src/HOL/Predicate.thy
author haftmann
Mon Nov 29 12:14:46 2010 +0100 (2010-11-29)
changeset 40813 f1fc2a1547eb
parent 40674 54dbe6a1c349
child 41075 4bed56dc95fb
permissions -rw-r--r--
moved generic definitions about relations from Quotient.thy to Predicate;
more natural deduction rules
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(*  Title:      HOL/Predicate.thy
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    Author:     Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen
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*)
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header {* Predicates as relations and enumerations *}
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theory Predicate
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imports Inductive Relation
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begin
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notation
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  inf (infixl "\<sqinter>" 70) and
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  sup (infixl "\<squnion>" 65) and
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  Inf ("\<Sqinter>_" [900] 900) and
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  Sup ("\<Squnion>_" [900] 900) and
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  top ("\<top>") and
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  bot ("\<bottom>")
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subsection {* Predicates as (complete) lattices *}
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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 ==> P <= Q ==> 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 ==> P <= Q ==> 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)) = (R = S)"
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  by (simp add: mem_def)
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lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"
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  by (simp add: 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) <= (\<lambda>x. x \<in> S)) = (R <= S)"
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  by (simp add: mem_def)
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lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) <= (\<lambda>x y. (x, y) \<in> S)) = (R <= S)"
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  by fast
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subsubsection {* Top and bottom elements *}
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lemma top1I [intro!]: "top x"
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  by (simp add: top_fun_eq top_bool_eq)
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lemma top2I [intro!]: "top x y"
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  by (simp add: top_fun_eq top_bool_eq)
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lemma bot1E [no_atp, elim!]: "bot x \<Longrightarrow> P"
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  by (simp add: bot_fun_eq bot_bool_eq)
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lemma bot2E [elim!]: "bot x y \<Longrightarrow> P"
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  by (simp add: bot_fun_eq bot_bool_eq)
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lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
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  by (auto simp add: fun_eq_iff)
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subsubsection {* Binary union *}
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lemma sup1I1 [elim?]: "A x \<Longrightarrow> sup A B x"
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  by (simp add: sup_fun_eq sup_bool_eq)
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lemma sup2I1 [elim?]: "A x y \<Longrightarrow> sup A B x y"
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  by (simp add: sup_fun_eq sup_bool_eq)
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lemma sup1I2 [elim?]: "B x \<Longrightarrow> sup A B x"
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  by (simp add: sup_fun_eq sup_bool_eq)
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lemma sup2I2 [elim?]: "B x y \<Longrightarrow> sup A B x y"
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  by (simp add: sup_fun_eq sup_bool_eq)
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lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
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  by (simp add: sup_fun_eq sup_bool_eq) iprover
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lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
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  by (simp add: sup_fun_eq sup_bool_eq) 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!]: "(~ B x ==> A x) ==> sup A B x"
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  by (auto simp add: sup_fun_eq sup_bool_eq)
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lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
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  by (auto simp add: sup_fun_eq sup_bool_eq)
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lemma sup_Un_eq: "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
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  by (simp add: sup_fun_eq sup_bool_eq mem_def)
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lemma sup_Un_eq2 [pred_set_conv]: "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
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  by (simp add: sup_fun_eq sup_bool_eq mem_def)
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subsubsection {* Binary intersection *}
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lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf1D1: "inf A B x ==> A x"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf2D1: "inf A B x y ==> A x y"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf1D2: "inf A B x ==> B x"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf2D2: "inf A B x y ==> B x y"
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  by (simp add: inf_fun_eq inf_bool_eq)
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lemma inf_Int_eq: "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
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  by (simp add: inf_fun_eq inf_bool_eq mem_def)
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lemma inf_Int_eq2 [pred_set_conv]: "inf (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
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  by (simp add: inf_fun_eq inf_bool_eq mem_def)
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subsubsection {* Unions of families *}
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lemma SUP1_iff: "(SUP x:A. B x) b = (EX x:A. B x b)"
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  by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
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lemma SUP2_iff: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
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  by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
