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(* Title: HOL/Predicate.thy
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Author: Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen
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*)
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header {* Predicates as relations and enumerations *}
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theory Predicate
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imports Inductive Relation
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begin
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notation
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inf (infixl "\<sqinter>" 70) and
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sup (infixl "\<squnion>" 65) and
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Inf ("\<Sqinter>_" [900] 900) and
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Sup ("\<Squnion>_" [900] 900) and
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top ("\<top>") and
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bot ("\<bottom>")
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subsection {* Predicates as (complete) lattices *}
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subsubsection {* 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: expand_fun_eq 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 [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: expand_fun_eq)
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lemma bot_empty_eq2: "bot = (\<lambda>x y. (x, y) \<in> {})"
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by (auto simp add: expand_fun_eq)
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subsubsection {* Binary union *}
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lemma 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 expand_fun_eq)
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lemma SUP_UN_eq2: "(SUP i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (UN i. r i))"
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by (simp add: SUP2_iff expand_fun_eq)
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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 expand_fun_eq)
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lemma INF_INT_eq2: "(INF i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (INT i. r i))"
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by (simp add: INF2_iff expand_fun_eq)
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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: expand_fun_eq 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: expand_fun_eq)
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lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
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by (iprover intro: order_antisym conversepI dest: conversepD)
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lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
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by (iprover intro: order_antisym conversepI pred_compI
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elim: pred_compE dest: conversepD)
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lemma converse_meet: "(inf r s)^--1 = inf r^--1 s^--1"
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by (simp add: inf_fun_eq inf_bool_eq)
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(iprover intro: conversepI ext dest: conversepD)
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lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
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by (simp add: sup_fun_eq sup_bool_eq)
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(iprover intro: conversepI ext dest: conversepD)
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lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
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by (auto simp add: expand_fun_eq)
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lemma conversep_eq [simp]: "(op =)^--1 = op ="
|
berghofe@22259
|
253 |
by (auto simp add: expand_fun_eq)
|
berghofe@22259
|
254 |
|
berghofe@22259
|
255 |
|
haftmann@30328
|
256 |
subsubsection {* Domain *}
|
berghofe@22259
|
257 |
|
berghofe@23741
|
258 |
inductive
|
berghofe@22259
|
259 |
DomainP :: "('a => 'b => bool) => 'a => bool"
|
berghofe@22259
|
260 |
for r :: "'a => 'b => bool"
|
berghofe@22259
|
261 |
where
|
berghofe@22259
|
262 |
DomainPI [intro]: "r a b ==> DomainP r a"
|
berghofe@22259
|
263 |
|
berghofe@23741
|
264 |
inductive_cases DomainPE [elim!]: "DomainP r a"
|
berghofe@22259
|
265 |
|
berghofe@23741
|
266 |
lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
|
berghofe@26797
|
267 |
by (blast intro!: Orderings.order_antisym predicate1I)
|
berghofe@22259
|
268 |
|
berghofe@22259
|
269 |
|
haftmann@30328
|
270 |
subsubsection {* Range *}
|
berghofe@22259
|
271 |
|
berghofe@23741
|
272 |
inductive
|
berghofe@22259
|
273 |
RangeP :: "('a => 'b => bool) => 'b => bool"
|
berghofe@22259
|
274 |
for r :: "'a => 'b => bool"
|
berghofe@22259
|
275 |
where
|
berghofe@22259
|
276 |
RangePI [intro]: "r a b ==> RangeP r b"
|
berghofe@22259
|
277 |
|
berghofe@23741
|
278 |
inductive_cases RangePE [elim!]: "RangeP r b"
|
berghofe@22259
|
279 |
|
berghofe@23741
