src/HOL/Predicate.thy
author kuncar
Fri Dec 09 18:07:04 2011 +0100 (2011-12-09)
changeset 45802 b16f976db515
parent 45630 0dd654a01217
child 45970 b6d0cff57d96
permissions -rw-r--r--
Quotient_Info stores only relation maps
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(*  Title:      HOL/Predicate.thy
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    Author:     Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen
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*)
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header {* Predicates as relations and enumerations *}
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theory Predicate
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imports Inductive Relation
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begin
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notation
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  bot ("\<bottom>") and
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  top ("\<top>") and
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  inf (infixl "\<sqinter>" 70) and
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  sup (infixl "\<squnion>" 65) and
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  Inf ("\<Sqinter>_" [900] 900) and
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  Sup ("\<Squnion>_" [900] 900)
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syntax (xsymbols)
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  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
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  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
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  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
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  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
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subsection {* Predicates as (complete) lattices *}
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text {*
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  Handy introduction and elimination rules for @{text "\<le>"}
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  on unary and binary predicates
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*}
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lemma predicate1I:
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  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
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  shows "P \<le> Q"
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  apply (rule le_funI)
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  apply (rule le_boolI)
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  apply (rule PQ)
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  apply assumption
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  done
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lemma predicate1D [Pure.dest?, dest?]:
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  "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
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  apply (erule le_funE)
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  apply (erule le_boolE)
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  apply assumption+
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  done
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lemma rev_predicate1D:
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  "P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
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  by (rule predicate1D)
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lemma predicate2I [Pure.intro!, intro!]:
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  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
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  shows "P \<le> Q"
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  apply (rule le_funI)+
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  apply (rule le_boolI)
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  apply (rule PQ)
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  apply assumption
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  done
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lemma predicate2D [Pure.dest, dest]:
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  "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
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  apply (erule le_funE)+
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  apply (erule le_boolE)
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  apply assumption+
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  done
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lemma rev_predicate2D:
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  "P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