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lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
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  by (auto simp add: SUP1_iff)
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lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
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  by (auto simp add: SUP2_iff)
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lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
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  by (auto simp add: SUP1_iff)
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lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
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  by (auto simp add: SUP2_iff)
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lemma SUP_UN_eq: "(SUP i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (UN i. r i))"
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  by (simp add: SUP1_iff fun_eq_iff)
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lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"
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  by (simp add: SUP2_iff fun_eq_iff)
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subsubsection {* Intersections of families *}
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lemma INF1_iff: "(INF x:A. B x) b = (ALL x:A. B x b)"
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  by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
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lemma INF2_iff: "(INF x:A. B x) b c = (ALL x:A. B x b c)"
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  by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
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lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b"
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  by (auto simp add: INF1_iff)
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lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c"
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  by (auto simp add: INF2_iff)
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lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b"
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  by (auto simp add: INF1_iff)
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lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c"
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  by (auto simp add: INF2_iff)
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lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R"
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  by (auto simp add: INF1_iff)
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lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R"
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  by (auto simp add: INF2_iff)
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lemma INF_INT_eq: "(INF i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (INT i. r i))"
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  by (simp add: INF1_iff fun_eq_iff)
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lemma INF_INT_eq2: "(INF i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (INT i. r i))"
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  by (simp add: INF2_iff fun_eq_iff)
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subsection {* Predicates as relations *}
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subsubsection {* Composition  *}
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inductive
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  pred_comp  :: "['a => 'b => bool, 'b => 'c => bool] => 'a => 'c => bool"
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    (infixr "OO" 75)
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  for r :: "'a => 'b => bool" and s :: "'b => 'c => bool"
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where
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  pred_compI [intro]: "r a b ==> s b c ==> (r OO s) a c"
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inductive_cases pred_compE [elim!]: "(r OO s) a c"
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lemma pred_comp_rel_comp_eq [pred_set_conv]:
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  "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
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  by (auto simp add: fun_eq_iff elim: pred_compE)
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subsubsection {* Converse *}
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inductive
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  conversep :: "('a => 'b => bool) => 'b => 'a => bool"
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    ("(_^--1)" [1000] 1000)
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  for r :: "'a => 'b => bool"
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where
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  conversepI: "r a b ==> r^--1 b a"
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notation (xsymbols)
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  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
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lemma conversepD:
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  assumes ab: "r^--1 a b"
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  shows "r b a" using ab
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  by cases simp
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lemma conversep_iff [iff]: "r^--1 a b = r b a"
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  by (iprover intro: conversepI dest: conversepD)
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lemma conversep_converse_eq [pred_set_conv]:
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  "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"
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  by (auto simp add: 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: "(inf r s)^--1 = inf r^--1 s^--1"
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  by (simp add: inf_fun_eq inf_bool_eq)
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    (iprover intro: conversepI ext dest: conversepD)
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lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
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  by (simp add: sup_fun_eq sup_bool_eq)
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    (iprover intro: conversepI ext dest: conversepD)
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lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
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  by (auto simp add: 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
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  DomainP :: "('a => 'b => bool) => 'a => bool"
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  for r :: "'a => 'b => bool"
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where
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  DomainPI [intro]: "r a b ==> DomainP r a"