|
280 |
lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
|
berghofe@26797
|
281 |
by (blast intro!: Orderings.order_antisym predicate1I)
|
berghofe@22259
|
282 |
|
berghofe@22259
|
283 |
|
haftmann@30328
|
284 |
subsubsection {* Inverse image *}
|
berghofe@22259
|
285 |
|
berghofe@22259
|
286 |
definition
|
berghofe@22259
|
287 |
inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where
|
berghofe@22259
|
288 |
"inv_imagep r f == %x y. r (f x) (f y)"
|
berghofe@22259
|
289 |
|
berghofe@23741
|
290 |
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
|
291 |
by (simp add: inv_image_def inv_imagep_def)
|
berghofe@22259
|
292 |
|
berghofe@22259
|
293 |
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
|
berghofe@22259
|
294 |
by (simp add: inv_imagep_def)
|
berghofe@22259
|
295 |
|
berghofe@22259
|
296 |
|
haftmann@30328
|
297 |
subsubsection {* Powerset *}
|
berghofe@23741
|
298 |
|
berghofe@23741
|
299 |
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
|
berghofe@23741
|
300 |
"Powp A == \<lambda>B. \<forall>x \<in> B. A x"
|
berghofe@23741
|
301 |
|
berghofe@23741
|
302 |
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
|
berghofe@23741
|
303 |
by (auto simp add: Powp_def expand_fun_eq)
|
berghofe@23741
|
304 |
|
berghofe@26797
|
305 |
lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]
|
berghofe@26797
|
306 |
|
berghofe@23741
|
307 |
|
haftmann@30328
|
308 |
subsubsection {* Properties of relations *}
|
berghofe@22259
|
309 |
|
berghofe@22259
|
310 |
abbreviation antisymP :: "('a => 'a => bool) => bool" where
|
berghofe@23741
|
311 |
"antisymP r == antisym {(x, y). r x y}"
|
berghofe@22259
|
312 |
|
berghofe@22259
|
313 |
abbreviation transP :: "('a => 'a => bool) => bool" where
|
berghofe@23741
|
314 |
"transP r == trans {(x, y). r x y}"
|
berghofe@22259
|
315 |
|
berghofe@22259
|
316 |
abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
|
berghofe@23741
|
317 |
"single_valuedP r == single_valued {(x, y). r x y}"
|
berghofe@22259
|
318 |
|
haftmann@30328
|
319 |
|
haftmann@30328
|
320 |
subsection {* Predicates as enumerations *}
|
haftmann@30328
|
321 |
|
haftmann@30328
|
322 |
subsubsection {* The type of predicate enumerations (a monad) *}
|
haftmann@30328
|
323 |
|
haftmann@30328
|
324 |
datatype 'a pred = Pred "'a \<Rightarrow> bool"
|
haftmann@30328
|
325 |
|
haftmann@30328
|
326 |
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
|
haftmann@30328
|
327 |
eval_pred: "eval (Pred f) = f"
|
haftmann@30328
|
328 |
|
haftmann@30328
|
329 |
lemma Pred_eval [simp]:
|
haftmann@30328
|
330 |
"Pred (eval x) = x"
|
haftmann@30328
|
331 |
by (cases x) simp
|
haftmann@30328
|
332 |
|
haftmann@30328
|
333 |
lemma eval_inject: "eval x = eval y \<longleftrightarrow> x = y"
|
haftmann@30328
|
334 |
by (cases x) auto
|
haftmann@30328
|
335 |
|
haftmann@30328
|
336 |
definition single :: "'a \<Rightarrow> 'a pred" where
|
haftmann@30328
|
337 |
"single x = Pred ((op =) x)"
|
haftmann@30328
|
338 |
|
haftmann@30328
|
339 |
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
|
haftmann@30328
|
340 |
"P \<guillemotright>= f = Pred (\<lambda>x. (\<exists>y. eval P y \<and> eval (f y) x))"
|
haftmann@30328
|
341 |
|
haftmann@32578
|
342 |
instantiation pred :: (type) "{complete_lattice, boolean_algebra}"
|
haftmann@30328
|
343 |
begin
|
haftmann@30328
|
344 |
|
haftmann@30328
|
345 |
definition
|
haftmann@30328
|
346 |
"P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
|
haftmann@30328
|
347 |
|
haftmann@30328
|
348 |
definition
|
haftmann@30328
|
349 |
"P < Q \<longleftrightarrow> eval P < eval Q"
|
haftmann@30328
|
350 |
|
haftmann@30328
|
351 |
definition
|
haftmann@30328
|
352 |
"\<bottom> = Pred \<bottom>"
|
haftmann@30328
|
353 |
|
haftmann@30328
|
354 |
definition
|
haftmann@30328
|
355 |
"\<top> = Pred \<top>"
|
haftmann@30328
|
356 |
|
haftmann@30328
|
357 |
definition
|
haftmann@30328
|
358 |
"P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
|
haftmann@30328
|
359 |
|
haftmann@30328
|
360 |
definition
|
haftmann@30328
|
361 |
"P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
|
haftmann@30328
|
362 |
|
haftmann@30328
|
363 |
definition
|
haftmann@31931
|
364 |
[code del]: "\<Sqinter>A = Pred (INFI A eval)"
|
haftmann@30328
|
365 |
|
haftmann@30328
|
366 |
definition
|
haftmann@31931
|
367 |
[code del]: "\<Squnion>A = Pred (SUPR A eval)"
|
haftmann@30328
|
368 |
|
haftmann@32578
|
369 |
definition
|
haftmann@32578
|
370 |
"- P = Pred (- eval P)"
|
haftmann@32578
|
371 |
|
haftmann@32578
|
372 |
definition
|
haftmann@32578
|
373 |
"P - Q = Pred (eval P - eval Q)"
|
haftmann@32578
|
374 |
|
haftmann@32578
|
375 |
instance proof
|
haftmann@32578
|
376 |
qed (auto simp add: less_eq_pred_def less_pred_def
|
haftmann@30328
|
377 |
inf_pred_def sup_pred_def bot_pred_def top_pred_def
|
haftmann@32578
|
378 |
Inf_pred_def Sup_pred_def uminus_pred_def minus_pred_def fun_Compl_def bool_Compl_def,
|
haftmann@30328
|
379 |
auto simp add: le_fun_def less_fun_def le_bool_def less_bool_def
|
haftmann@30328
|
380 |
eval_inject mem_def)
|
haftmann@30328
|
381 |
|
berghofe@22259
|
382 |
end
|
haftmann@30328
|
383 |
|
haftmann@30328
|
384 |
lemma bind_bind:
|
haftmann@30328
|
385 |
"(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
|
haftmann@30328
|
386 |
by (auto simp add: bind_def expand_fun_eq)
|
haftmann@30328