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  by (rule predicate2D)
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subsubsection {* Equality *}
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lemma pred_equals_eq: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R = S)"
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  by (simp add: set_eq_iff fun_eq_iff mem_def)
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lemma pred_equals_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R = S)"
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  by (simp add: set_eq_iff fun_eq_iff mem_def)
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subsubsection {* Order relation *}
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lemma pred_subset_eq: "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"
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  by (simp add: subset_iff le_fun_def mem_def)
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lemma pred_subset_eq2 [pred_set_conv]: "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) \<longleftrightarrow> (R \<subseteq> S)"
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  by (simp add: subset_iff le_fun_def mem_def)
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subsubsection {* Top and bottom elements *}
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lemma bot1E [no_atp, elim!]: "\<bottom> x \<Longrightarrow> P"
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  by (simp add: bot_fun_def)
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lemma bot2E [elim!]: "\<bottom> x y \<Longrightarrow> P"
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  by (simp add: bot_fun_def)
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lemma bot_empty_eq: "\<bottom> = (\<lambda>x. x \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma bot_empty_eq2: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma top1I [intro!]: "\<top> x"
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  by (simp add: top_fun_def)
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lemma top2I [intro!]: "\<top> x y"
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  by (simp add: top_fun_def)
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subsubsection {* Binary intersection *}
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lemma inf1I [intro!]: "A x \<Longrightarrow> B x \<Longrightarrow> (A \<sqinter> B) x"
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  by (simp add: inf_fun_def)
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lemma inf2I [intro!]: "A x y \<Longrightarrow> B x y \<Longrightarrow> (A \<sqinter> B) x y"
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  by (simp add: inf_fun_def)
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lemma inf1E [elim!]: "(A \<sqinter> B) x \<Longrightarrow> (A x \<Longrightarrow> B x \<Longrightarrow> P) \<Longrightarrow> P"
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  by (simp add: inf_fun_def)
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lemma inf2E [elim!]: "(A \<sqinter> B) x y \<Longrightarrow> (A x y \<Longrightarrow> B x y \<Longrightarrow> P) \<Longrightarrow> P"
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  by (simp add: inf_fun_def)
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lemma inf1D1: "(A \<sqinter> B) x \<Longrightarrow> A x"
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  by (simp add: inf_fun_def)
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lemma inf2D1: "(A \<sqinter> B) x y \<Longrightarrow> A x y"
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  by (simp add: inf_fun_def)
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lemma inf1D2: "(A \<sqinter> B) x \<Longrightarrow> B x"
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  by (simp add: inf_fun_def)
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lemma inf2D2: "(A \<sqinter> B) x y \<Longrightarrow> B x y"
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  by (simp add: inf_fun_def)
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lemma inf_Int_eq: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
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  by (simp add: inf_fun_def mem_def)
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lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
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  by (simp add: inf_fun_def mem_def)
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subsubsection {* Binary union *}
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lemma sup1I1 [intro?]: "A x \<Longrightarrow> (A \<squnion> B) x"
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  by (simp add: sup_fun_def)
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lemma sup2I1 [intro?]: "A x y \<Longrightarrow> (A \<squnion> B) x y"