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inductive_cases DomainPE [elim!]: "DomainP r a"
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lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
berghofe@26797
   314
  by (blast intro!: Orderings.order_antisym predicate1I)
berghofe@22259
   315
berghofe@22259
   316
haftmann@30328
   317
subsubsection {* Range *}
berghofe@22259
   318
berghofe@23741
   319
inductive
berghofe@22259
   320
  RangeP :: "('a => 'b => bool) => 'b => bool"
berghofe@22259
   321
  for r :: "'a => 'b => bool"
berghofe@22259
   322
where
berghofe@22259
   323
  RangePI [intro]: "r a b ==> RangeP r b"
berghofe@22259
   324
berghofe@23741
   325
inductive_cases RangePE [elim!]: "RangeP r b"
berghofe@22259
   326
berghofe@23741
   327
lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
berghofe@26797
   328
  by (blast intro!: Orderings.order_antisym predicate1I)
berghofe@22259
   329
berghofe@22259
   330
haftmann@30328
   331
subsubsection {* Inverse image *}
berghofe@22259
   332
berghofe@22259
   333
definition
berghofe@22259
   334
  inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where
berghofe@22259
   335
  "inv_imagep r f == %x y. r (f x) (f y)"
berghofe@22259
   336
berghofe@23741
   337
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
berghofe@22259
   338
  by (simp add: inv_image_def inv_imagep_def)
berghofe@22259
   339
berghofe@22259
   340
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
berghofe@22259
   341
  by (simp add: inv_imagep_def)
berghofe@22259
   342
berghofe@22259
   343
haftmann@30328
   344
subsubsection {* Powerset *}
berghofe@23741
   345
berghofe@23741
   346
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
berghofe@23741
   347
  "Powp A == \<lambda>B. \<forall>x \<in> B. A x"
berghofe@23741
   348
berghofe@23741
   349
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
nipkow@39302
   350
  by (auto simp add: Powp_def fun_eq_iff)
berghofe@23741
   351
berghofe@26797
   352
lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]
berghofe@26797
   353
berghofe@23741
   354
haftmann@30328
   355
subsubsection {* Properties of relations *}
berghofe@22259
   356
berghofe@22259
   357
abbreviation antisymP :: "('a => 'a => bool) => bool" where
berghofe@23741
   358
  "antisymP r == antisym {(x, y). r x y}"
berghofe@22259
   359
berghofe@22259
   360
abbreviation transP :: "('a => 'a => bool) => bool" where
berghofe@23741
   361
  "transP r == trans {(x, y). r x y}"
berghofe@22259
   362
berghofe@22259
   363
abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
berghofe@23741
   364
  "single_valuedP r == single_valued {(x, y). r x y}"
berghofe@22259
   365
haftmann@40813
   366
(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
haftmann@40813
   367
haftmann@40813
   368
definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40813
   369
  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
haftmann@40813
   370
haftmann@40813
   371
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40813
   372
  "symp r \<longleftrightarrow> sym {(x, y). r x y}"
haftmann@40813
   373
haftmann@40813
   374
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40813
   375
  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
haftmann@40813
   376
haftmann@40813
   377
lemma reflpI:
haftmann@40813
   378
  "(\<And>x. r x x) \<Longrightarrow> reflp r"
haftmann@40813
   379
  by (auto intro: refl_onI simp add: reflp_def)
haftmann@40813
   380
haftmann@40813
   381
lemma reflpE:
haftmann@40813
   382
  assumes "reflp r"
haftmann@40813
   383
  obtains "r x x"
haftmann@40813
   384
  using assms by (auto dest: refl_onD simp add: reflp_def)
haftmann@40813
   385
haftmann@40813
   386
lemma sympI:
haftmann@40813
   387
  "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
haftmann@40813
   388
  by (auto intro: symI simp add: symp_def)
haftmann@40813
   389
haftmann@40813
   390
lemma sympE:
haftmann@40813
   391
  assumes "symp r" and "r x y"
haftmann@40813
   392
  obtains "r y x"
haftmann@40813
   393
  using assms by (auto dest: symD simp add: symp_def)
haftmann@40813
   394
haftmann@40813
   395
lemma transpI:
haftmann@40813
   396
  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
haftmann@40813
   397
  by (auto intro: transI simp add: transp_def)
haftmann@40813
   398
  
haftmann@40813
   399
lemma transpE:
haftmann@40813
   400
  assumes "transp r" and "r x y" and "r y z"
haftmann@40813
   401
  obtains "r x z"
haftmann@40813
   402
  using assms by (auto dest: transD simp add: transp_def)
haftmann@40813
   403
haftmann@30328
   404
haftmann@30328
   405
subsection {* Predicates as enumerations *}
haftmann@30328
   406
haftmann@30328
   407
subsubsection {* The type of predicate enumerations (a monad) *}
haftmann@30328
   408
haftmann@30328
   409
datatype 'a pred = Pred "'a \<Rightarrow> bool"
haftmann@30328
   410
haftmann@30328
   411
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
haftmann@30328
   412
  eval_pred: "eval (Pred f) = f"
haftmann@30328
   413
haftmann@30328
   414
lemma Pred_eval [simp]:
haftmann@30328
   415
  "Pred (eval x) = x"
haftmann@30328
   416
  by (cases x) simp
haftmann@30328
   417
haftmann@40616
   418
lemma pred_eqI:
haftmann@40616
   419
  "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"
haftmann@40616
   420
  by (cases P, cases Q) (auto simp add: fun_eq_iff)
haftmann@30328
   421
haftmann@40616
   422
lemma eval_mem [simp]:
haftmann@40616
   423
  "x \<in> eval P \<longleftrightarrow> eval P x"
haftmann@40616
   424
  by (simp add: mem_def)
haftmann@30328
   425
haftmann@40616
   426
lemma eq_mem [simp]:
haftmann@40616
   427
  "x \<in> (op =) y \<longleftrightarrow> x = y"
haftmann@40616
   428
  by (auto simp add: mem_def)
haftmann@30328
   429
haftmann@32578
   430
instantiation pred :: (type) "{complete_lattice, boolean_algebra}"
haftmann@30328
   431
begin
haftmann@30328
   432
haftmann@30328
   433
definition
haftmann@30328
   434
  "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
haftmann@30328
   435
haftmann@30328
   436
definition
haftmann@30328
   437
  "P < Q \<longleftrightarrow> eval P < eval Q"
haftmann@30328
   438
haftmann@30328
   439
definition
haftmann@30328
   440
  "\<bottom> = Pred \<bottom>"
haftmann@30328
   441
haftmann@40616
   442
lemma eval_bot [simp]:
haftmann@40616
   443
  "eval \<bottom>  = \<bottom>"
haftmann@40616
   444
  by (simp add: bot_pred_def)
haftmann@40616
   445
haftmann@30328
   446
definition
haftmann@30328
   447
  "\<top> = Pred \<top>"
haftmann@30328
   448
haftmann@40616
   449
lemma eval_top [simp]:
haftmann@40616
   450
  "eval \<top>  = \<top>"
haftmann@40616
   451
  by (simp add: top_pred_def)
haftmann@40616
   452
haftmann@30328
   453
definition
haftmann@30328
   454
  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
haftmann@30328
   455
haftmann@40616
   456
lemma eval_inf [simp]:
haftmann@40616
   457
  "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
haftmann@40616
   458
  by (simp add: inf_pred_def)
haftmann@40616
   459
haftmann@30328
   460
definition
haftmann@30328
   461
  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
haftmann@30328
   462
haftmann@40616
   463
lemma eval_sup [simp]:
haftmann@40616
   464
