src/HOL/Predicate.thy
author haftmann
Tue Oct 06 18:44:06 2009 +0200 (2009-10-06)
changeset 32883 7cbd93dacef3
parent 32782 faf347097852
child 33104 560372b461e5
permissions -rw-r--r--
inf/sup1/2_iff are mere duplicates of underlying definitions: dropped
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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 ="
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  by (auto simp add: expand_fun_eq)
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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)"
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  by (blast intro!: Orderings.order_antisym predicate1I)
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subsubsection {* Range *}
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inductive
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  RangeP :: "('a => 'b => bool) => 'b => bool"
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  for r :: "'a => 'b => bool"
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where
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  RangePI [intro]: "r a b ==> RangeP r b"
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inductive_cases RangePE [elim!]: "RangeP r b"
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lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
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  by (blast intro!: Orderings.order_antisym predicate1I)
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subsubsection {* Inverse image *}
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definition
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  inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where
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  "inv_imagep r f == %x y. r (f x) (f y)"
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lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
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  by (simp add: inv_image_def inv_imagep_def)
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lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
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  by (simp add: inv_imagep_def)
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subsubsection {* Powerset *}
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definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
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  "Powp A == \<lambda>B. \<forall>x \<in> B. A x"
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lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
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  by (auto simp add: Powp_def expand_fun_eq)
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lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]
berghofe@26797
   306
berghofe@23741
   307
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   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]:
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   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@31932
   364
  [code del]: "\<Sqinter>A = Pred (INFI A eval)"
haftmann@30328
   365
haftmann@30328
   366
definition
haftmann@31932
   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
haftmann@32578
   474
definition singleton :: "'a pred \<Rightarrow> 'a" where
haftmann@32578
   475
  "singleton A = (if \<exists>!x. eval A x then THE x. eval A x else undefined)"
haftmann@32578
   476
haftmann@32578
   477
lemma singleton_eqI:
haftmann@32578
   478
  "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton A = x"
haftmann@32578
   479
  by (auto simp add: singleton_def)
haftmann@32578
   480
haftmann@32578
   481
lemma eval_singletonI:
haftmann@32578
   482
  "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton 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" ..
haftmann@32578
   486
  moreover with assm have "singleton 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:
haftmann@32578
   491
  "\<exists>!x. eval A x \<Longrightarrow> single (singleton A) = A"
haftmann@32578
   492
proof -
haftmann@32578
   493
  assume assm: "\<exists>!x. eval A x"
haftmann@32578
   494
  then have "eval A (singleton A)"
haftmann@32578
   495
    by (rule eval_singletonI)