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  by (simp add: sup_fun_def)
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lemma sup1I2 [intro?]: "B x \<Longrightarrow> (A \<squnion> B) x"
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  by (simp add: sup_fun_def)
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lemma sup2I2 [intro?]: "B x y \<Longrightarrow> (A \<squnion> B) x y"
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  by (simp add: sup_fun_def)
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lemma sup1E [elim!]: "(A \<squnion> B) x \<Longrightarrow> (A x \<Longrightarrow> P) \<Longrightarrow> (B x \<Longrightarrow> P) \<Longrightarrow> P"
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  by (simp add: sup_fun_def) iprover
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lemma sup2E [elim!]: "(A \<squnion> B) x y \<Longrightarrow> (A x y \<Longrightarrow> P) \<Longrightarrow> (B x y \<Longrightarrow> P) \<Longrightarrow> P"
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  by (simp add: sup_fun_def) iprover
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text {*
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  \medskip Classical introduction rule: no commitment to @{text A} vs
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  @{text B}.
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*}
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lemma sup1CI [intro!]: "(\<not> B x \<Longrightarrow> A x) \<Longrightarrow> (A \<squnion> B) x"
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  by (auto simp add: sup_fun_def)
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lemma sup2CI [intro!]: "(\<not> B x y \<Longrightarrow> A x y) \<Longrightarrow> (A \<squnion> B) x y"
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  by (auto simp add: sup_fun_def)
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lemma sup_Un_eq: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
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  by (simp add: sup_fun_def mem_def)
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lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
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  by (simp add: sup_fun_def mem_def)
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subsubsection {* Intersections of families *}
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lemma INF1_iff: "(\<Sqinter>x\<in>A. B x) b = (\<forall>x\<in>A. B x b)"
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  by (simp add: INF_apply)
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lemma INF2_iff: "(\<Sqinter>x\<in>A. B x) b c = (\<forall>x\<in>A. B x b c)"
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  by (simp add: INF_apply)
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lemma INF1_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b"
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  by (auto simp add: INF_apply)
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lemma INF2_I [intro!]: "(\<And>x. x \<in> A \<Longrightarrow> B x b c) \<Longrightarrow> (\<Sqinter>x\<in>A. B x) b c"
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  by (auto simp add: INF_apply)
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lemma INF1_D [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> a \<in> A \<Longrightarrow> B a b"
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  by (auto simp add: INF_apply)
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lemma INF2_D [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> a \<in> A \<Longrightarrow> B a b c"
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  by (auto simp add: INF_apply)
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lemma INF1_E [elim]: "(\<Sqinter>x\<in>A. B x) b \<Longrightarrow> (B a b \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
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  by (auto simp add: INF_apply)
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lemma INF2_E [elim]: "(\<Sqinter>x\<in>A. B x) b c \<Longrightarrow> (B a b c \<Longrightarrow> R) \<Longrightarrow> (a \<notin> A \<Longrightarrow> R) \<Longrightarrow> R"
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  by (auto simp add: INF_apply)
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lemma INF_INT_eq: "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Sqinter>i. r i))"
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  by (simp add: INF_apply fun_eq_iff)
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lemma INF_INT_eq2: "(\<Sqinter>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Sqinter>i. r i))"
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  by (simp add: INF_apply fun_eq_iff)
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subsubsection {* Unions of families *}
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lemma SUP1_iff: "(\<Squnion>x\<in>A. B x) b = (\<exists>x\<in>A. B x b)"