  "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
haftmann@40616
   465
  by (simp add: sup_pred_def)
haftmann@40616
   466
haftmann@30328
   467
definition
haftmann@37767
   468
  "\<Sqinter>A = Pred (INFI A eval)"
haftmann@30328
   469
haftmann@40616
   470
lemma eval_Inf [simp]:
haftmann@40616
   471
  "eval (\<Sqinter>A) = INFI A eval"
haftmann@40616
   472
  by (simp add: Inf_pred_def)
haftmann@40616
   473
haftmann@30328
   474
definition
haftmann@37767
   475
  "\<Squnion>A = Pred (SUPR A eval)"
haftmann@30328
   476
haftmann@40616
   477
lemma eval_Sup [simp]:
haftmann@40616
   478
  "eval (\<Squnion>A) = SUPR A eval"
haftmann@40616
   479
  by (simp add: Sup_pred_def)
haftmann@40616
   480
haftmann@32578
   481
definition
haftmann@32578
   482
  "- P = Pred (- eval P)"
haftmann@32578
   483
haftmann@40616
   484
lemma eval_compl [simp]:
haftmann@40616
   485
  "eval (- P) = - eval P"
haftmann@40616
   486
  by (simp add: uminus_pred_def)
haftmann@40616
   487
haftmann@32578
   488
definition
haftmann@32578
   489
  "P - Q = Pred (eval P - eval Q)"
haftmann@32578
   490
haftmann@40616
   491
lemma eval_minus [simp]:
haftmann@40616
   492
  "eval (P - Q) = eval P - eval Q"
haftmann@40616
   493
  by (simp add: minus_pred_def)
haftmann@40616
   494
haftmann@32578
   495
instance proof
haftmann@40616
   496
qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def
haftmann@40616
   497
  fun_Compl_def fun_diff_def bool_Compl_def bool_diff_def)
haftmann@30328
   498
berghofe@22259
   499
end
haftmann@30328
   500
haftmann@40616
   501
lemma eval_INFI [simp]:
haftmann@40616
   502
  "eval (INFI A f) = INFI A (eval \<circ> f)"
haftmann@40616
   503
  by (unfold INFI_def) simp
haftmann@40616
   504
haftmann@40616
   505
lemma eval_SUPR [simp]:
haftmann@40616
   506
  "eval (SUPR A f) = SUPR A (eval \<circ> f)"
haftmann@40616
   507
  by (unfold SUPR_def) simp
haftmann@40616
   508
haftmann@40616
   509
definition single :: "'a \<Rightarrow> 'a pred" where
haftmann@40616
   510
  "single x = Pred ((op =) x)"
haftmann@40616
   511
haftmann@40616
   512
lemma eval_single [simp]:
haftmann@40616
   513
  "eval (single x) = (op =) x"
haftmann@40616
   514
  by (simp add: single_def)
haftmann@40616
   515
haftmann@40616
   516
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
haftmann@40674
   517
  "P \<guillemotright>= f = (SUPR {x. Predicate.eval P x} f)"
haftmann@40616
   518
haftmann@40616
   519
lemma eval_bind [simp]:
haftmann@40674
   520
  "eval (P \<guillemotright>= f) = Predicate.eval (SUPR {x. Predicate.eval P x} f)"
haftmann@40616
   521
  by (simp add: bind_def)
haftmann@40616
   522
haftmann@30328
   523
lemma bind_bind:
haftmann@30328
   524
  "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
haftmann@40674
   525
  by (rule pred_eqI) auto
haftmann@30328
   526
haftmann@30328
   527
lemma bind_single:
haftmann@30328
   528
  "P \<guillemotright>= single = P"
haftmann@40616
   529
  by (rule pred_eqI) auto
haftmann@30328
   530
haftmann@30328
   531
lemma single_bind:
haftmann@30328
   532
  "single x \<guillemotright>= P = P x"
haftmann@40616
   533
  by (rule pred_eqI) auto
haftmann@30328
   534
haftmann@30328
   535
lemma bottom_bind:
haftmann@30328
   536
  "\<bottom> \<guillemotright>= P = \<bottom>"
haftmann@40674
   537
  by (rule pred_eqI) auto
haftmann@30328
   538
haftmann@30328
   539
lemma sup_bind:
haftmann@30328
   540
  "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
haftmann@40674
   541
  by (rule pred_eqI) auto
haftmann@30328
   542
haftmann@40616
   543
lemma Sup_bind:
haftmann@40616
   544
  "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
haftmann@40674
   545
  by (rule pred_eqI) auto
haftmann@30328
   546
haftmann@30328
   547
lemma pred_iffI:
haftmann@30328
   548
  assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
haftmann@30328
   549
  and "\<And>x. eval B x \<Longrightarrow> eval A x"
haftmann@30328
   550
  shows "A = B"
haftmann@40616
   551
  using assms by (auto intro: pred_eqI)
haftmann@30328
   552
  
haftmann@30328
   553
lemma singleI: "eval (single x) x"
haftmann@40616
   554
  by simp
haftmann@30328
   555
haftmann@30328
   556
lemma singleI_unit: "eval (single ()) x"
haftmann@40616
   557
  by simp
haftmann@30328
   558
haftmann@30328
   559
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   560
  by simp
haftmann@30328
   561
haftmann@30328
   562
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   563
  by simp
haftmann@30328
   564
haftmann@30328
   565
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
haftmann@40616
   566
  by auto
haftmann@30328
   567
haftmann@30328
   568
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   569
  by auto
haftmann@30328
   570
haftmann@30328
   571
lemma botE: "eval \<bottom> x \<Longrightarrow> P"
haftmann@40616
   572
  by auto
haftmann@30328
   573
haftmann@30328
   574
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
haftmann@40616
   575
  by auto
haftmann@30328
   576
haftmann@30328
   577
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 
haftmann@40616
   578
  by auto
haftmann@30328
   579
haftmann@30328
   580
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   581
  by auto
haftmann@30328
   582
haftmann@32578
   583
lemma single_not_bot [simp]:
haftmann@32578
   584
  "single x \<noteq> \<bottom>"
nipkow@39302
   585
  by (auto simp add: single_def bot_pred_def fun_eq_iff)
haftmann@32578
   586
haftmann@32578
   587
lemma not_bot:
haftmann@32578
   588
  assumes "A \<noteq> \<bottom>"
haftmann@32578
   589
  obtains x where "eval A x"
haftmann@40616
   590
  using assms by (cases A)
haftmann@40616
   591
    (auto simp add: bot_pred_def, auto simp add: mem_def)
haftmann@32578
   592
  
haftmann@32578
   593
haftmann@32578
   594
subsubsection {* Emptiness check and definite choice *}
haftmann@32578
   595
haftmann@32578
   596
definition is_empty :: "'a pred \<Rightarrow> bool" where
haftmann@32578
   597
  "is_empty A \<longleftrightarrow> A = \<bottom>"
haftmann@32578
   598
haftmann@32578
   599
lemma is_empty_bot:
haftmann@32578
   600
  "is_empty \<bottom>"
haftmann@32578
   601
  by (simp add: is_empty_def)
haftmann@32578
   602
haftmann@32578
   603
lemma not_is_empty_single:
haftmann@32578
   604
  "\<not> is_empty (single x)"
nipkow@39302
   605
  by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)
haftmann@32578
   606
haftmann@32578
   607
lemma is_empty_sup:
haftmann@32578
   608
  "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
huffman@36008
   609
  by (auto simp add: is_empty_def)
haftmann@32578
   610
haftmann@40616
   611
definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
bulwahn@33111
   612
  "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"
haftmann@32578
   613
haftmann@32578
   614
lemma singleton_eqI:
bulwahn@33110
   615
  "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"
haftmann@32578
   616
  by (auto simp add: singleton_def)
haftmann@32578
   617
haftmann@32578
   618
lemma eval_singletonI:
bulwahn@33110
   619
  "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"
haftmann@32578
   620
proof -
haftmann@32578
   621
  assume assm: "\<exists>!x. eval A x"
haftmann@32578
   622
  then obtain x where "eval A x" ..