|
387 |
|
haftmann@30328
|
388 |
lemma bind_single:
|
haftmann@30328
|
389 |
"P \<guillemotright>= single = P"
|
haftmann@30328
|
390 |
by (simp add: bind_def single_def)
|
haftmann@30328
|
391 |
|
haftmann@30328
|
392 |
lemma single_bind:
|
haftmann@30328
|
393 |
"single x \<guillemotright>= P = P x"
|
haftmann@30328
|
394 |
by (simp add: bind_def single_def)
|
haftmann@30328
|
395 |
|
haftmann@30328
|
396 |
lemma bottom_bind:
|
haftmann@30328
|
397 |
"\<bottom> \<guillemotright>= P = \<bottom>"
|
haftmann@30328
|
398 |
by (auto simp add: bot_pred_def bind_def expand_fun_eq)
|
haftmann@30328
|
399 |
|
haftmann@30328
|
400 |
lemma sup_bind:
|
haftmann@30328
|
401 |
"(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
|
haftmann@30328
|
402 |
by (auto simp add: bind_def sup_pred_def expand_fun_eq)
|
haftmann@30328
|
403 |
|
haftmann@30328
|
404 |
lemma Sup_bind: "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
|
haftmann@32601
|
405 |
by (auto simp add: bind_def Sup_pred_def SUP1_iff expand_fun_eq)
|
haftmann@30328
|
406 |
|
haftmann@30328
|
407 |
lemma pred_iffI:
|
haftmann@30328
|
408 |
assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
|
haftmann@30328
|
409 |
and "\<And>x. eval B x \<Longrightarrow> eval A x"
|
haftmann@30328
|
410 |
shows "A = B"
|
haftmann@30328
|
411 |
proof -
|
haftmann@30328
|
412 |
from assms have "\<And>x. eval A x \<longleftrightarrow> eval B x" by blast
|
haftmann@30328
|
413 |
then show ?thesis by (cases A, cases B) (simp add: expand_fun_eq)
|
haftmann@30328
|
414 |
qed
|
haftmann@30328
|
415 |
|
haftmann@30328
|
416 |
lemma singleI: "eval (single x) x"
|
haftmann@30328
|
417 |
unfolding single_def by simp
|
haftmann@30328
|
418 |
|
haftmann@30328
|
419 |
lemma singleI_unit: "eval (single ()) x"
|
haftmann@30328
|
420 |
by simp (rule singleI)
|
haftmann@30328
|
421 |
|
haftmann@30328
|
422 |
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
|
haftmann@30328
|
423 |
unfolding single_def by simp
|
haftmann@30328
|
424 |
|
haftmann@30328
|
425 |
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
|
haftmann@30328
|
426 |
by (erule singleE) simp
|
haftmann@30328
|
427 |
|
haftmann@30328
|
428 |
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
|
haftmann@30328
|
429 |
unfolding bind_def by auto
|
haftmann@30328
|
430 |
|
haftmann@30328
|
431 |
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
|
haftmann@30328
|
432 |
unfolding bind_def by auto
|
haftmann@30328
|
433 |
|
haftmann@30328
|
434 |
lemma botE: "eval \<bottom> x \<Longrightarrow> P"
|
haftmann@30328
|
435 |
unfolding bot_pred_def by auto
|
haftmann@30328
|
436 |
|
haftmann@30328
|
437 |
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
|
haftmann@32883
|
438 |
unfolding sup_pred_def by (simp add: sup_fun_eq sup_bool_eq)
|
haftmann@30328
|
439 |
|
haftmann@30328
|
440 |
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x"
|
haftmann@32883
|
441 |
unfolding sup_pred_def by (simp add: sup_fun_eq sup_bool_eq)
|
haftmann@30328
|
442 |
|
haftmann@30328
|
443 |
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
|
haftmann@30328
|
444 |
unfolding sup_pred_def by auto
|
haftmann@30328
|
445 |
|
haftmann@32578
|
446 |
lemma single_not_bot [simp]:
|
haftmann@32578
|
447 |
"single x \<noteq> \<bottom>"
|
haftmann@32578
|
448 |
by (auto simp add: single_def bot_pred_def expand_fun_eq)
|
haftmann@32578
|
449 |
|
haftmann@32578
|
450 |
lemma not_bot:
|
haftmann@32578
|
451 |
assumes "A \<noteq> \<bottom>"
|
haftmann@32578
|
452 |
obtains x where "eval A x"
|
haftmann@32578
|
453 |
using assms by (cases A)
|
haftmann@32578
|
454 |
(auto simp add: bot_pred_def, auto simp add: mem_def)
|
haftmann@32578
|
455 |
|
haftmann@32578
|
456 |
|
haftmann@32578
|
457 |
subsubsection {* Emptiness check and definite choice *}
|
haftmann@32578
|
458 |
|
haftmann@32578
|
459 |
definition is_empty :: "'a pred \<Rightarrow> bool" where
|
haftmann@32578
|
460 |
"is_empty A \<longleftrightarrow> A = \<bottom>"
|
haftmann@32578
|
461 |
|
haftmann@32578
|
462 |
lemma is_empty_bot:
|
haftmann@32578
|
463 |
"is_empty \<bottom>"
|
haftmann@32578
|
464 |
by (simp add: is_empty_def)
|
haftmann@32578
|
465 |
|
haftmann@32578
|
466 |
lemma not_is_empty_single:
|
haftmann@32578
|
467 |
"\<not> is_empty (single x)"
|
haftmann@32578
|
468 |
by (auto simp add: is_empty_def single_def bot_pred_def expand_fun_eq)
|
haftmann@32578
|
469 |
|
haftmann@32578
|
470 |
lemma is_empty_sup:
|
haftmann@32578
|
471 |
"is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
|
haftmann@32578
|
472 |
by (auto simp add: is_empty_def intro: sup_eq_bot_eq1 sup_eq_bot_eq2)
|
haftmann@32578
|
473 |
|
bulwahn@33111
|
474 |
definition singleton :: "(unit => 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
|
bulwahn@33111
|
475 |
"singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"
|
haftmann@32578
|
476 |
|
haftmann@32578
|
477 |
lemma singleton_eqI:
|
bulwahn@33110
|
478 |
"\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"
|
haftmann@32578
|
479 |
by (auto simp add: singleton_def)
|
haftmann@32578
|
480 |
|
haftmann@32578
|
481 |
lemma eval_singletonI:
|
bulwahn@33110
|
482 |
"\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"
|
haftmann@32578
|
483 |
proof -
|
haftmann@32578
|
484 |
assume assm: "\<exists>!x. eval A x"
|
haftmann@32578
|
485 |
then obtain x where "eval A x" ..