haftmann@32578
   496
  moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton A = x"
haftmann@32578
   497
    by (rule singleton_eqI)
haftmann@32578
   498
  ultimately have "eval (single (singleton 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:
haftmann@32578
   504
  "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton A = undefined"
haftmann@32578
   505
  by (simp add: singleton_def)
haftmann@32578
   506
haftmann@32578
   507
lemma singleton_bot:
haftmann@32578
   508
  "singleton \<bottom> = undefined"
haftmann@32578
   509
  by (auto simp add: bot_pred_def intro: singleton_undefinedI)
haftmann@32578
   510
haftmann@32578
   511
lemma singleton_single:
haftmann@32578
   512
  "singleton (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:
haftmann@32578
   516
  "singleton (single x \<squnion> single y) = (if x = y then x else undefined)"
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
haftmann@32578
   526
  then have "singleton (single x \<squnion> single y) = undefined"
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:
haftmann@32578
   532
  "singleton (A \<squnion> B) = (if A = \<bottom> then singleton B
haftmann@32578
   533
    else if B = \<bottom> then singleton A
haftmann@32578
   534
    else singleton
haftmann@32578
   535
      (single (singleton A) \<squnion> single (singleton 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:
haftmann@32578
   541
    "singleton A = undefined \<or> singleton B = undefined"
haftmann@32578
   542
    by (auto intro!: singleton_undefinedI)
haftmann@32578
   543
  then have rhs: "singleton
haftmann@32578
   544
    (single (singleton A) \<squnion> single (singleton B)) = undefined"
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)
haftmann@32578
   554
    then have "singleton (A \<squnion> B) = undefined" 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:
haftmann@32578
   562
  "singleton (A \<squnion> B) = (if A = \<bottom> then singleton B
haftmann@32578
   563
    else if B = \<bottom> then singleton A
haftmann@32578
   564
    else if singleton A = singleton B then singleton A else undefined)"
haftmann@32578
   565
using singleton_sup_aux [of 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
haftmann@30328
   573
definition not_pred :: "unit pred \<Rightarrow> unit pred" where
haftmann@30328
   574
  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
haftmann@30328
   575
haftmann@30328
   576
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
haftmann@30328
   577
  unfolding if_pred_eq by (auto intro: singleI)
haftmann@30328
   578
haftmann@30328
   579
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30328
   580
  unfolding if_pred_eq by (cases b) (auto elim: botE)
haftmann@30328
   581
haftmann@30328
   582
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
haftmann@30328
   583
  unfolding not_pred_eq eval_pred by (auto intro: singleI)
haftmann@30328
   584
haftmann@30328
   585
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
haftmann@30328
   586
  unfolding not_pred_eq by (auto intro: singleI)
haftmann@30328
   587
haftmann@30328
   588
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   589
  unfolding not_pred_eq
haftmann@30328
   590
  by (auto split: split_if_asm elim: botE)
haftmann@30328
   591
haftmann@30328
   592
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   593
  unfolding not_pred_eq
haftmann@30328
   594
  by (auto split: split_if_asm elim: botE)
haftmann@30328
   595
haftmann@30328
   596
haftmann@30328
   597
subsubsection {* Implementation *}
haftmann@30328
   598
haftmann@30328
   599
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
haftmann@30328
   600
haftmann@30328
   601
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
haftmann@30328
   602
    "pred_of_seq Empty = \<bottom>"
haftmann@30328
   603
  | "pred_of_seq (Insert x P) = single x \<squnion> P"
haftmann@30328
   604
  | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
haftmann@30328
   605
haftmann@30328
   606
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
haftmann@30328
   607
  "Seq f = pred_of_seq (f ())"
haftmann@30328
   608
haftmann@30328
   609
code_datatype Seq
haftmann@30328
   610
haftmann@30328
   611
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
