src/HOL/Predicate.thy
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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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  bot ("\<bottom>") and
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  top ("\<top>") and
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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)
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syntax (xsymbols)
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  "_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)
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  "_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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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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subsubsection {* Equality *}
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lemma pred_equals_eq: "((\<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 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)) \<longleftrightarrow> (R = S)"
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  by (simp add: set_eq_iff fun_eq_iff mem_def)
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subsubsection {* Order relation *}
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lemma pred_subset_eq: "((\<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 mem_def)
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lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"
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  by (simp add: subset_iff le_fun_def mem_def)
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subsubsection {* Top and bottom elements *}
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lemma bot1E [no_atp, elim!]: "\<bottom> x \<Longrightarrow> P"
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  by (simp add: bot_fun_def)
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lemma bot2E [elim!]: "\<bottom> x y \<Longrightarrow> P"
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  by (simp add: bot_fun_def)
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lemma bot_empty_eq: "\<bottom> = (\<lambda>x. x \<in> {})"
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  by (auto simp add: fun_eq_iff)
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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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lemma top1I [intro!]: "\<top> x"
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  by (simp add: top_fun_def)
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lemma top2I [intro!]: "\<top> x y"
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  by (simp add: top_fun_def)
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subsubsection {* Binary intersection *}
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lemma inf1I [intro!]: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x"
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  by (simp add: inf_fun_def)
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lemma inf2I [intro!]: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y"
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  by (simp add: inf_fun_def)
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lemma inf1E [elim!]: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P"
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  by (simp add: inf_fun_def)
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lemma inf2E [elim!]: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P"
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  by (simp add: inf_fun_def)
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lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x"
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  by (simp add: inf_fun_def)
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lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y"
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  by (simp add: inf_fun_def)
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lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x"
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  by (simp add: inf_fun_def)
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lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y"
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  by (simp add: inf_fun_def)
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lemma inf_Int_eq: "(\<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 mem_def)
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lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
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  by (simp add: inf_fun_def mem_def)
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subsubsection {* Binary union *}
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lemma sup1I1 [elim?]: "A x \<Longrightarrow> (A \<squnion> B) x"
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  by (simp add: sup_fun_def)
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lemma sup2I1 [elim?]: "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 [elim?]: "B x \<Longrightarrow> (A \<squnion> B) x"
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  by (simp add: sup_fun_def)
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lemma sup2I2 [elim?]: "B x y \<Longrightarrow> (A \<squnion> B) x y"
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  by (simp add: sup_fun_def)
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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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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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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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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: "(\<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 mem_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 mem_def)
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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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subsubsection {* Unions of families *}
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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: "(\<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: "(\<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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subsection {* Predicates as relations *}
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subsubsection {* Composition  *}
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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
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  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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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
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  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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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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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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subsubsection {* Domain *}
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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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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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subsubsection {* Range *}
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inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
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  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  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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subsubsection {* Inverse image *}
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definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
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  "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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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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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 pred_subset_eq]
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subsubsection {* Properties of relations *}
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abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "antisymP r \<equiv> antisym {(x, y). r x y}"
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abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "transP r \<equiv> trans {(x, y). r x y}"
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abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
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  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
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(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
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definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
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definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "symp r \<longleftrightarrow> sym {(x, y). r x y}"
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definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
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lemma reflpI:
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  "(\<And>x. r x x) \<Longrightarrow> reflp r"