|
bulwahn@33110
|
486 |
moreover with assm have "singleton dfault A = x" by (rule singleton_eqI)
|
haftmann@32578
|
487 |
ultimately show ?thesis by simp
|
haftmann@32578
|
488 |
qed
|
haftmann@32578
|
489 |
|
haftmann@32578
|
490 |
lemma single_singleton:
|
bulwahn@33110
|
491 |
"\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"
|
haftmann@32578
|
492 |
proof -
|
haftmann@32578
|
493 |
assume assm: "\<exists>!x. eval A x"
|
bulwahn@33110
|
494 |
then have "eval A (singleton dfault A)"
|
haftmann@32578
|
495 |
by (rule eval_singletonI)
|
bulwahn@33110
|
496 |
moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"
|
haftmann@32578
|
497 |
by (rule singleton_eqI)
|
bulwahn@33110
|
498 |
ultimately have "eval (single (singleton dfault A)) = eval A"
|
haftmann@32578
|
499 |
by (simp (no_asm_use) add: single_def expand_fun_eq) blast
|
haftmann@32578
|
500 |
then show ?thesis by (simp add: eval_inject)
|
haftmann@32578
|
501 |
qed
|
haftmann@32578
|
502 |
|
haftmann@32578
|
503 |
lemma singleton_undefinedI:
|
bulwahn@33111
|
504 |
"\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"
|
haftmann@32578
|
505 |
by (simp add: singleton_def)
|
haftmann@32578
|
506 |
|
haftmann@32578
|
507 |
lemma singleton_bot:
|
bulwahn@33111
|
508 |
"singleton dfault \<bottom> = dfault ()"
|
haftmann@32578
|
509 |
by (auto simp add: bot_pred_def intro: singleton_undefinedI)
|
haftmann@32578
|
510 |
|
haftmann@32578
|
511 |
lemma singleton_single:
|
bulwahn@33110
|
512 |
"singleton dfault (single x) = x"
|
haftmann@32578
|
513 |
by (auto simp add: intro: singleton_eqI singleI elim: singleE)
|
haftmann@32578
|
514 |
|
haftmann@32578
|
515 |
lemma singleton_sup_single_single:
|
bulwahn@33111
|
516 |
"singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"
|
haftmann@32578
|
517 |
proof (cases "x = y")
|
haftmann@32578
|
518 |
case True then show ?thesis by (simp add: singleton_single)
|
haftmann@32578
|
519 |
next
|
haftmann@32578
|
520 |
case False
|
haftmann@32578
|
521 |
have "eval (single x \<squnion> single y) x"
|
haftmann@32578
|
522 |
and "eval (single x \<squnion> single y) y"
|
haftmann@32578
|
523 |
by (auto intro: supI1 supI2 singleI)
|
haftmann@32578
|
524 |
with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
|
haftmann@32578
|
525 |
by blast
|
bulwahn@33111
|
526 |
then have "singleton dfault (single x \<squnion> single y) = dfault ()"
|
haftmann@32578
|
527 |
by (rule singleton_undefinedI)
|
haftmann@32578
|
528 |
with False show ?thesis by simp
|
haftmann@32578
|
529 |
qed
|
haftmann@32578
|
530 |
|
haftmann@32578
|
531 |
lemma singleton_sup_aux:
|
bulwahn@33110
|
532 |
"singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
|
bulwahn@33110
|
533 |
else if B = \<bottom> then singleton dfault A
|
bulwahn@33110
|
534 |
else singleton dfault
|
bulwahn@33110
|
535 |
(single (singleton dfault A) \<squnion> single (singleton dfault B)))"
|
haftmann@32578
|
536 |
proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
|
haftmann@32578
|
537 |
case True then show ?thesis by (simp add: single_singleton)
|
haftmann@32578
|
538 |
next
|
haftmann@32578
|
539 |
case False
|
haftmann@32578
|
540 |
from False have A_or_B:
|
bulwahn@33111
|
541 |
"singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
|
haftmann@32578
|
542 |
by (auto intro!: singleton_undefinedI)
|
bulwahn@33110
|
543 |
then have rhs: "singleton dfault
|
bulwahn@33111
|
544 |
(single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
|
haftmann@32578
|
545 |
by (auto simp add: singleton_sup_single_single singleton_single)
|
haftmann@32578
|
546 |
from False have not_unique:
|
haftmann@32578
|
547 |
"\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
|
haftmann@32578
|
548 |
show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
|
haftmann@32578
|
549 |
case True
|
haftmann@32578
|
550 |
then obtain a b where a: "eval A a" and b: "eval B b"
|
haftmann@32578
|
551 |
by (blast elim: not_bot)
|
haftmann@32578
|
552 |
with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
|
haftmann@32578
|
553 |
by (auto simp add: sup_pred_def bot_pred_def)
|
bulwahn@33111
|
554 |
then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)
|
haftmann@32578
|
555 |
with True rhs show ?thesis by simp
|
haftmann@32578
|
556 |
next
|
haftmann@32578
|
557 |
case False then show ?thesis by auto
|
haftmann@32578
|
558 |
qed
|
haftmann@32578
|
559 |
qed
|
haftmann@32578
|
560 |
|
haftmann@32578
|
561 |
lemma singleton_sup:
|
bulwahn@33110
