: null_is_empty Seq_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

940 

33111  941 
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where 
942 
[code del]: "the_only dfault Empty = dfault ()" 

44415  943 
 "the_only dfault (Insert x P) = (if is_empty P then x else let y = singleton dfault P in if x = y then x else dfault ())" 
944 
 "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P 

33110  945 
else let x = singleton dfault P; y = the_only dfault xq in 
33111  946 
if x = y then x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

947 

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

948 
lemma the_only_singleton: 
33110  949 
"the_only dfault xq = singleton dfault (pred_of_seq xq)" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

950 
by (induct xq) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

951 
(auto simp add: singleton_bot singleton_single is_empty_def 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

952 
null_is_empty Let_def singleton_sup) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

953 

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

954 
lemma singleton_code [code]: 
33110  955 
"singleton dfault (Seq f) = (case f () 
33111  956 
of Empty \<Rightarrow> dfault () 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

957 
 Insert x P \<Rightarrow> if is_empty P then x 
33110  958 
else let y = singleton dfault P in 
33111  959 
if x = y then x else dfault () 
33110  960 
 Join P xq \<Rightarrow> if is_empty P then the_only dfault xq 
961 
else if null xq then singleton dfault P 

962 
else let x = singleton dfault P; y = the_only dfault xq in 

33111  963 
if x = y then x else dfault ())" 
32578
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

964 
by (cases "f ()") 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

965 
(auto simp add: Seq_def the_only_singleton is_empty_def 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

966 
null_is_empty singleton_bot singleton_single singleton_sup Let_def) 
22117a76f943
added emptiness check predicate and singleton projection
haftmann
parents:
32372
diff
changeset

967 

44414  968 
definition the :: "'a pred \<Rightarrow> 'a" where 
37767  969 
"the A = (THE x. eval A x)" 
33111  970 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

971 
lemma the_eqI: 
41080  972 
"(THE x. eval P x) = x \<Longrightarrow> the P = x" 
40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

973 
by (simp add: the_def) 
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

974 

44414  975 
definition not_unique :: "'a pred \<Rightarrow> 'a" where 
976 
[code del]: "not_unique A = (THE x. eval A x)" 

977 

978 
code_abort not_unique 

979 

40674
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

980 
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A" 
54dbe6a1c349
adhere established Collect/mem convention more closely
haftmann
parents:
40616
diff
changeset

981 
by (rule the_eqI) (simp add: singleton_def not_unique_def) 
33110  982 

36531
19f6e3b0d9b6
code_reflect: specify module name directly after keyword
haftmann
parents:
36513
diff
changeset

983 
code_reflect Predicate 
36513  984 
datatypes pred = Seq and seq = Empty  Insert  Join 
985 
functions map 

986 

30948  987 
ML {* 
988 
signature PREDICATE = 

989 
sig 

990 
datatype 'a pred = Seq of (unit > 'a seq) 

991 
and 'a seq = Empty  Insert of 'a * 'a pred  Join of 'a pred * 'a seq 

30959
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

992 
val yield: 'a pred > ('a * 'a pred) option 
458e55fd0a33
fixed compilation of predicate types in ML environment
haftmann
parents:
30948
diff
changeset

993 
val yieldn: int > 'a pred > 'a list * 'a pred 
31222  994 
val map: ('a > 'b) > 'a pred > 'b pred 
30948  995 
end; 
996 

997 
structure Predicate : PREDICATE = 

998 
struct 

999 

36513  1000 
datatype pred = datatype Predicate.pred 
1001 
datatype seq = datatype Predicate.seq 

1002 

1003 
fun map f = Predicate.map f; 

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1004 

36513  1005 
fun yield (Seq f) = next (f ()) 
1006 
and next Empty = NONE 

1007 
 next (Insert (x, P)) = SOME (x, P) 

1008 
 next (Join (P, xq)) = (case yield P 

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1009 
of NONE => next xq 
36513  1010 
 SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq)))); 
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1011 

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1012 
fun anamorph f k x = (if k = 0 then ([], x) 
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1013 
else case f x 
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1014 
of NONE => ([], x) 
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1015 
 SOME (v, y) => let 
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1016 
val (vs, z) = anamorph f (k  1) y 
33607  1017 
in (v :: vs, z) end); 
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1018 

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1019 
fun yieldn P = anamorph yield P; 
30948  1020 

1021 
end; 

1022 
*} 

1023 

46038
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1024 
text {* Conversion from and to sets *} 
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1025 

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1026 
definition pred_of_set :: "'a set \<Rightarrow> 'a pred" where 
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1027 
"pred_of_set = Pred \<circ> (\<lambda>A x. x \<in> A)" 
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1028 

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1029 
lemma eval_pred_of_set [simp]: 
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1030 
"eval (pred_of_set A) x \<longleftrightarrow> x \<in>A" 
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1031 
by (simp add: pred_of_set_def) 
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1032 

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1033 
definition set_of_pred :: "'a pred \<Rightarrow> 'a set" where 
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1034 
"set_of_pred = Collect \<circ> eval" 
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1035 

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1036 
lemma member_set_of_pred [simp]: 
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1037 
"x \<in> set_of_pred P \<longleftrightarrow> Predicate.eval P x" 
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1038 
by (simp add: set_of_pred_def) 
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1039 

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1040 
definition set_of_seq :: "'a seq \<Rightarrow> 'a set" where 
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1041 
"set_of_seq = set_of_pred \<circ> pred_of_seq" 
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1042 

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1043 
lemma member_set_of_seq [simp]: 
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1044 
"x \<in> set_of_seq xq = Predicate.member xq x" 
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1045 
by (simp add: set_of_seq_def eval_member) 
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1046 

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1047 
lemma of_pred_code [code]: 
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1048 
"set_of_pred (Predicate.Seq f) = (case f () of 
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1049 
Predicate.Empty \<Rightarrow> {} 
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1050 
 Predicate.Insert x P \<Rightarrow> insert x (set_of_pred P) 
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1051 
 Predicate.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)" 
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1052 
by (auto split: seq.split simp add: eval_code) 
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1053 

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1054 
lemma of_seq_code [code]: 
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1055 
"set_of_seq Predicate.Empty = {}" 
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1056 
"set_of_seq (Predicate.Insert x P) = insert x (set_of_pred P)" 
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1057 
"set_of_seq (Predicate.Join P xq) = set_of_pred P \<union> set_of_seq xq" 
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1058 
by auto 
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1059 

30328  1060 
no_notation 
41082  1061 
bot ("\<bottom>") and 
1062 
top ("\<top>") and 

30328  1063 
inf (infixl "\<sqinter>" 70) and 
1064 
sup (infixl "\<squnion>" 65) and 

1065 
Inf ("\<Sqinter>_" [900] 900) and 

1066 
Sup ("\<Squnion>_" [900] 900) and 

1067 
bind (infixl "\<guillemotright>=" 70) 

1068 

41080  1069 
no_syntax (xsymbols) 
41082  1070 
"_INF1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_./ _)" [0, 10] 10) 
1071 
"_INF" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10) 

41080  1072 
"_SUP1" :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_./ _)" [0, 10] 10) 
1073 
"_SUP" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10) 

1074 

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1075 
hide_type (open) pred seq 
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1076 
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds 
33111  1077 
Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the 
30328  1078 

1079 
end 