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  by (auto intro: refl_onI simp add: reflp_def)
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lemma reflpE:
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  assumes "reflp r"
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  obtains "r x x"
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  using assms by (auto dest: refl_onD simp add: reflp_def)
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lemma sympI:
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  "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
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  by (auto intro: symI simp add: symp_def)
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lemma sympE:
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  assumes "symp r" and "r x y"
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  obtains "r y x"
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haftmann
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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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haftmann
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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
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   397
subsection {* Predicates as enumerations *}
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   398
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   399
subsubsection {* The type of predicate enumerations (a monad) *}
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   400
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   401
datatype 'a pred = Pred "'a \<Rightarrow> bool"
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   402
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   403
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
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   404
  eval_pred: "eval (Pred f) = f"
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haftmann
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diff changeset
   405
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   406
lemma Pred_eval [simp]:
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haftmann
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diff changeset
   407
  "Pred (eval x) = x"
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haftmann
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   408
  by (cases x) simp
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   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)
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   413
44033
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   414
instantiation pred :: (type) complete_lattice
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   415
begin
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   416
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   417
definition
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   418
  "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
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   419
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   420
definition
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   421
  "P < Q \<longleftrightarrow> eval P < eval Q"
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haftmann
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diff changeset
   422
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   423
definition
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   424
  "\<bottom> = Pred \<bottom>"
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diff changeset
   425
40616
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   426
lemma eval_bot [simp]:
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haftmann
parents: 39302
diff changeset
   427
  "eval \<bottom>  = \<bottom>"
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haftmann
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diff changeset
   428
  by (simp add: bot_pred_def)
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haftmann
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diff changeset
   429
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   430
definition
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haftmann
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   431
  "\<top> = Pred \<top>"
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diff changeset
   432
40616
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diff changeset
   433
lemma eval_top [simp]:
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haftmann
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diff changeset
   434
  "eval \<top>  = \<top>"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   435
  by (simp add: top_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   436
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   437
definition
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   438
  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   439
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   440
lemma eval_inf [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   441
  "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   442
  by (simp add: inf_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   443
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   444
definition
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   445
  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   446
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   447
lemma eval_sup [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   448
  "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   449
  by (simp add: sup_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   450
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   451
definition
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37549
diff changeset
   452
  "\<Sqinter>A = Pred (INFI A eval)"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   453
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   454
lemma eval_Inf [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   455
  "eval (\<Sqinter>A) = INFI A eval"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   456
  by (simp add: Inf_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   457
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   458
definition
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37549
diff changeset
   459
  "\<Squnion>A = Pred (SUPR A eval)"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   460
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   461
lemma eval_Sup [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   462
  "eval (\<Squnion>A) = SUPR A eval"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   463
  by (simp add: Sup_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   464
44033
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   465
instance proof
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   466
qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)
44033
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   467
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   468
end
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   469
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   470
lemma eval_INFI [simp]:
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   471
  "eval (INFI A f) = INFI A (eval \<circ> f)"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44415
diff changeset
   472
  by (simp only: INF_def eval_Inf image_compose)
44033
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   473
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   474
lemma eval_SUPR [simp]:
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   475
  "eval (SUPR A f) = SUPR A (eval \<circ> f)"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44415
diff changeset
   476
  by (simp only: SUP_def eval_Sup image_compose)
44033
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   477
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   478
instantiation pred :: (type) complete_boolean_algebra
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   479
begin
bc45393f497b more fine-granular instantiation
haftmann
parents: 44026
diff changeset
   480
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   481
definition
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   482
  "- P = Pred (- eval P)"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   483
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   484
lemma eval_compl [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   485
  "eval (- P) = - eval P"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   486
  by (simp add: uminus_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   487
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   488
definition
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   489
  "P - Q = Pred (eval P - eval Q)"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   490
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   491
lemma eval_minus [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   492
  "eval (P - Q) = eval P - eval Q"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   493
  by (simp add: minus_pred_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   494
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   495
instance proof
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   496
qed (auto intro!: pred_eqI simp add: uminus_apply minus_apply INF_apply SUP_apply)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   497
22259
476604be7d88 New theory for converting between predicates and sets.