|
562 |
"singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
|
bulwahn@33110
|
563 |
else if B = \<bottom> then singleton dfault A
|
bulwahn@33111
|
564 |
else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
|
bulwahn@33110
|
565 |
using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
|
haftmann@32578
|
566 |
|
haftmann@30328
|
567 |
|
haftmann@30328
|
568 |
subsubsection {* Derived operations *}
|
haftmann@30328
|
569 |
|
haftmann@30328
|
570 |
definition if_pred :: "bool \<Rightarrow> unit pred" where
|
haftmann@30328
|
571 |
if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
|
haftmann@30328
|
572 |
|
bulwahn@33754
|
573 |
definition holds :: "unit pred \<Rightarrow> bool" where
|
bulwahn@33754
|
574 |
holds_eq: "holds P = eval P ()"
|
bulwahn@33754
|
575 |
|
haftmann@30328
|
576 |
definition not_pred :: "unit pred \<Rightarrow> unit pred" where
|
haftmann@30328
|
577 |
not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
|
haftmann@30328
|
578 |
|
haftmann@30328
|
579 |
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
|
haftmann@30328
|
580 |
unfolding if_pred_eq by (auto intro: singleI)
|
haftmann@30328
|
581 |
|
haftmann@30328
|
582 |
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
|
haftmann@30328
|
583 |
unfolding if_pred_eq by (cases b) (auto elim: botE)
|
haftmann@30328
|
584 |
|
haftmann@30328
|
585 |
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
|
haftmann@30328
|
586 |
unfolding not_pred_eq eval_pred by (auto intro: singleI)
|
haftmann@30328
|
587 |
|
haftmann@30328
|
588 |
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
|
haftmann@30328
|
589 |
unfolding not_pred_eq by (auto intro: singleI)
|
haftmann@30328
|
590 |
|
haftmann@30328
|
591 |
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
|
haftmann@30328
|
592 |
unfolding not_pred_eq
|
haftmann@30328
|
593 |
by (auto split: split_if_asm elim: botE)
|
haftmann@30328
|
594 |
|
haftmann@30328
|
595 |
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
|
haftmann@30328
|
596 |
unfolding not_pred_eq
|
haftmann@30328
|
597 |
by (auto split: split_if_asm elim: botE)
|
bulwahn@33754
|
598 |
lemma "f () = False \<or> f () = True"
|
bulwahn@33754
|
599 |
by simp
|
haftmann@30328
|
600 |
|
bulwahn@33754
|
601 |
lemma closure_of_bool_cases:
|
bulwahn@33754
|
602 |
assumes "(f :: unit \<Rightarrow> bool) = (%u. False) \<Longrightarrow> P f"
|
bulwahn@33754
|
603 |
assumes "f = (%u. True) \<Longrightarrow> P f"
|
bulwahn@33754
|
604 |
shows "P f"
|
bulwahn@33754
|
605 |
proof -
|
bulwahn@33754
|
606 |
have "f = (%u. False) \<or> f = (%u. True)"
|
bulwahn@33754
|
607 |
apply (cases "f ()")
|
bulwahn@33754
|
608 |
apply (rule disjI2)
|
bulwahn@33754
|
609 |
apply (rule ext)
|
bulwahn@33754
|
610 |
apply (simp add: unit_eq)
|
bulwahn@33754
|
611 |
apply (rule disjI1)
|
bulwahn@33754
|
612 |
apply (rule ext)
|
bulwahn@33754
|
613 |
apply (simp add: unit_eq)
|
bulwahn@33754
|
614 |
done
|
bulwahn@33754
|
615 |
from this prems show ?thesis by blast
|
bulwahn@33754
|
616 |
qed
|
bulwahn@33754
|
617 |
|
bulwahn@33754
|
618 |
lemma unit_pred_cases:
|
bulwahn@33754
|
619 |
assumes "P \<bottom>"
|
bulwahn@33754
|
620 |
assumes "P (single ())"
|
bulwahn@33754
|
621 |
shows "P Q"
|
bulwahn@33754
|
622 |
using assms
|
bulwahn@33754
|
623 |
unfolding bot_pred_def Collect_def empty_def single_def
|
bulwahn@33754
|
624 |
apply (cases Q)
|
bulwahn@33754
|
625 |
apply simp
|
bulwahn@33754
|
626 |
apply (rule_tac f="fun" in closure_of_bool_cases)
|
bulwahn@33754
|
627 |
apply auto
|
bulwahn@33754
|
628 |
apply (subgoal_tac "(%x. () = x) = (%x. True)")
|
bulwahn@33754
|
629 |
apply auto
|
bulwahn@33754
|
630 |
done
|
bulwahn@33754
|
631 |
|
bulwahn@33754
|
632 |
lemma holds_if_pred:
|
bulwahn@33754
|
633 |
"holds (if_pred b) = b"
|
bulwahn@33754
|
634 |
unfolding if_pred_eq holds_eq
|
bulwahn@33754
|
635 |
by (cases b) (auto intro: singleI elim: botE)
|
bulwahn@33754
|
636 |
|
bulwahn@33754
|
637 |
lemma if_pred_holds:
|
bulwahn@33754
|
638 |
"if_pred (holds P) = P"
|
bulwahn@33754
|
639 |
unfolding if_pred_eq holds_eq
|
bulwahn@33754
|
640 |
by (rule unit_pred_cases) (auto intro: singleI elim: botE)
|
bulwahn@33754
|
641 |
|
bulwahn@33754
|
642 |
lemma is_empty_holds:
|
bulwahn@33754
|
643 |
"is_empty P \<longleftrightarrow> \<not> holds P"
|
bulwahn@33754
|
644 |
unfolding is_empty_def holds_eq
|
bulwahn@33754
|
645 |
by (rule unit_pred_cases) (auto elim: botE intro: singleI)
|
haftmann@30328
|
646 |