bulwahn@33110
   623
  moreover with assm have "singleton dfault A = x" by (rule singleton_eqI)
haftmann@32578
   624
  ultimately show ?thesis by simp 
haftmann@32578
   625
qed
haftmann@32578
   626
haftmann@32578
   627
lemma single_singleton:
bulwahn@33110
   628
  "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"
haftmann@32578
   629
proof -
haftmann@32578
   630
  assume assm: "\<exists>!x. eval A x"
bulwahn@33110
   631
  then have "eval A (singleton dfault A)"
haftmann@32578
   632
    by (rule eval_singletonI)
bulwahn@33110
   633
  moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"
haftmann@32578
   634
    by (rule singleton_eqI)
bulwahn@33110
   635
  ultimately have "eval (single (singleton dfault A)) = eval A"
nipkow@39302
   636
    by (simp (no_asm_use) add: single_def fun_eq_iff) blast
haftmann@40616
   637
  then have "\<And>x. eval (single (singleton dfault A)) x = eval A x"
haftmann@40616
   638
    by simp
haftmann@40616
   639
  then show ?thesis by (rule pred_eqI)
haftmann@32578
   640
qed
haftmann@32578
   641
haftmann@32578
   642
lemma singleton_undefinedI:
bulwahn@33111
   643
  "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"
haftmann@32578
   644
  by (simp add: singleton_def)
haftmann@32578
   645
haftmann@32578
   646
lemma singleton_bot:
bulwahn@33111
   647
  "singleton dfault \<bottom> = dfault ()"
haftmann@32578
   648
  by (auto simp add: bot_pred_def intro: singleton_undefinedI)
haftmann@32578
   649
haftmann@32578
   650
lemma singleton_single:
bulwahn@33110
   651
  "singleton dfault (single x) = x"
haftmann@32578
   652
  by (auto simp add: intro: singleton_eqI singleI elim: singleE)
haftmann@32578
   653
haftmann@32578
   654
lemma singleton_sup_single_single:
bulwahn@33111
   655
  "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"
haftmann@32578
   656
proof (cases "x = y")
haftmann@32578
   657
  case True then show ?thesis by (simp add: singleton_single)
haftmann@32578
   658
next
haftmann@32578
   659
  case False
haftmann@32578
   660
  have "eval (single x \<squnion> single y) x"
haftmann@32578
   661
    and "eval (single x \<squnion> single y) y"
haftmann@32578
   662
  by (auto intro: supI1 supI2 singleI)
haftmann@32578
   663
  with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
haftmann@32578
   664
    by blast
bulwahn@33111
   665
  then have "singleton dfault (single x \<squnion> single y) = dfault ()"
haftmann@32578
   666
    by (rule singleton_undefinedI)
haftmann@32578
   667
  with False show ?thesis by simp
haftmann@32578
   668
qed
haftmann@32578
   669
haftmann@32578
   670
lemma singleton_sup_aux:
bulwahn@33110
   671
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
bulwahn@33110
   672
    else if B = \<bottom> then singleton dfault A
bulwahn@33110
   673
    else singleton dfault
bulwahn@33110
   674
      (single (singleton dfault A) \<squnion> single (singleton dfault B)))"
haftmann@32578
   675
proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
haftmann@32578
   676
  case True then show ?thesis by (simp add: single_singleton)
haftmann@32578
   677
next
haftmann@32578
   678
  case False
haftmann@32578
   679
  from False have A_or_B:
bulwahn@33111
   680
    "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
haftmann@32578
   681
    by (auto intro!: singleton_undefinedI)
bulwahn@33110
   682
  then have rhs: "singleton dfault
bulwahn@33111
   683
    (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
haftmann@32578
   684
    by (auto simp add: singleton_sup_single_single singleton_single)
haftmann@32578
   685
  from False have not_unique:
haftmann@32578
   686
    "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
haftmann@32578
   687
  show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
haftmann@32578
   688
    case True
haftmann@32578
   689
    then obtain a b where a: "eval A a" and b: "eval B b"
haftmann@32578
   690
      by (blast elim: not_bot)
haftmann@32578
   691
    with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
haftmann@32578
   692
      by (auto simp add: sup_pred_def bot_pred_def)
bulwahn@33111
   693
    then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)
haftmann@32578
   694
    with True rhs show ?thesis by simp
haftmann@32578
   695
  next
haftmann@32578
   696
    case False then show ?thesis by auto
haftmann@32578
   697
  qed
haftmann@32578
   698
qed
haftmann@32578
   699
haftmann@32578
   700
lemma singleton_sup:
bulwahn@33110
   701
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
bulwahn@33110
   702
    else if B = \<bottom> then singleton dfault A
bulwahn@33111
   703
    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
bulwahn@33110
   704
using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
haftmann@32578
   705
haftmann@30328
   706
haftmann@30328
   707
subsubsection {* Derived operations *}
haftmann@30328
   708
haftmann@30328
   709
definition if_pred :: "bool \<Rightarrow> unit pred" where
haftmann@30328
   710
  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
haftmann@30328
   711
bulwahn@33754
   712
definition holds :: "unit pred \<Rightarrow> bool" where
bulwahn@33754
   713
  holds_eq: "holds P = eval P ()"
bulwahn@33754
   714
haftmann@30328
   715
definition not_pred :: "unit pred \<Rightarrow> unit pred" where
haftmann@30328
   716
  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
haftmann@30328
   717
haftmann@30328