berghofe
parents:
diff changeset
   498
end
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   499
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   500
definition single :: "'a \<Rightarrow> 'a pred" where
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   501
  "single x = Pred ((op =) x)"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   502
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   503
lemma eval_single [simp]:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   504
  "eval (single x) = (op =) x"
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   505
  by (simp add: single_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   506
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   507
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   508
  "P \<guillemotright>= f = (SUPR {x. eval P x} f)"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   509
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   510
lemma eval_bind [simp]:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   511
  "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   512
  by (simp add: bind_def)
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   513
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   514
lemma bind_bind:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   515
  "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   516
  by (rule pred_eqI) (auto simp add: SUP_apply)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   517
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   518
lemma bind_single:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   519
  "P \<guillemotright>= single = P"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   520
  by (rule pred_eqI) auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   521
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   522
lemma single_bind:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   523
  "single x \<guillemotright>= P = P x"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   524
  by (rule pred_eqI) auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   525
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   526
lemma bottom_bind:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   527
  "\<bottom> \<guillemotright>= P = \<bottom>"
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   528
  by (rule pred_eqI) auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   529
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   530
lemma sup_bind:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   531
  "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   532
  by (rule pred_eqI) auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   533
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   534
lemma Sup_bind:
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   535
  "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   536
  by (rule pred_eqI) (auto simp add: SUP_apply)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   537
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   538
lemma pred_iffI:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   539
  assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   540
  and "\<And>x. eval B x \<Longrightarrow> eval A x"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   541
  shows "A = B"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   542
  using assms by (auto intro: pred_eqI)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   543
  
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   544
lemma singleI: "eval (single x) x"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   545
  by simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   546
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   547
lemma singleI_unit: "eval (single ()) x"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   548
  by simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   549
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   550
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
   551
  by simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   552
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   553
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
   554
  by simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   555
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   556
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
   557
  by auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   558
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   559
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
   560
  by auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   561
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   562
lemma botE: "eval \<bottom> x \<Longrightarrow> P"
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   563
  by auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   564
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   565
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
   566
  by auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   567
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   568
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
   569
  by auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   570
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   571
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
   572
  by auto
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   573
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   574
lemma single_not_bot [simp]:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   575
  "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
   576
  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
   577
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   578
lemma not_bot:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   579
  assumes "A \<noteq> \<bottom>"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   580
  obtains x where "eval A x"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   581
  using assms by (cases A) (auto simp add: bot_pred_def, simp add: mem_def)
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   582
  
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   583
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   584
subsubsection {* Emptiness check and definite choice *}
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   585
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   586
definition is_empty :: "'a pred \<Rightarrow> bool" where
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   587
  "is_empty A \<longleftrightarrow> A = \<bottom>"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   588
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   589
lemma is_empty_bot:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   590
  "is_empty \<bottom>"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   591
  by (simp add: is_empty_def)
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 not_is_empty_single:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   594
  "\<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
   595
  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
   596
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   597
lemma is_empty_sup:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   598
  "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
36008
23dfa8678c7c add/change some lemmas about lattices
huffman
parents: 34065
diff changeset
   599
  by (auto simp add: is_empty_def)
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   600
40616
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   601
definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   602
  "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
   603
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   604
lemma singleton_eqI:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   605
  "\<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
   606
  by (auto simp add: singleton_def)
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 eval_singletonI:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   609
  "\<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
   610
proof -
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   611
  assume assm: "\<exists>!x. eval A x"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   612
  then obtain x where "eval A x" ..