haftmann@30328
   612
  "member Empty x \<longleftrightarrow> False"
haftmann@30328
   613
  | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
haftmann@30328
   614
  | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
haftmann@30328
   615
haftmann@30328
   616
lemma eval_member:
haftmann@30328
   617
  "member xq = eval (pred_of_seq xq)"
haftmann@30328
   618
proof (induct xq)
haftmann@30328
   619
  case Empty show ?case
haftmann@30328
   620
  by (auto simp add: expand_fun_eq elim: botE)
haftmann@30328
   621
next
haftmann@30328
   622
  case Insert show ?case
haftmann@30328
   623
  by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI)
haftmann@30328
   624
next
haftmann@30328
   625
  case Join then show ?case
haftmann@30328
   626
  by (auto simp add: expand_fun_eq elim: supE intro: supI1 supI2)
haftmann@30328
   627
qed
haftmann@30328
   628
haftmann@30328
   629
lemma eval_code [code]: "eval (Seq f) = member (f ())"
haftmann@30328
   630
  unfolding Seq_def by (rule sym, rule eval_member)
haftmann@30328
   631
haftmann@30328
   632
lemma single_code [code]:
haftmann@30328
   633
  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
haftmann@30328
   634
  unfolding Seq_def by simp
haftmann@30328
   635
haftmann@30328
   636
primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
haftmann@30328
   637
    "apply f Empty = Empty"
haftmann@30328
   638
  | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
haftmann@30328
   639
  | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
haftmann@30328
   640
haftmann@30328
   641
lemma apply_bind:
haftmann@30328
   642
  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
haftmann@30328
   643
proof (induct xq)
haftmann@30328
   644
  case Empty show ?case
haftmann@30328
   645
    by (simp add: bottom_bind)
haftmann@30328
   646
next
haftmann@30328
   647
  case Insert show ?case
haftmann@30328
   648
    by (simp add: single_bind sup_bind)
haftmann@30328
   649
next
haftmann@30328
   650
  case Join then show ?case
haftmann@30328
   651
    by (simp add: sup_bind)
haftmann@30328
   652
qed
haftmann@30328
   653
  
haftmann@30328
   654
lemma bind_code [code]:
haftmann@30328
   655
  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
haftmann@30328
   656
  unfolding Seq_def by (rule sym, rule apply_bind)
haftmann@30328
   657
haftmann@30328
   658
lemma bot_set_code [code]:
haftmann@30328
   659
  "\<bottom> = Seq (\<lambda>u. Empty)"
haftmann@30328
   660
  unfolding Seq_def by simp
haftmann@30328
   661
haftmann@30376
   662
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
haftmann@30376
   663
    "adjunct P Empty = Join P Empty"
haftmann@30376
   664
  | "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
haftmann@30376
   665
  | "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
haftmann@30376
   666
haftmann@30376
   667
lemma adjunct_sup:
haftmann@30376
   668
  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
haftmann@30376
   669
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
haftmann@30376
   670
haftmann@30328
   671
lemma sup_code [code]:
haftmann@30328
   672
  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
haftmann@30328
   673
    of Empty \<Rightarrow> g ()
haftmann@30328
   674
     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
haftmann@30376
   675
     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
haftmann@30328
   676
proof (cases "f ()")
haftmann@30328
   677
  case Empty
haftmann@30328
   678
  thus ?thesis
haftmann@30376
   679
    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"]  sup_bot)
haftmann@30328
   680
next
haftmann@30328
   681
  case Insert
haftmann@30328
   682
  thus ?thesis
haftmann@30328
   683
    unfolding Seq_def by (simp add: sup_assoc)
haftmann@30328
   684
next
haftmann@30328
   685
  case Join
haftmann@30328
   686
  thus ?thesis
haftmann@30376
   687
    unfolding Seq_def
haftmann@30376
   688
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
haftmann@30328
   689
qed
haftmann@30328
   690
haftmann@30430
   691
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
haftmann@30430
   692
    "contained Empty Q \<longleftrightarrow> True"
haftmann@30430
   693
  | "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
haftmann@30430
   694