|
haftmann@30328
|
647 |
subsubsection {* Implementation *}
|
haftmann@30328
|
648 |
|
haftmann@30328
|
649 |
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
|
haftmann@30328
|
650 |
|
haftmann@30328
|
651 |
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
|
haftmann@30328
|
652 |
"pred_of_seq Empty = \<bottom>"
|
haftmann@30328
|
653 |
| "pred_of_seq (Insert x P) = single x \<squnion> P"
|
haftmann@30328
|
654 |
| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
|
haftmann@30328
|
655 |
|
haftmann@30328
|
656 |
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
|
haftmann@30328
|
657 |
"Seq f = pred_of_seq (f ())"
|
haftmann@30328
|
658 |
|
haftmann@30328
|
659 |
code_datatype Seq
|
haftmann@30328
|
660 |
|
haftmann@30328
|
661 |
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool" where
|
haftmann@30328
|
662 |
"member Empty x \<longleftrightarrow> False"
|
haftmann@30328
|
663 |
| "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
|
haftmann@30328
|
664 |
| "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
|
haftmann@30328
|
665 |
|
haftmann@30328
|
666 |
lemma eval_member:
|
haftmann@30328
|
667 |
"member xq = eval (pred_of_seq xq)"
|
haftmann@30328
|
668 |
proof (induct xq)
|
haftmann@30328
|
669 |
case Empty show ?case
|
haftmann@30328
|
670 |
by (auto simp add: expand_fun_eq elim: botE)
|
haftmann@30328
|
671 |
next
|
haftmann@30328
|
672 |
case Insert show ?case
|
haftmann@30328
|
673 |
by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI)
|
haftmann@30328
|
674 |
next
|
haftmann@30328
|
675 |
case Join then show ?case
|
haftmann@30328
|
676 |
by (auto simp add: expand_fun_eq elim: supE intro: supI1 supI2)
|
haftmann@30328
|
677 |
qed
|
haftmann@30328
|
678 |
|
haftmann@30328
|
679 |
lemma eval_code [code]: "eval (Seq f) = member (f ())"
|
haftmann@30328
|
680 |
unfolding Seq_def by (rule sym, rule eval_member)
|
haftmann@30328
|
681 |
|
haftmann@30328
|
682 |
lemma single_code [code]:
|
haftmann@30328
|
683 |
"single x = Seq (\<lambda>u. Insert x \<bottom>)"
|
haftmann@30328
|
684 |
unfolding Seq_def by simp
|
haftmann@30328
|
685 |
|
haftmann@30328
|
686 |
primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
|
haftmann@30328
|
687 |
"apply f Empty = Empty"
|
haftmann@30328
|
688 |
| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
|
haftmann@30328
|
689 |
| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
|
haftmann@30328
|
690 |
|
haftmann@30328
|
691 |
lemma apply_bind:
|
haftmann@30328
|
692 |
"pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
|
haftmann@30328
|
693 |
proof (induct xq)
|
haftmann@30328
|
694 |
case Empty show ?case
|
haftmann@30328
|
695 |
by (simp add: bottom_bind)
|
haftmann@30328
|
696 |
next
|
haftmann@30328
|
697 |
case Insert show ?case
|
haftmann@30328
|
698 |
by (simp add: single_bind sup_bind)
|
haftmann@30328
|
699 |
next
|
haftmann@30328
|
700 |
case Join then show ?case
|
haftmann@30328
|
701 |
by (simp add: sup_bind)
|
haftmann@30328
|
702 |
qed
|
haftmann@30328
|
703 |
|
haftmann@30328
|
704 |
lemma bind_code [code]:
|
haftmann@30328
|
705 |
"Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
|
haftmann@30328
|
706 |
unfolding Seq_def by (rule sym, rule apply_bind)
|
haftmann@30328
|
707 |
|
haftmann@30328
|
708 |
lemma bot_set_code [code]:
|
haftmann@30328
|
709 |
"\<bottom> = Seq (\<lambda>u. Empty)"
|
haftmann@30328
|
710 |
unfolding Seq_def by simp
|
haftmann@30328
|
711 |
|
haftmann@30376
|
712 |
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
|
haftmann@30376
|
713 |
"adjunct P Empty = Join P Empty"
|
haftmann@30376
|
714 |
| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
|
haftmann@30376
|
715 |
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
|
haftmann@30376
|
716 |
|
haftmann@30376
|
717 |
lemma adjunct_sup:
|
haftmann@30376
|
718 |
"pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
|
haftmann@30376
|
719 |
by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
|
haftmann@30376
|
720 |
|
haftmann@30328
|
721 |
lemma sup_code [code]:
|
haftmann@30328
|
722 |
"Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
|
haftmann@30328
|
723 |
of Empty \<Rightarrow> g ()
|
haftmann@30328
|
724 |
| Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
|
haftmann@30376
|
725 |
| Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
|
haftmann@30328
|
726 |
proof (cases "f ()")
|
haftmann@30328
|
727 |
case Empty
|
haftmann@30328
|
728 |
thus ?thesis
|
haftmann@30376