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   613
  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
   614
  ultimately show ?thesis by simp 
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   615
qed
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   616
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   617
lemma single_singleton:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   618
  "\<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
   619
proof -
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   620
  assume assm: "\<exists>!x. eval A x"
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   621
  then have "eval A (singleton dfault A)"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   622
    by (rule eval_singletonI)
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   623
  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
   624
    by (rule singleton_eqI)
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   625
  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
   626
    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
   627
  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
   628
    by simp
c5ee1e06d795 eval simp rules for predicate type, simplify primitive proofs
haftmann
parents: 39302
diff changeset
   629
  then show ?thesis by (rule pred_eqI)
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   630
qed
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   631
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   632
lemma singleton_undefinedI:
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   633
  "\<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
   634
  by (simp add: singleton_def)
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_bot:
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   637
  "singleton dfault \<bottom> = dfault ()"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   638
  by (auto simp add: bot_pred_def intro: singleton_undefinedI)
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_single:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   641
  "singleton dfault (single x) = x"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   642
  by (auto simp add: intro: singleton_eqI singleI elim: singleE)
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_sup_single_single:
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   645
  "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
   646
proof (cases "x = y")
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   647
  case True then show ?thesis by (simp add: singleton_single)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   648
next
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   649
  case False
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   650
  have "eval (single x \<squnion> single y) x"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   651
    and "eval (single x \<squnion> single y) y"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   652
  by (auto intro: supI1 supI2 singleI)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   653
  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
   654
    by blast
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   655
  then have "singleton dfault (single x \<squnion> single y) = dfault ()"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   656
    by (rule singleton_undefinedI)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   657
  with False show ?thesis by simp
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   658
qed
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   659
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   660
lemma singleton_sup_aux:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   661
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   662
    else if B = \<bottom> then singleton dfault A
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   663
    else singleton dfault
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   664
      (single (singleton dfault A) \<squnion> single (singleton dfault B)))"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   665
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
   666
  case True then show ?thesis by (simp add: single_singleton)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   667
next
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   668
  case False
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   669
  from False have A_or_B:
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   670
    "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   671
    by (auto intro!: singleton_undefinedI)
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   672
  then have rhs: "singleton dfault
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   673
    (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   674
    by (auto simp add: singleton_sup_single_single singleton_single)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   675
  from False have not_unique:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   676
    "\<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
   677
  show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   678
    case True
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   679
    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
   680
      by (blast elim: not_bot)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   681
    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
   682
      by (auto simp add: sup_pred_def bot_pred_def)
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   683
    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
   684
    with True rhs show ?thesis by simp
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   685
  next
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   686
    case False then show ?thesis by auto
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   687
  qed
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   688
qed
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   689
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   690
lemma singleton_sup:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   691
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   692
    else if B = \<bottom> then singleton dfault A
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   693
    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   694
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
   695
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   696
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   697
subsubsection {* Derived operations *}
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   698
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   699
definition if_pred :: "bool \<Rightarrow> unit pred" where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   700
  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   701
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   702
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
   703
  holds_eq: "holds P = eval P ()"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   704
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   705
definition not_pred :: "unit pred \<Rightarrow> unit pred" where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   706
  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   707
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   708
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   709
  unfolding if_pred_eq by (auto intro: singleI)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   710
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   711
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   712
  unfolding if_pred_eq by (cases b) (auto elim: botE)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   713
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   714
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   715
  unfolding not_pred_eq eval_pred by (auto intro: singleI)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   716
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   717
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   718
  unfolding not_pred_eq by (auto intro: singleI)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   719
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   720
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   721
  unfolding not_pred_eq
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   722
  by (auto split: split_if_asm elim: botE)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   723
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   724
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   725
  unfolding not_pred_eq