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  by (simp add: SUP_apply)
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lemma SUP2_iff: "(\<Squnion>x\<in>A. B x) b c = (\<exists>x\<in>A. B x b c)"
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  by (simp add: SUP_apply)
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lemma SUP1_I [intro]: "a \<in> A \<Longrightarrow> B a b \<Longrightarrow> (\<Squnion>x\<in>A. B x) b"
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  by (auto simp add: SUP_apply)
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lemma SUP2_I [intro]: "a \<in> A \<Longrightarrow> B a b c \<Longrightarrow> (\<Squnion>x\<in>A. B x) b c"
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  by (auto simp add: SUP_apply)
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lemma SUP1_E [elim!]: "(\<Squnion>x\<in>A. B x) b \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b \<Longrightarrow> R) \<Longrightarrow> R"
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  by (auto simp add: SUP_apply)
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lemma SUP2_E [elim!]: "(\<Squnion>x\<in>A. B x) b c \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x b c \<Longrightarrow> R) \<Longrightarrow> R"
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  by (auto simp add: SUP_apply)
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lemma SUP_UN_eq: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i. r i))"
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  by (simp add: SUP_apply fun_eq_iff)
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lemma SUP_UN_eq2: "(\<Squnion>i. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i. r i))"
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  by (simp add: SUP_apply fun_eq_iff)
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subsection {* Predicates as relations *}
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subsubsection {* Composition  *}
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inductive pred_comp  :: "['a \<Rightarrow> 'b \<Rightarrow> bool, 'b \<Rightarrow> 'c \<Rightarrow> bool] \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" (infixr "OO" 75)
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  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool" where
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  pred_compI [intro]: "r a b \<Longrightarrow> s b c \<Longrightarrow> (r OO s) a c"
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inductive_cases pred_compE [elim!]: "(r OO s) a c"
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lemma pred_comp_rel_comp_eq [pred_set_conv]:
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  "((\<lambda>x y. (x, y) \<in> r) OO (\<lambda>x y. (x, y) \<in> s)) = (\<lambda>x y. (x, y) \<in> r O s)"
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  by (auto simp add: fun_eq_iff)
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subsubsection {* Converse *}
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inductive conversep :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(_^--1)" [1000] 1000)
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  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  conversepI: "r a b \<Longrightarrow> r^--1 b a"
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notation (xsymbols)
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  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
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lemma conversepD:
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  assumes ab: "r^--1 a b"
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  shows "r b a" using ab
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  by cases simp
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lemma conversep_iff [iff]: "r^--1 a b = r b a"
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  by (iprover intro: conversepI dest: conversepD)
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lemma conversep_converse_eq [pred_set_conv]:
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  "(\<lambda>x y. (x, y) \<in> r)^--1 = (\<lambda>x y. (x, y) \<in> r^-1)"
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  by (auto simp add: fun_eq_iff)
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lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
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  by (iprover intro: order_antisym conversepI dest: conversepD)
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lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
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  by (iprover intro: order_antisym conversepI pred_compI
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    elim: pred_compE dest: conversepD)
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lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