   718
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
haftmann@30328
   719
  unfolding if_pred_eq by (auto intro: singleI)
haftmann@30328
   720
haftmann@30328
   721
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30328
   722
  unfolding if_pred_eq by (cases b) (auto elim: botE)
haftmann@30328
   723
haftmann@30328
   724
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
haftmann@30328
   725
  unfolding not_pred_eq eval_pred by (auto intro: singleI)
haftmann@30328
   726
haftmann@30328
   727
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
haftmann@30328
   728
  unfolding not_pred_eq by (auto intro: singleI)
haftmann@30328
   729
haftmann@30328
   730
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   731
  unfolding not_pred_eq
haftmann@30328
   732
  by (auto split: split_if_asm elim: botE)
haftmann@30328
   733
haftmann@30328
   734
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   735
  unfolding not_pred_eq
haftmann@30328
   736
  by (auto split: split_if_asm elim: botE)
bulwahn@33754
   737
lemma "f () = False \<or> f () = True"
bulwahn@33754
   738
by simp
haftmann@30328
   739
blanchet@37549
   740
lemma closure_of_bool_cases [no_atp]:
bulwahn@33754
   741
assumes "(f :: unit \<Rightarrow> bool) = (%u. False) \<Longrightarrow> P f"
bulwahn@33754
   742
assumes "f = (%u. True) \<Longrightarrow> P f"
bulwahn@33754
   743
shows "P f"
bulwahn@33754
   744
proof -
bulwahn@33754
   745
  have "f = (%u. False) \<or> f = (%u. True)"
bulwahn@33754
   746
    apply (cases "f ()")
bulwahn@33754
   747
    apply (rule disjI2)
bulwahn@33754
   748
    apply (rule ext)
bulwahn@33754
   749
    apply (simp add: unit_eq)
bulwahn@33754
   750
    apply (rule disjI1)
bulwahn@33754
   751
    apply (rule ext)
bulwahn@33754
   752
    apply (simp add: unit_eq)
bulwahn@33754
   753
    done
bulwahn@33754
   754
  from this prems show ?thesis by blast
bulwahn@33754
   755
qed
bulwahn@33754
   756
bulwahn@33754
   757
lemma unit_pred_cases:
bulwahn@33754
   758
assumes "P \<bottom>"
bulwahn@33754
   759
assumes "P (single ())"
bulwahn@33754
   760
shows "P Q"
bulwahn@33754
   761
using assms
bulwahn@33754
   762
unfolding bot_pred_def Collect_def empty_def single_def
bulwahn@33754
   763
apply (cases Q)
bulwahn@33754
   764
apply simp
bulwahn@33754
   765
apply (rule_tac f="fun" in closure_of_bool_cases)
bulwahn@33754
   766
apply auto
bulwahn@33754
   767
apply (subgoal_tac "(%x. () = x) = (%x. True)") 
bulwahn@33754
   768
apply auto
bulwahn@33754
   769
done
bulwahn@33754
   770
bulwahn@33754
   771
lemma holds_if_pred:
bulwahn@33754
   772
  "holds (if_pred b) = b"
bulwahn@33754
   773
unfolding if_pred_eq holds_eq
bulwahn@33754
   774
by (cases b) (auto intro: singleI elim: botE)
bulwahn@33754
   775
bulwahn@33754
   776
lemma if_pred_holds:
bulwahn@33754
   777
  "if_pred (holds P) = P"
bulwahn@33754
   778
unfolding if_pred_eq holds_eq
bulwahn@33754
   779
by (rule unit_pred_cases) (auto intro: singleI elim: botE)
bulwahn@33754
   780
bulwahn@33754
   781
lemma is_empty_holds:
bulwahn@33754
   782
  "is_empty P \<longleftrightarrow> \<not> holds P"
bulwahn@33754
   783
unfolding is_empty_def holds_eq
bulwahn@33754
   784
by (rule unit_pred_cases) (auto elim: botE intro: singleI)
haftmann@30328
   785
haftmann@30328
   786
subsubsection {* Implementation *}
haftmann@30328
   787
haftmann@30328
   788
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
haftmann@30328
   789
haftmann@30328
   790
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
haftmann@30328
   791
    "pred_of_seq Empty = \<bottom>"
haftmann@30328
   792
  | "pred_of_seq (Insert x P) = single x \<squnion> P"
haftmann@30328
   793
  | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
haftmann@30328
   794
haftmann@30328
   795
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
haftmann@30328
   796
  "Seq f = pred_of_seq (f ())"
haftmann@30328
   797
haftmann@30328
   798
code_datatype Seq
haftmann@30328
   799
haftmann@30328
   800
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
haftmann@30328
   801
  "member Empty x \<longleftrightarrow> False"
haftmann@30328
   802
  | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
haftmann@30328
   803
  | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
haftmann@30328
   804
haftmann@30328
   805
lemma eval_member:
haftmann@30328
   806
  "member xq = eval (pred_of_seq xq)"
haftmann@30328
   807
proof (induct xq)
haftmann@30328
   808
  case Empty show ?case
nipkow@39302
   809
  by (auto simp add: fun_eq_iff elim: botE)
haftmann@30328
   810
next
haftmann@30328
   811
  case Insert show ?case
nipkow@39302
   812
  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
haftmann@30328
   813
next
haftmann@30328
   814
  case Join then show ?case
nipkow@39302
   815
  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
haftmann@30328
   816
qed
haftmann@30328
   817
haftmann@30328
   818
lemma eval_code [code]: "eval (Seq f) = member (f ())"
haftmann@30328
   819
  unfolding Seq_def by (rule sym, rule eval_member)
haftmann@30328
   820
haftmann@30328
   821
lemma single_code [code]:
haftmann@30328
   822
  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
haftmann@30328
   823
  unfolding Seq_def by simp
haftmann@30328
   824
haftmann@30328
   825
primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
haftmann@30328
   826
    "apply f Empty = Empty"
haftmann@30328