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   726
  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
   727
lemma "f () = False \<or> f () = True"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   728
by simp
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   729
37549
a62f742f1d58 yields ill-typed ATP/metis proofs -- raus!
blanchet
parents: 36531
diff changeset
   730
lemma closure_of_bool_cases [no_atp]:
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   731
  fixes f :: "unit \<Rightarrow> bool"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   732
  assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   733
  assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   734
  shows "P f"
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   735
proof -
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   736
  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
   737
    apply (cases "f ()")
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   738
    apply (rule disjI2)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   739
    apply (rule ext)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   740
    apply (simp add: unit_eq)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   741
    apply (rule disjI1)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   742
    apply (rule ext)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   743
    apply (simp add: unit_eq)
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   744
    done
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 41505
diff changeset
   745
  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
   746
qed
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   747
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   748
lemma unit_pred_cases:
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   749
  assumes "P \<bottom>"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   750
  assumes "P (single ())"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   751
  shows "P Q"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   752
using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
44007
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   753
  fix f
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   754
  assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   755
  then have "P (Pred f)" 
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   756
    by (cases _ f rule: closure_of_bool_cases) simp_all
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   757
  moreover assume "Q = Pred f"
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   758
  ultimately show "P Q" by simp
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   759
qed
b5e7594061ce tuned proofs
haftmann
parents: 41550
diff changeset
   760
  
33754
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   761
lemma holds_if_pred:
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   762
  "holds (if_pred b) = b"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   763
unfolding if_pred_eq holds_eq
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   764
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
   765
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   766
lemma if_pred_holds:
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   767
  "if_pred (holds P) = P"
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   768
unfolding if_pred_eq holds_eq
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   769
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
   770
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   771
lemma is_empty_holds:
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   772
  "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
   773
unfolding is_empty_def holds_eq
f2957bd46faf adding derived constant Predicate.holds to Predicate theory; adopting the predicate compiler
bulwahn
parents: 33622
diff changeset
   774
by (rule unit_pred_cases) (auto elim: botE intro: singleI)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   775
41311
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   776
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   777
  "map f P = P \<guillemotright>= (single o f)"
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   778
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   779
lemma eval_map [simp]:
44363
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
   780
  "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   781
  by (auto simp add: map_def comp_def)
41311
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   782
41505
6d19301074cf "enriched_type" replaces less specific "type_lifting"
haftmann
parents: 41372
diff changeset
   783
enriched_type map: map
44363
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
   784
  by (rule ext, rule pred_eqI, auto)+
41311
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   785
de0c906dfe60 type_lifting for predicates
haftmann
parents: 41082
diff changeset
   786
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   787
subsubsection {* Implementation *}
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   788
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   789
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   790
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   791
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   792
  "pred_of_seq Empty = \<bottom>"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   793
| "pred_of_seq (Insert x P) = single x \<squnion> P"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   794
| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   795
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   796
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   797
  "Seq f = pred_of_seq (f ())"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   798
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   799
code_datatype Seq
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   800
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   801
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   802
  "member Empty x \<longleftrightarrow> False"
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   803
| "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   804
| "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   805
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   806
lemma eval_member:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   807
  "member xq = eval (pred_of_seq xq)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   808
proof (induct xq)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   809
  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
   810
  by (auto simp add: fun_eq_iff elim: botE)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   811
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   812
  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
   813
  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   814
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   815
  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
   816
  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   817
qed
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   818
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   819
lemma eval_code [code]: "eval (Seq f) = member (f ())"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   820
  unfolding Seq_def by (rule sym, rule eval_member)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   821
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   822
lemma single_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   823
  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   824
  unfolding Seq_def by simp
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   825
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   826
primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   827
  "apply f Empty = Empty"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   828
| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   829
| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   830
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   831
lemma apply_bind:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   832
  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   833
proof (induct xq)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   834
  case Empty show ?case
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   835
    by (simp add: bottom_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   836
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   837
  case Insert show ?case
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   838
    by (simp add: single_bind sup_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   839