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  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
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lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
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  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
berghofe@22259
   292
haftmann@44414
   293
lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
nipkow@39302
   294
  by (auto simp add: fun_eq_iff)
berghofe@22259
   295
berghofe@22259
   296
lemma conversep_eq [simp]: "(op =)^--1 = op ="
nipkow@39302
   297
  by (auto simp add: fun_eq_iff)
berghofe@22259
   298
berghofe@22259
   299
haftmann@30328
   300
subsubsection {* Domain *}
berghofe@22259
   301
haftmann@44414
   302
inductive DomainP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@44414
   303
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
haftmann@44414
   304
  DomainPI [intro]: "r a b \<Longrightarrow> DomainP r a"
berghofe@22259
   305
berghofe@23741
   306
inductive_cases DomainPE [elim!]: "DomainP r a"
berghofe@22259
   307
berghofe@23741
   308
lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Domain r)"
berghofe@26797
   309
  by (blast intro!: Orderings.order_antisym predicate1I)
berghofe@22259
   310
berghofe@22259
   311
haftmann@30328
   312
subsubsection {* Range *}
berghofe@22259
   313
haftmann@44414
   314
inductive RangeP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"
haftmann@44414
   315
  for r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" where
haftmann@44414
   316
  RangePI [intro]: "r a b \<Longrightarrow> RangeP r b"
berghofe@22259
   317
berghofe@23741
   318
inductive_cases RangePE [elim!]: "RangeP r b"
berghofe@22259
   319
berghofe@23741
   320
lemma RangeP_Range_eq [pred_set_conv]: "RangeP (\<lambda>x y. (x, y) \<in> r) = (\<lambda>x. x \<in> Range r)"
berghofe@26797
   321
  by (blast intro!: Orderings.order_antisym predicate1I)
berghofe@22259
   322
berghofe@22259
   323
haftmann@30328
   324
subsubsection {* Inverse image *}
berghofe@22259
   325
haftmann@44414
   326
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" where
haftmann@44414
   327
  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
berghofe@22259
   328
berghofe@23741
   329
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
   330
  by (simp add: inv_image_def inv_imagep_def)
berghofe@22259
   331
berghofe@22259
   332
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
berghofe@22259
   333
  by (simp add: inv_imagep_def)
berghofe@22259
   334
berghofe@22259
   335
haftmann@30328
   336
subsubsection {* Powerset *}
berghofe@23741
   337
berghofe@23741
   338
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
haftmann@44414
   339
  "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
berghofe@23741
   340
berghofe@23741
   341
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
nipkow@39302
   342
  by (auto simp add: Powp_def fun_eq_iff)
berghofe@23741
   343
berghofe@26797
   344
lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]
berghofe@26797
   345
berghofe@23741
   346
haftmann@30328
   347
subsubsection {* Properties of relations *}
berghofe@22259
   348
haftmann@44414
   349
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@44414
   350
  "antisymP r \<equiv> antisym {(x, y). r x y}"
berghofe@22259
   351
haftmann@44414
   352
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@44414
   353
  "transP r \<equiv> trans {(x, y). r x y}"
berghofe@22259
   354
haftmann@44414
   355
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@44414
   356
  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
berghofe@22259
   357
haftmann@40813
   358
(*FIXME inconsistencies: abbreviations vs. definitions, suffix `P` vs. suffix `p`*)
haftmann@40813
   359
haftmann@40813
   360
definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40813
   361
  "reflp r \<longleftrightarrow> refl {(x, y). r x y}"
haftmann@40813
   362
haftmann@40813
   363
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40813
   364
  "symp r \<longleftrightarrow> sym {(x, y). r x y}"
haftmann@40813
   365
haftmann@40813
   366
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@40813
   367
  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
haftmann@40813
   368
haftmann@40813
   369
lemma reflpI:
haftmann@40813
   370
  "(\<And>x. r x x) \<Longrightarrow> reflp r"
haftmann@40813
   371
  by (auto intro: refl_onI simp add: reflp_def)
haftmann@40813
   372
haftmann@40813
   373
lemma reflpE:
haftmann@40813
   374
  assumes "reflp r"
haftmann@40813
   375
  obtains "r x x"
haftmann@40813
   376
  using assms by (auto dest: refl_onD simp add: reflp_def)
haftmann@40813
   377
haftmann@40813
   378
lemma sympI:
haftmann@40813
   379
  "(\<And>x y. r x y \<Longrightarrow> r y x) \<Longrightarrow> symp r"
haftmann@40813
   380
  by (auto intro: symI simp add: symp_def)