   827
  | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
haftmann@30328
   828
  | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
haftmann@30328
   829
haftmann@30328
   830
lemma apply_bind:
haftmann@30328
   831
  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
haftmann@30328
   832
proof (induct xq)
haftmann@30328
   833
  case Empty show ?case
haftmann@30328
   834
    by (simp add: bottom_bind)
haftmann@30328
   835
next
haftmann@30328
   836
  case Insert show ?case
haftmann@30328
   837
    by (simp add: single_bind sup_bind)
haftmann@30328
   838
next
haftmann@30328
   839
  case Join then show ?case
haftmann@30328
   840
    by (simp add: sup_bind)
haftmann@30328
   841
qed
haftmann@30328
   842
  
haftmann@30328
   843
lemma bind_code [code]:
haftmann@30328
   844
  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
haftmann@30328
   845
  unfolding Seq_def by (rule sym, rule apply_bind)
haftmann@30328
   846
haftmann@30328
   847
lemma bot_set_code [code]:
haftmann@30328
   848
  "\<bottom> = Seq (\<lambda>u. Empty)"
haftmann@30328
   849
  unfolding Seq_def by simp
haftmann@30328
   850
haftmann@30376
   851
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
haftmann@30376
   852
    "adjunct P Empty = Join P Empty"
haftmann@30376
   853
  | "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
haftmann@30376
   854
  | "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
haftmann@30376
   855
haftmann@30376
   856
lemma adjunct_sup:
haftmann@30376
   857
  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
haftmann@30376
   858
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
haftmann@30376
   859
haftmann@30328
   860
lemma sup_code [code]:
haftmann@30328
   861
  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
haftmann@30328
   862
    of Empty \<Rightarrow> g ()
haftmann@30328
   863
     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
haftmann@30376
   864
     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
haftmann@30328
   865
proof (cases "f ()")
haftmann@30328
   866
  case Empty
haftmann@30328
   867
  thus ?thesis
haftmann@34007
   868
    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
haftmann@30328
   869
next
haftmann@30328
   870
  case Insert
haftmann@30328
   871
  thus ?thesis
haftmann@30328
   872
    unfolding Seq_def by (simp add: sup_assoc)
haftmann@30328
   873
next
haftmann@30328
   874
  case Join
haftmann@30328
   875
  thus ?thesis
haftmann@30376
   876
    unfolding Seq_def
haftmann@30376
   877
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
haftmann@30328
   878
qed
haftmann@30328
   879
haftmann@30430
   880
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
haftmann@30430
   881
    "contained Empty Q \<longleftrightarrow> True"
haftmann@30430
   882
  | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
haftmann@30430
   883
  | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
haftmann@30430
   884
haftmann@30430
   885
lemma single_less_eq_eval:
haftmann@30430
   886
  "single x \<le> P \<longleftrightarrow> eval P x"
haftmann@30430
   887
  by (auto simp add: single_def less_eq_pred_def mem_def)
haftmann@30430
   888
haftmann@30430
   889
lemma contained_less_eq:
haftmann@30430
   890
  "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
haftmann@30430
   891
  by (induct xq) (simp_all add: single_less_eq_eval)
haftmann@30430
   892
haftmann@30430
   893
lemma less_eq_pred_code [code]:
haftmann@30430
   894
  "Seq f \<le> Q = (case f ()
haftmann@30430
   895
   of Empty \<Rightarrow> True
haftmann@30430
   896
    | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
haftmann@30430
   897
    | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
haftmann@30430
   898
  by (cases "f ()")
haftmann@30430
   899
    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
haftmann@30430
   900
haftmann@30430
   901
lemma eq_pred_code [code]:
haftmann@31133
   902
  fixes P Q :: "'a pred"
haftmann@38857
   903
  shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
haftmann@38857
   904
  by (auto simp add: equal)
haftmann@38857
   905
haftmann@38857
   906
lemma [code nbe]:
haftmann@38857
   907
  "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
haftmann@38857
   908
  by (fact equal_refl)
haftmann@30430
   909
haftmann@30430
   910
lemma [code]:
haftmann@30430
   911
  "pred_case f P = f (eval P)"
haftmann@30430
   912
  by (cases P) simp
haftmann@30430
   913
haftmann@30430
   914
lemma [code]:
haftmann@30430
   915
  "pred_rec f P = f (eval P)"
haftmann@30430
   916
  by (cases P) simp
haftmann@30328
   917
bulwahn@31105
   918
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
bulwahn@31105
   919
bulwahn@31105
   920
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
haftmann@31108
   921
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
haftmann@30948
   922
haftmann@31216
   923
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
haftmann@31216
   924
  "map f P = P \<guillemotright>= (single o f)"
haftmann@31216
   925
haftmann@32578
   926
primrec null :: "'a seq \<Rightarrow> bool" where
haftmann@32578
   927
    "null Empty \<longleftrightarrow> True"
haftmann@32578
   928
  | "null (Insert x P) \<longleftrightarrow> False"
haftmann@32578
   929
  | "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
haftmann@32578
   930
haftmann@32578
   931
lemma null_is_empty:
haftmann@32578
   932