|
729 |
unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"] sup_bot)
|
haftmann@30328
|
730 |
next
|
haftmann@30328
|
731 |
case Insert
|
haftmann@30328
|
732 |
thus ?thesis
|
haftmann@30328
|
733 |
unfolding Seq_def by (simp add: sup_assoc)
|
haftmann@30328
|
734 |
next
|
haftmann@30328
|
735 |
case Join
|
haftmann@30328
|
736 |
thus ?thesis
|
haftmann@30376
|
737 |
unfolding Seq_def
|
haftmann@30376
|
738 |
by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
|
haftmann@30328
|
739 |
qed
|
haftmann@30328
|
740 |
|
haftmann@30430
|
741 |
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
|
haftmann@30430
|
742 |
"contained Empty Q \<longleftrightarrow> True"
|
haftmann@30430
|
743 |
| "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
|
haftmann@30430
|
744 |
| "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
|
haftmann@30430
|
745 |
|
haftmann@30430
|
746 |
lemma single_less_eq_eval:
|
haftmann@30430
|
747 |
"single x \<le> P \<longleftrightarrow> eval P x"
|
haftmann@30430
|
748 |
by (auto simp add: single_def less_eq_pred_def mem_def)
|
haftmann@30430
|
749 |
|
haftmann@30430
|
750 |
lemma contained_less_eq:
|
haftmann@30430
|
751 |
"contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
|
haftmann@30430
|
752 |
by (induct xq) (simp_all add: single_less_eq_eval)
|
haftmann@30430
|
753 |
|
haftmann@30430
|
754 |
lemma less_eq_pred_code [code]:
|
haftmann@30430
|
755 |
"Seq f \<le> Q = (case f ()
|
haftmann@30430
|
756 |
of Empty \<Rightarrow> True
|
haftmann@30430
|
757 |
| Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
|
haftmann@30430
|
758 |
| Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
|
haftmann@30430
|
759 |
by (cases "f ()")
|
haftmann@30430
|
760 |
(simp_all add: Seq_def single_less_eq_eval contained_less_eq)
|
haftmann@30430
|
761 |
|
haftmann@30430
|
762 |
lemma eq_pred_code [code]:
|
haftmann@31133
|
763 |
fixes P Q :: "'a pred"
|
haftmann@30430
|
764 |
shows "eq_class.eq P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
|
haftmann@30430
|
765 |
unfolding eq by auto
|
haftmann@30430
|
766 |
|
haftmann@30430
|
767 |
lemma [code]:
|
haftmann@30430
|
768 |
"pred_case f P = f (eval P)"
|
haftmann@30430
|
769 |
by (cases P) simp
|
haftmann@30430
|
770 |
|
haftmann@30430
|
771 |
lemma [code]:
|
haftmann@30430
|
772 |
"pred_rec f P = f (eval P)"
|
haftmann@30430
|
773 |
by (cases P) simp
|
haftmann@30328
|
774 |
|
bulwahn@31105
|
775 |
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
|
bulwahn@31105
|
776 |
|
bulwahn@31105
|
777 |
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
|
haftmann@31108
|
778 |
by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
|
haftmann@30948
|
779 |
|
haftmann@31216
|
780 |
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
|
haftmann@31216
|
781 |
"map f P = P \<guillemotright>= (single o f)"
|
haftmann@31216
|
782 |
|
haftmann@32578
|
783 |
primrec null :: "'a seq \<Rightarrow> bool" where
|
haftmann@32578
|
784 |
"null Empty \<longleftrightarrow> True"
|
haftmann@32578
|
785 |
| "null (Insert x P) \<longleftrightarrow> False"
|
haftmann@32578
|
786 |
| "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
|
haftmann@32578
|
787 |
|
haftmann@32578
|
788 |
lemma null_is_empty:
|
haftmann@32578
|
789 |
"null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
|
haftmann@32578
|
790 |
by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
|
haftmann@32578
|
791 |
|
haftmann@32578
|
792 |
lemma is_empty_code [code]:
|
haftmann@32578
|
793 |
"is_empty (Seq f) \<longleftrightarrow> null (f ())"
|
haftmann@32578
|
794 |
by (simp add: null_is_empty Seq_def)
|
haftmann@32578
|
795 |
|
bulwahn@33111
|
796 |
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
|
bulwahn@33111
|
797 |
[code del]: "the_only dfault Empty = dfault ()"
|
bulwahn@33111
|
798 |
| "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
|
799 |
| "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
|
800 |
else let x = singleton dfault P; y = the_only dfault xq in
|
bulwahn@33111
|
801 |
if x = y then x else dfault ())"
|
haftmann@32578
|
802 |
|
haftmann@32578
|
803 |
lemma the_only_singleton:
|
bulwahn@33110
|
804 |
"the_only dfault xq = singleton dfault (pred_of_seq xq)"
|
haftmann@32578
|
805 |
by (induct xq)
|
haftmann@32578
|
806 |
(auto simp add: singleton_bot singleton_single is_empty_def
|
haftmann@32578
|
807 |
null_is_empty Let_def singleton_sup)
|
haftmann@32578
|
808 |
|
haftmann@32578
|
809 |