haftmann@40813
   381
haftmann@40813
   382
lemma sympE:
haftmann@40813
   383
  assumes "symp r" and "r x y"
haftmann@40813
   384
  obtains "r y x"
haftmann@40813
   385
  using assms by (auto dest: symD simp add: symp_def)
haftmann@40813
   386
haftmann@40813
   387
lemma transpI:
haftmann@40813
   388
  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
haftmann@40813
   389
  by (auto intro: transI simp add: transp_def)
haftmann@40813
   390
  
haftmann@40813
   391
lemma transpE:
haftmann@40813
   392
  assumes "transp r" and "r x y" and "r y z"
haftmann@40813
   393
  obtains "r x z"
haftmann@40813
   394
  using assms by (auto dest: transD simp add: transp_def)
haftmann@40813
   395
haftmann@30328
   396
haftmann@30328
   397
subsection {* Predicates as enumerations *}
haftmann@30328
   398
haftmann@30328
   399
subsubsection {* The type of predicate enumerations (a monad) *}
haftmann@30328
   400
haftmann@30328
   401
datatype 'a pred = Pred "'a \<Rightarrow> bool"
haftmann@30328
   402
haftmann@30328
   403
primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
haftmann@30328
   404
  eval_pred: "eval (Pred f) = f"
haftmann@30328
   405
haftmann@30328
   406
lemma Pred_eval [simp]:
haftmann@30328
   407
  "Pred (eval x) = x"
haftmann@30328
   408
  by (cases x) simp
haftmann@30328
   409
haftmann@40616
   410
lemma pred_eqI:
haftmann@40616
   411
  "(\<And>w. eval P w \<longleftrightarrow> eval Q w) \<Longrightarrow> P = Q"
haftmann@40616
   412
  by (cases P, cases Q) (auto simp add: fun_eq_iff)
haftmann@30328
   413
haftmann@44033
   414
instantiation pred :: (type) complete_lattice
haftmann@30328
   415
begin
haftmann@30328
   416
haftmann@30328
   417
definition
haftmann@30328
   418
  "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
haftmann@30328
   419
haftmann@30328
   420
definition
haftmann@30328
   421
  "P < Q \<longleftrightarrow> eval P < eval Q"
haftmann@30328
   422
haftmann@30328
   423
definition
haftmann@30328
   424
  "\<bottom> = Pred \<bottom>"
haftmann@30328
   425
haftmann@40616
   426
lemma eval_bot [simp]:
haftmann@40616
   427
  "eval \<bottom>  = \<bottom>"
haftmann@40616
   428
  by (simp add: bot_pred_def)
haftmann@40616
   429
haftmann@30328
   430
definition
haftmann@30328
   431
  "\<top> = Pred \<top>"
haftmann@30328
   432
haftmann@40616
   433
lemma eval_top [simp]:
haftmann@40616
   434
  "eval \<top>  = \<top>"
haftmann@40616
   435
  by (simp add: top_pred_def)
haftmann@40616
   436
haftmann@30328
   437
definition
haftmann@30328
   438
  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
haftmann@30328
   439
haftmann@40616
   440
lemma eval_inf [simp]:
haftmann@40616
   441
  "eval (P \<sqinter> Q) = eval P \<sqinter> eval Q"
haftmann@40616
   442
  by (simp add: inf_pred_def)
haftmann@40616
   443
haftmann@30328
   444
definition
haftmann@30328
   445
  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
haftmann@30328
   446
haftmann@40616
   447
lemma eval_sup [simp]:
haftmann@40616
   448
  "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
haftmann@40616
   449
  by (simp add: sup_pred_def)
haftmann@40616
   450
haftmann@30328
   451
definition
haftmann@37767
   452
  "\<Sqinter>A = Pred (INFI A eval)"
haftmann@30328
   453
haftmann@40616
   454
lemma eval_Inf [simp]:
haftmann@40616
   455
  "eval (\<Sqinter>A) = INFI A eval"
haftmann@40616
   456
  by (simp add: Inf_pred_def)
haftmann@40616
   457
haftmann@30328
   458
definition
haftmann@37767
   459
  "\<Squnion>A = Pred (SUPR A eval)"
haftmann@30328
   460
haftmann@40616
   461
lemma eval_Sup [simp]:
haftmann@40616
   462
  "eval (\<Squnion>A) = SUPR A eval"
haftmann@40616
   463
  by (simp add: Sup_pred_def)
haftmann@40616
   464
haftmann@44033
   465
instance proof
haftmann@44415
   466
qed (auto intro!: pred_eqI simp add: less_eq_pred_def less_pred_def le_fun_def less_fun_def)
haftmann@44033
   467
haftmann@44033
   468
end
haftmann@44033
   469
haftmann@44033
   470
lemma eval_INFI [simp]:
haftmann@44033
   471
  "eval (INFI A f) = INFI A (eval \<circ> f)"
hoelzl@44928
   472
  by (simp only: INF_def eval_Inf image_compose)
haftmann@44033
   473
haftmann@44033
   474
lemma eval_SUPR [simp]:
haftmann@44033
   475
  "eval (SUPR A f) = SUPR A (eval \<circ> f)"
hoelzl@44928
   476
  by (simp only: SUP_def eval_Sup image_compose)
haftmann@44033
   477
haftmann@44033
   478
instantiation pred :: (type) complete_boolean_algebra
haftmann@44033
   479
begin
haftmann@44033
   480
haftmann@32578
   481
definition
haftmann@32578
   482
  "- P = Pred (- eval P)"
haftmann@32578
   483
haftmann@40616
   484
lemma eval_compl [simp]:
haftmann@40616
   485
  "eval (- P) = - eval P"
haftmann@40616
   486
  by (simp add: uminus_pred_def)
haftmann@40616
   487
haftmann@32578
   488
definition
haftmann@32578
   489
  "P - Q = Pred (eval P - eval Q)"
haftmann@32578
   490
haftmann@40616
   491
lemma eval_minus [simp]:
haftmann@40616
   492