  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
haftmann@32578
   933
  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
haftmann@32578
   934
haftmann@32578
   935
lemma is_empty_code [code]:
haftmann@32578
   936
  "is_empty (Seq f) \<longleftrightarrow> null (f ())"
haftmann@32578
   937
  by (simp add: null_is_empty Seq_def)
haftmann@32578
   938
bulwahn@33111
   939
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
bulwahn@33111
   940
  [code del]: "the_only dfault Empty = dfault ()"
bulwahn@33111
   941
  | "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 ())"
bulwahn@33110
   942
  | "the_only dfault (Join P xq) = (if is_empty P then the_only dfault xq else if null xq then singleton dfault P
bulwahn@33110
   943
       else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111
   944
       if x = y then x else dfault ())"
haftmann@32578
   945
haftmann@32578
   946
lemma the_only_singleton:
bulwahn@33110
   947
  "the_only dfault xq = singleton dfault (pred_of_seq xq)"
haftmann@32578
   948
  by (induct xq)
haftmann@32578
   949
    (auto simp add: singleton_bot singleton_single is_empty_def
haftmann@32578
   950
    null_is_empty Let_def singleton_sup)
haftmann@32578
   951
haftmann@32578
   952
lemma singleton_code [code]:
bulwahn@33110
   953
  "singleton dfault (Seq f) = (case f ()
bulwahn@33111
   954
   of Empty \<Rightarrow> dfault ()
haftmann@32578
   955
    | Insert x P \<Rightarrow> if is_empty P then x
bulwahn@33110
   956
        else let y = singleton dfault P in
bulwahn@33111
   957
          if x = y then x else dfault ()
bulwahn@33110
   958
    | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
bulwahn@33110
   959
        else if null xq then singleton dfault P
bulwahn@33110
   960
        else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111
   961
          if x = y then x else dfault ())"
haftmann@32578
   962
  by (cases "f ()")
haftmann@32578
   963
   (auto simp add: Seq_def the_only_singleton is_empty_def
haftmann@32578
   964
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)
haftmann@32578
   965
bulwahn@33110
   966
definition not_unique :: "'a pred => 'a"
bulwahn@33110
   967
where
bulwahn@33111
   968
  [code del]: "not_unique A = (THE x. eval A x)"
bulwahn@33110
   969
bulwahn@33111
   970
definition the :: "'a pred => 'a"
bulwahn@33111
   971
where
haftmann@37767
   972
  "the A = (THE x. eval A x)"
bulwahn@33111
   973
haftmann@40674
   974
lemma the_eqI:
haftmann@40674
   975
  "(THE x. Predicate.eval P x) = x \<Longrightarrow> Predicate.the P = x"
haftmann@40674
   976
  by (simp add: the_def)
haftmann@40674
   977
haftmann@40674
   978
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"
haftmann@40674
   979
  by (rule the_eqI) (simp add: singleton_def not_unique_def)
bulwahn@33110
   980
haftmann@33988
   981
code_abort not_unique
haftmann@33988
   982
haftmann@36531
   983
code_reflect Predicate
haftmann@36513
   984
  datatypes pred = Seq and seq = Empty | Insert | Join
haftmann@36513
   985
  functions map
haftmann@36513
   986
haftmann@30948
   987
ML {*
haftmann@30948
   988
signature PREDICATE =
haftmann@30948
   989
sig
haftmann@30948
   990
  datatype 'a pred = Seq of (unit -> 'a seq)
haftmann@30948
   991
  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
haftmann@30959
   992
  val yield: 'a pred -> ('a * 'a pred) option
haftmann@30959
   993
  val yieldn: int -> 'a pred -> 'a list * 'a pred
haftmann@31222
   994
  val map: ('a -> 'b) -> 'a pred -> 'b pred
haftmann@30948
   995
end;
haftmann@30948
   996
haftmann@30948
   997
structure Predicate : PREDICATE =
haftmann@30948
   998
struct
haftmann@30948
   999
haftmann@36513
  1000
datatype pred = datatype Predicate.pred
haftmann@36513
  1001
datatype seq = datatype Predicate.seq
haftmann@36513
  1002
haftmann@36513
  1003
fun map f = Predicate.map f;
haftmann@30959
  1004
haftmann@36513
  1005
fun yield (Seq f) = next (f ())
haftmann@36513
  1006
and next Empty = NONE
haftmann@36513
  1007
  | next (Insert (x, P)) = SOME (x, P)
haftmann@36513
  1008
  | next (Join (P, xq)) = (case yield P
haftmann@30959
  1009
     of NONE => next xq
haftmann@36513
  1010
      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
haftmann@30959
  1011
haftmann@30959
  1012
fun anamorph f k x = (if k = 0 then ([], x)
haftmann@30959
  1013
  else case f x
haftmann@30959
  1014
   of NONE => ([], x)
haftmann@30959
  1015
    | SOME (v, y) => let
haftmann@30959
  1016
        val (vs, z) = anamorph f (k - 1) y
haftmann@33607
  1017
      in (v :: vs, z) end);
haftmann@30959
  1018
haftmann@30959
  1019
fun yieldn P = anamorph yield P;
haftmann@30948
  1020
haftmann@30948
  1021
end;
haftmann@30948
  1022
*}
haftmann@30948
  1023
haftmann@30328
  1024
no_notation
haftmann@30328
  1025
  inf (infixl "\<sqinter>" 70) and
haftmann@30328
  1026
  sup (infixl "\<squnion>" 65) and
haftmann@30328
  1027
  Inf ("\<Sqinter>_" [900] 900) and
haftmann@30328
  1028
  Sup ("\<Squnion>_" [900] 900) and
haftmann@30328
  1029
  top ("\<top>") and
haftmann@30328
  1030
  bot ("\<bottom>") and
haftmann@30328
  1031
  bind (infixl "\<guillemotright>=" 70)
haftmann@30328
  1032
wenzelm@36176
  1033
hide_type (open) pred seq
wenzelm@36176
  1034
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
bulwahn@33111
  1035
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the
haftmann@30328
  1036
haftmann@30328
  1037
end