lemma singleton_code [code]:
|
bulwahn@33110
|
810 |
"singleton dfault (Seq f) = (case f ()
|
bulwahn@33111
|
811 |
of Empty \<Rightarrow> dfault ()
|
haftmann@32578
|
812 |
| Insert x P \<Rightarrow> if is_empty P then x
|
bulwahn@33110
|
813 |
else let y = singleton dfault P in
|
bulwahn@33111
|
814 |
if x = y then x else dfault ()
|
bulwahn@33110
|
815 |
| Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
|
bulwahn@33110
|
816 |
else if null xq then singleton dfault P
|
bulwahn@33110
|
817 |
else let x = singleton dfault P; y = the_only dfault xq in
|
bulwahn@33111
|
818 |
if x = y then x else dfault ())"
|
haftmann@32578
|
819 |
by (cases "f ()")
|
haftmann@32578
|
820 |
(auto simp add: Seq_def the_only_singleton is_empty_def
|
haftmann@32578
|
821 |
null_is_empty singleton_bot singleton_single singleton_sup Let_def)
|
haftmann@32578
|
822 |
|
bulwahn@33110
|
823 |
definition not_unique :: "'a pred => 'a"
|
bulwahn@33110
|
824 |
where
|
bulwahn@33111
|
825 |
[code del]: "not_unique A = (THE x. eval A x)"
|
bulwahn@33110
|
826 |
|
bulwahn@33111
|
827 |
definition the :: "'a pred => 'a"
|
bulwahn@33111
|
828 |
where
|
bulwahn@33111
|
829 |
[code del]: "the A = (THE x. eval A x)"
|
bulwahn@33111
|
830 |
|
bulwahn@33111
|
831 |
lemma the_eq[code]: "the A = singleton (\<lambda>x. not_unique A) A"
|
bulwahn@33111
|
832 |
by (auto simp add: the_def singleton_def not_unique_def)
|
bulwahn@33110
|
833 |
|
haftmann@33985
|
834 |
code_abort not_unique
|
haftmann@33985
|
835 |
|
haftmann@30948
|
836 |
ML {*
|
haftmann@30948
|
837 |
signature PREDICATE =
|
haftmann@30948
|
838 |
sig
|
haftmann@30948
|
839 |
datatype 'a pred = Seq of (unit -> 'a seq)
|
haftmann@30948
|
840 |
and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
|
haftmann@30959
|
841 |
val yield: 'a pred -> ('a * 'a pred) option
|
haftmann@30959
|
842 |
val yieldn: int -> 'a pred -> 'a list * 'a pred
|
haftmann@31222
|
843 |
val map: ('a -> 'b) -> 'a pred -> 'b pred
|
haftmann@30948
|
844 |
end;
|
haftmann@30948
|
845 |
|
haftmann@30948
|
846 |
structure Predicate : PREDICATE =
|
haftmann@30948
|
847 |
struct
|
haftmann@30948
|
848 |
|
haftmann@30959
|
849 |
@{code_datatype pred = Seq};
|
haftmann@30959
|
850 |
@{code_datatype seq = Empty | Insert | Join};
|
haftmann@30959
|
851 |
|
haftmann@32372
|
852 |
fun yield (@{code Seq} f) = next (f ())
|
haftmann@30959
|
853 |
and next @{code Empty} = NONE
|
haftmann@30959
|
854 |
| next (@{code Insert} (x, P)) = SOME (x, P)
|
haftmann@30959
|
855 |
| next (@{code Join} (P, xq)) = (case yield P
|
haftmann@30959
|
856 |
of NONE => next xq
|
haftmann@33607
|
857 |
| SOME (x, Q) => SOME (x, @{code Seq} (fn _ => @{code Join} (Q, xq))));
|
haftmann@30959
|
858 |
|
haftmann@30959
|
859 |
fun anamorph f k x = (if k = 0 then ([], x)
|
haftmann@30959
|
860 |
else case f x
|
haftmann@30959
|
861 |
of NONE => ([], x)
|
haftmann@30959
|
862 |
| SOME (v, y) => let
|
haftmann@30959
|
863 |
val (vs, z) = anamorph f (k - 1) y
|
haftmann@33607
|
864 |
in (v :: vs, z) end);
|
haftmann@30959
|
865 |
|
haftmann@30959
|
866 |
fun yieldn P = anamorph yield P;
|
haftmann@30948
|
867 |
|
haftmann@31222
|
868 |
fun map f = @{code map} f;
|
haftmann@31222
|
869 |
|
haftmann@30948
|
870 |
end;
|
haftmann@30948
|
871 |
*}
|
haftmann@30948
|
872 |
|
haftmann@30948
|
873 |
code_reserved Eval Predicate
|
haftmann@30948
|
874 |
|
haftmann@30948
|
875 |
code_type pred and seq
|
haftmann@30948
|
876 |
(Eval "_/ Predicate.pred" and "_/ Predicate.seq")
|
haftmann@30948
|
877 |
|
haftmann@30948
|
878 |
code_const Seq and Empty and Insert and Join
|
haftmann@30948
|
879 |
(Eval "Predicate.Seq" and "Predicate.Empty" and "Predicate.Insert/ (_,/ _)" and "Predicate.Join/ (_,/ _)")
|
haftmann@30948
|
880 |
|
haftmann@30328
|
881 |
no_notation
|
haftmann@30328
|
882 |
inf (infixl "\<sqinter>" 70) and
|
haftmann@30328
|
883 |
sup (infixl "\<squnion>" 65) and
|
haftmann@30328
|
884 |
Inf ("\<Sqinter>_" [900] 900) and
|
haftmann@30328
|
885 |
Sup ("\<Squnion>_" [900] 900) and
|
haftmann@30328
|
886 |
top ("\<top>") and
|
haftmann@30328
|
887 |
bot ("\<bottom>") and
|
haftmann@30328
|
888 |
bind (infixl "\<guillemotright>=" 70)
|
haftmann@30328
|
889 |
|
haftmann@30328
|
890 |
hide (open) type pred seq
|
bulwahn@33754
|
891 |
hide (open) const Pred eval single bind is_empty singleton if_pred not_pred holds
|
bulwahn@33111
|
892 |
Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the
|
haftmann@30328
|
893 |
|
haftmann@30328
|
894 |
end
|