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   840
  case Join then show ?case
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   841
    by (simp add: sup_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   842
qed
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   843
  
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   844
lemma bind_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   845
  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   846
  unfolding Seq_def by (rule sym, rule apply_bind)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   847
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   848
lemma bot_set_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   849
  "\<bottom> = Seq (\<lambda>u. Empty)"
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   850
  unfolding Seq_def by simp
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   851
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   852
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   853
  "adjunct P Empty = Join P Empty"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   854
| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   855
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   856
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   857
lemma adjunct_sup:
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   858
  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   859
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   860
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   861
lemma sup_code [code]:
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   862
  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   863
    of Empty \<Rightarrow> g ()
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   864
     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   865
     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   866
proof (cases "f ()")
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   867
  case Empty
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   868
  thus ?thesis
34007
aea892559fc5 tuned lattices theory fragements; generlized some lemmas from sets to lattices
haftmann
parents: 33988
diff changeset
   869
    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   870
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   871
  case Insert
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   872
  thus ?thesis
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   873
    unfolding Seq_def by (simp add: sup_assoc)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   874
next
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   875
  case Join
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   876
  thus ?thesis
30376
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   877
    unfolding Seq_def
e8cc806a3755 refined enumeration implementation
haftmann
parents: 30328
diff changeset
   878
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   879
qed
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   880
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   881
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   882
  "contained Empty Q \<longleftrightarrow> True"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   883
| "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   884
| "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
   885
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   886
lemma single_less_eq_eval:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   887
  "single x \<le> P \<longleftrightarrow> eval P x"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   888
  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
   889
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   890
lemma contained_less_eq:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   891
  "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
   892
  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
   893
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   894
lemma less_eq_pred_code [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   895
  "Seq f \<le> Q = (case f ()
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   896
   of Empty \<Rightarrow> True
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   897
    | 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
   898
    | 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
   899
  by (cases "f ()")
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   900
    (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
   901
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   902
lemma eq_pred_code [code]:
31133
a9f728dc5c8e dropped sort constraint on predicate equality
haftmann
parents: 31122
diff changeset
   903
  fixes P Q :: "'a pred"
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   904
  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
   905
  by (auto simp add: equal)
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   906
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   907
lemma [code nbe]:
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   908
  "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38651
diff changeset
   909
  by (fact equal_refl)
30430
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   910
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   911
lemma [code]:
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   912
  "pred_case f P = f (eval P)"
42ea5d85edcc explicit code equations for some rarely used pred operations
haftmann
parents: 30378
diff changeset
   913
  by (cases P) simp
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_rec 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
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
   918
31105
95f66b234086 added general preprocessing of equality in predicates for code generation
bulwahn
parents: 30430
diff changeset
   919
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
   920
95f66b234086 added general preprocessing of equality in predicates for code generation
bulwahn
parents: 30430
diff changeset
   921
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
31108
haftmann
parents: 31106 30959
diff changeset
   922
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   923
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   924
primrec null :: "'a seq \<Rightarrow> bool" where
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   925
  "null Empty \<longleftrightarrow> True"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   926
| "null (Insert x P) \<longleftrightarrow> False"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   927
| "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
   928
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   929
lemma null_is_empty:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   930
  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   931
  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
   932
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   933
lemma is_empty_code [code]:
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   934
  "is_empty (Seq f) \<longleftrightarrow> null (f ())"
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   935
  by (simp add: null_is_empty Seq_def)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   936
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   937
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   938
  [code del]: "the_only dfault Empty = dfault ()"
44415
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   939
| "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 ())"
ce6cd1b2344b tuned specifications, syntax and proofs
haftmann
parents: 44414
diff changeset
   940
| "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   941
       else let x = singleton dfault P; y = the_only dfault xq in
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   942
       if x = y then x else dfault ())"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   943
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   944
lemma the_only_singleton:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   945
  "the_only dfault xq = singleton dfault (pred_of_seq xq)"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   946
  by (induct xq)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   947
    (auto simp add: singleton_bot singleton_single is_empty_def
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   948
    null_is_empty Let_def singleton_sup)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   949
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   950
lemma singleton_code [code]:
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   951