  | "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
haftmann@30430
   695
haftmann@30430
   696
lemma single_less_eq_eval:
haftmann@30430
   697
  "single x \<le> P \<longleftrightarrow> eval P x"
haftmann@30430
   698
  by (auto simp add: single_def less_eq_pred_def mem_def)
haftmann@30430
   699
haftmann@30430
   700
lemma contained_less_eq:
haftmann@30430
   701
  "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
haftmann@30430
   702
  by (induct xq) (simp_all add: single_less_eq_eval)
haftmann@30430
   703
haftmann@30430
   704
lemma less_eq_pred_code [code]:
haftmann@30430
   705
  "Seq f \<le> Q = (case f ()
haftmann@30430
   706
   of Empty \<Rightarrow> True
haftmann@30430
   707
    | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
haftmann@30430
   708
    | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
haftmann@30430
   709
  by (cases "f ()")
haftmann@30430
   710
    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
haftmann@30430
   711
haftmann@30430
   712
lemma eq_pred_code [code]:
haftmann@31133
   713
  fixes P Q :: "'a pred"
haftmann@30430
   714
  shows "eq_class.eq P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
haftmann@30430
   715
  unfolding eq by auto
haftmann@30430
   716
haftmann@30430
   717
lemma [code]:
haftmann@30430
   718
  "pred_case f P = f (eval P)"
haftmann@30430
   719
  by (cases P) simp
haftmann@30430
   720
haftmann@30430
   721
lemma [code]:
haftmann@30430
   722
  "pred_rec f P = f (eval P)"
haftmann@30430
   723
  by (cases P) simp
haftmann@30328
   724
bulwahn@31105
   725
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
bulwahn@31105
   726
bulwahn@31105
   727
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
haftmann@31108
   728
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
haftmann@30948
   729
haftmann@31216
   730
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
haftmann@31216
   731
  "map f P = P \<guillemotright>= (single o f)"
haftmann@31216
   732
haftmann@32578
   733
primrec null :: "'a seq \<Rightarrow> bool" where
haftmann@32578
   734
    "null Empty \<longleftrightarrow> True"
haftmann@32578
   735
  | "null (Insert x P) \<longleftrightarrow> False"
haftmann@32578
   736
  | "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
haftmann@32578
   737
haftmann@32578
   738
lemma null_is_empty:
haftmann@32578
   739
  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
haftmann@32578
   740
  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
haftmann@32578
   741
haftmann@32578
   742
lemma is_empty_code [code]:
haftmann@32578
   743
  "is_empty (Seq f) \<longleftrightarrow> null (f ())"
haftmann@32578
   744
  by (simp add: null_is_empty Seq_def)
haftmann@32578
   745
haftmann@32578
   746
primrec the_only :: "'a seq \<Rightarrow> 'a" where
haftmann@32578
   747
  [code del]: "the_only Empty = undefined"
haftmann@32578
   748
  | "the_only (Insert x P) = (if is_empty P then x else let y = singleton P in if x = y then x else undefined)"
haftmann@32578
   749
  | "the_only (Join P xq) = (if is_empty P then the_only xq else if null xq then singleton P
haftmann@32578
   750
       else let x = singleton P; y = the_only xq in
haftmann@32578
   751
       if x = y then x else undefined)"
haftmann@32578
   752
haftmann@32578
   753
lemma the_only_singleton:
haftmann@32578
   754
  "the_only xq = singleton (pred_of_seq xq)"
haftmann@32578
   755
  by (induct xq)
haftmann@32578
   756
    (auto simp add: singleton_bot singleton_single is_empty_def
haftmann@32578
   757
    null_is_empty Let_def singleton_sup)
haftmann@32578
   758
haftmann@32578
   759
lemma singleton_code [code]:
haftmann@32578
   760
  "singleton (Seq f) = (case f ()
haftmann@32578
   761
   of Empty \<Rightarrow> undefined
haftmann@32578
   762
    | Insert x P \<Rightarrow> if is_empty P then x
haftmann@32578
   763
        else let y = singleton P in
haftmann@32578
   764
          if x = y then x else undefined
haftmann@32578
   765
    | Join P xq \<Rightarrow> if is_empty P then the_only xq
haftmann@32578
   766
        else if null xq then singleton P
haftmann@32578
   767
        else let x = singleton P; y = the_only xq in
haftmann@32578
   768
          if x = y then x else undefined)"
haftmann@32578
   769
  by (cases "f ()")
haftmann@32578
   770
   (auto simp add: Seq_def the_only_singleton is_empty_def
haftmann@32578
   771
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)
haftmann@32578
   772
bulwahn@32668
   773
lemma meta_fun_cong:
bulwahn@32668
   774
"f == g ==> f x == g x"
bulwahn@32668
   775
by simp
bulwahn@32668
   776
haftmann@30948
   777
ML {*
haftmann@30948
   778
signature PREDICATE =
haftmann@30948
   779
sig
haftmann@30948
   780
  datatype 'a pred = Seq of (unit -> 'a seq)
haftmann@30948
   781
  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
haftmann@30959
   782
  val yield: 'a pred -> ('a * 'a pred) option
haftmann@30959
   783
  val yieldn: int -> 'a pred -> 'a list * 'a pred
haftmann@31222
   784
  val map: ('a -> 'b) -> 'a pred -> 'b pred
haftmann@30948
   785
end;
haftmann@30948
   786
haftmann@30948
   787
structure Predicate : PREDICATE =
haftmann@30948
   788
struct
haftmann@30948
   789
haftmann@30959
   790
@{code_datatype pred = Seq};
haftmann@30959
   791
@{code_datatype seq = Empty | Insert | Join};
haftmann@30959
   792
haftmann@32372
   793
fun yield (@{code Seq} f) = next (f ())
haftmann@30959
   794
and next @{code Empty} = NONE
haftmann@30959
   795
  | next (@{code Insert} (x, P)) = SOME (x, P)
haftmann@30959
   796
  | next (@{code Join} (P, xq)) = (case yield P
haftmann@30959
   797
     of NONE => next xq
haftmann@30959
   798
      | SOME (x, Q) => SOME (x, @{code Seq} (fn _ => @{code Join} (Q, xq))))
haftmann@30959
   799
haftmann@30959
   800
fun anamorph f k x = (if k = 0 then ([], x)
haftmann@30959
   801
  else case f x
haftmann@30959
   802
   of NONE => ([], x)
haftmann@30959
   803
    | SOME (v, y) => let
haftmann@30959
   804
        val (vs, z) = anamorph f (k - 1) y
haftmann@30959
   805
      in (v :: vs, z) end)
haftmann@30959
   806
haftmann@30959
   807
fun yieldn P = anamorph yield P;
haftmann@30948
   808
haftmann@31222
   809
fun map f = @{code map} f;
haftmann@31222
   810
haftmann@30948
   811
end;
haftmann@30948
   812
*}
haftmann@30948
   813
haftmann@30948
   814
code_reserved Eval Predicate
haftmann@30948
   815
haftmann@30948
   816
code_type pred and seq
haftmann@30948
   817
  (Eval "_/ Predicate.pred" and "_/ Predicate.seq")
haftmann@30948
   818
haftmann@30948
   819
code_const Seq and Empty and Insert and Join
haftmann@30948
   820
  (Eval "Predicate.Seq" and "Predicate.Empty" and "Predicate.Insert/ (_,/ _)" and "Predicate.Join/ (_,/ _)")
haftmann@30948
   821
haftmann@31122
   822
text {* dummy setup for @{text code_pred} and @{text values} keywords *}
haftmann@31108
   823
haftmann@31108
   824
ML {*
haftmann@31122
   825
local
haftmann@31122
   826
haftmann@31122
   827
structure P = OuterParse;
haftmann@31122
   828
haftmann@31122
   829
val opt_modes = Scan.optional (P.$$$ "(" |-- P.!!! (Scan.repeat1 P.xname --| P.$$$ ")")) [];
haftmann@31122
   830
haftmann@31122
   831
in
haftmann@31122
   832
haftmann@31122
   833
val _ = OuterSyntax.local_theory_to_proof "code_pred" "sets up goal for cases rule from given introduction rules and compiles predicate"
haftmann@31122
   834
  OuterKeyword.thy_goal (P.term_group >> (K (Proof.theorem_i NONE (K I) [[]])));
haftmann@31122
   835
haftmann@31216
   836
val _ = OuterSyntax.improper_command "values" "enumerate and print comprehensions"
haftmann@31122
   837
  OuterKeyword.diag ((opt_modes -- P.term)
haftmann@31122
   838
    >> (fn (modes, t) => Toplevel.no_timing o Toplevel.keep
haftmann@31122
   839
        (K ())));
haftmann@31122
   840
haftmann@31122
   841
end
haftmann@31108
   842
*}
haftmann@30959
   843
haftmann@30328
   844
no_notation
haftmann@30328
   845
  inf (infixl "\<sqinter>" 70) and
haftmann@30328
   846
  sup (infixl "\<squnion>" 65) and
haftmann@30328
   847
  Inf ("\<Sqinter>_" [900] 900) and
haftmann@30328
   848
  Sup ("\<Squnion>_" [900] 900) and
haftmann@30328
   849
  top ("\<top>") and
haftmann@30328
   850
  bot ("\<bottom>") and
haftmann@30328
   851
  bind (infixl "\<guillemotright>=" 70)
haftmann@30328
   852
haftmann@30328
   853
hide (open) type pred seq
haftmann@32582
   854
hide (open) const Pred eval single bind is_empty singleton if_pred not_pred
haftmann@32582
   855
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map
haftmann@30328
   856
haftmann@30328
   857
end