  "singleton dfault (Seq f) = (case f ()
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   952
   of Empty \<Rightarrow> dfault ()
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   953
    | Insert x P \<Rightarrow> if is_empty P then x
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   954
        else let y = singleton dfault P in
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   955
          if x = y then x else dfault ()
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   956
    | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   957
        else if null xq then singleton dfault P
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   958
        else let x = singleton dfault P; y = the_only dfault xq in
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   959
          if x = y then x else dfault ())"
32578
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   960
  by (cases "f ()")
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   961
   (auto simp add: Seq_def the_only_singleton is_empty_def
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   962
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)
22117a76f943 added emptiness check predicate and singleton projection
haftmann
parents: 32372
diff changeset
   963
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   964
definition the :: "'a pred \<Rightarrow> 'a" where
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 37549
diff changeset
   965
  "the A = (THE x. eval A x)"
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
   966
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   967
lemma the_eqI:
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
   968
  "(THE x. eval P x) = x \<Longrightarrow> the P = x"
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   969
  by (simp add: the_def)
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   970
44414
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   971
definition not_unique :: "'a pred \<Rightarrow> 'a" where
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   972
  [code del]: "not_unique A = (THE x. eval A x)"
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   973
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   974
code_abort not_unique
fb25c131bd73 tuned specifications and syntax
haftmann
parents: 44363
diff changeset
   975
40674
54dbe6a1c349 adhere established Collect/mem convention more closely
haftmann
parents: 40616
diff changeset
   976
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
   977
  by (rule the_eqI) (simp add: singleton_def not_unique_def)
33110
16f2814653ed generalizing singleton with a default value
bulwahn
parents: 33104
diff changeset
   978
36531
19f6e3b0d9b6 code_reflect: specify module name directly after keyword
haftmann
parents: 36513
diff changeset
   979
code_reflect Predicate
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   980
  datatypes pred = Seq and seq = Empty | Insert | Join
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   981
  functions map
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   982
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   983
ML {*
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   984
signature PREDICATE =
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   985
sig
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   986
  datatype 'a pred = Seq of (unit -> 'a seq)
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   987
  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
   988
  val yield: 'a pred -> ('a * 'a pred) option
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
   989
  val yieldn: int -> 'a pred -> 'a list * 'a pred
31222
4a84ae57b65f added Predicate.map in SML environment
haftmann
parents: 31216
diff changeset
   990
  val map: ('a -> 'b) -> 'a pred -> 'b pred
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   991
end;
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   992
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   993
structure Predicate : PREDICATE =
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   994
struct
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
   995
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   996
datatype pred = datatype Predicate.pred
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   997
datatype seq = datatype Predicate.seq
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   998
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
   999
fun map f = Predicate.map f;
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1000
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
  1001
fun yield (Seq f) = next (f ())
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
  1002
and next Empty = NONE
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
  1003
  | next (Insert (x, P)) = SOME (x, P)
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
  1004
  | next (Join (P, xq)) = (case yield P
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1005
     of NONE => next xq
36513
70096cbdd4e0 avoid code_datatype antiquotation
haftmann
parents: 36176
diff changeset
  1006
      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1007
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1008
fun anamorph f k x = (if k = 0 then ([], x)
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1009
  else case f x
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1010
   of NONE => ([], x)
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1011
    | SOME (v, y) => let
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1012
        val (vs, z) = anamorph f (k - 1) y
33607
haftmann
parents: 33111
diff changeset
  1013
      in (v :: vs, z) end);
30959
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1014
458e55fd0a33 fixed compilation of predicate types in ML environment
haftmann
parents: 30948
diff changeset
  1015
fun yieldn P = anamorph yield P;
30948
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
  1016
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
  1017
end;
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
  1018
*}
7f699568a877 static compilation of enumeration type
haftmann
parents: 30430
diff changeset
  1019
44363
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1020
lemma eval_mem [simp]:
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1021
  "x \<in> eval P \<longleftrightarrow> eval P x"
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1022
  by (simp add: mem_def)
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1023
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1024
lemma eq_mem [simp]:
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1025
  "x \<in> (op =) y \<longleftrightarrow> x = y"
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1026
  by (auto simp add: mem_def)
53f4f8287606 avoid pred/set mixture
haftmann
parents: 44033
diff changeset
  1027
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1028
no_notation
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1029
  bot ("\<bottom>") and
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1030
  top ("\<top>") and
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1031
  inf (infixl "\<sqinter>" 70) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1032
  sup (infixl "\<squnion>" 65) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1033
  Inf ("\<Sqinter>_" [900] 900) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1034
  Sup ("\<Squnion>_" [900] 900) and
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1035
  bind (infixl "\<guillemotright>=" 70)
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1036
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1037
no_syntax (xsymbols)
41082
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1038
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
9ff94e7cc3b3 bot comes before top, inf before sup etc.
haftmann
parents: 41080
diff changeset
  1039
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
41080
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1040
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1041
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
294956ff285b nice syntax for lattice INFI, SUPR;
haftmann
parents: 41075
diff changeset
  1042
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 36008
diff changeset
  1043
hide_type (open) pred seq
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 36008
diff changeset
  1044
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
33111
db5af7b86a2f developing an executable the operator
bulwahn
parents: 33110
diff changeset
  1045
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the
30328
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1046
ab47f43f7581 added enumeration of predicates
haftmann
parents: 26797
diff changeset
  1047
end