  "eval (P - Q) = eval P - eval Q"
haftmann@40616
   493
  by (simp add: minus_pred_def)
haftmann@40616
   494
haftmann@32578
   495
instance proof
haftmann@44415
   496
qed (auto intro!: pred_eqI simp add: uminus_apply minus_apply INF_apply SUP_apply)
haftmann@30328
   497
berghofe@22259
   498
end
haftmann@30328
   499
haftmann@40616
   500
definition single :: "'a \<Rightarrow> 'a pred" where
haftmann@40616
   501
  "single x = Pred ((op =) x)"
haftmann@40616
   502
haftmann@40616
   503
lemma eval_single [simp]:
haftmann@40616
   504
  "eval (single x) = (op =) x"
haftmann@40616
   505
  by (simp add: single_def)
haftmann@40616
   506
haftmann@40616
   507
definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
haftmann@41080
   508
  "P \<guillemotright>= f = (SUPR {x. eval P x} f)"
haftmann@40616
   509
haftmann@40616
   510
lemma eval_bind [simp]:
haftmann@41080
   511
  "eval (P \<guillemotright>= f) = eval (SUPR {x. eval P x} f)"
haftmann@40616
   512
  by (simp add: bind_def)
haftmann@40616
   513
haftmann@30328
   514
lemma bind_bind:
haftmann@30328
   515
  "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
haftmann@44415
   516
  by (rule pred_eqI) (auto simp add: SUP_apply)
haftmann@30328
   517
haftmann@30328
   518
lemma bind_single:
haftmann@30328
   519
  "P \<guillemotright>= single = P"
haftmann@40616
   520
  by (rule pred_eqI) auto
haftmann@30328
   521
haftmann@30328
   522
lemma single_bind:
haftmann@30328
   523
  "single x \<guillemotright>= P = P x"
haftmann@40616
   524
  by (rule pred_eqI) auto
haftmann@30328
   525
haftmann@30328
   526
lemma bottom_bind:
haftmann@30328
   527
  "\<bottom> \<guillemotright>= P = \<bottom>"
haftmann@40674
   528
  by (rule pred_eqI) auto
haftmann@30328
   529
haftmann@30328
   530
lemma sup_bind:
haftmann@30328
   531
  "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
haftmann@40674
   532
  by (rule pred_eqI) auto
haftmann@30328
   533
haftmann@40616
   534
lemma Sup_bind:
haftmann@40616
   535
  "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
haftmann@44415
   536
  by (rule pred_eqI) (auto simp add: SUP_apply)
haftmann@30328
   537
haftmann@30328
   538
lemma pred_iffI:
haftmann@30328
   539
  assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
haftmann@30328
   540
  and "\<And>x. eval B x \<Longrightarrow> eval A x"
haftmann@30328
   541
  shows "A = B"
haftmann@40616
   542
  using assms by (auto intro: pred_eqI)
haftmann@30328
   543
  
haftmann@30328
   544
lemma singleI: "eval (single x) x"
haftmann@40616
   545
  by simp
haftmann@30328
   546
haftmann@30328
   547
lemma singleI_unit: "eval (single ()) x"
haftmann@40616
   548
  by simp
haftmann@30328
   549
haftmann@30328
   550
lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   551
  by simp
haftmann@30328
   552
haftmann@30328
   553
lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   554
  by simp
haftmann@30328
   555
haftmann@30328
   556
lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
haftmann@40616
   557
  by auto
haftmann@30328
   558
haftmann@30328
   559
lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   560
  by auto
haftmann@30328
   561
haftmann@30328
   562
lemma botE: "eval \<bottom> x \<Longrightarrow> P"
haftmann@40616
   563
  by auto
haftmann@30328
   564
haftmann@30328
   565
lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
haftmann@40616
   566
  by auto
haftmann@30328
   567
haftmann@30328
   568
lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x" 
haftmann@40616
   569
  by auto
haftmann@30328
   570
haftmann@30328
   571
lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@40616
   572
  by auto
haftmann@30328
   573
haftmann@32578
   574
lemma single_not_bot [simp]:
haftmann@32578
   575
  "single x \<noteq> \<bottom>"
nipkow@39302
   576
  by (auto simp add: single_def bot_pred_def fun_eq_iff)
haftmann@32578
   577
haftmann@32578
   578
lemma not_bot:
haftmann@32578
   579
  assumes "A \<noteq> \<bottom>"
haftmann@32578
   580
  obtains x where "eval A x"
haftmann@44415
   581
  using assms by (cases A) (auto simp add: bot_pred_def, simp add: mem_def)
haftmann@32578
   582
  
haftmann@32578
   583
haftmann@32578
   584
subsubsection {* Emptiness check and definite choice *}
haftmann@32578
   585
haftmann@32578
   586
definition is_empty :: "'a pred \<Rightarrow> bool" where
haftmann@32578
   587
  "is_empty A \<longleftrightarrow> A = \<bottom>"
haftmann@32578
   588
haftmann@32578
   589
lemma is_empty_bot:
haftmann@32578
   590
  "is_empty \<bottom>"
haftmann@32578
   591
  by (simp add: is_empty_def)
haftmann@32578
   592
haftmann@32578
   593
lemma not_is_empty_single:
haftmann@32578
   594
  "\<not> is_empty (single x)"
nipkow@39302
   595
  by (auto simp add: is_empty_def single_def bot_pred_def fun_eq_iff)
haftmann@32578
   596
haftmann@32578
   597
lemma is_empty_sup:
haftmann@32578
   598
  "is_empty (A \<squnion> B) \<longleftrightarrow> is_empty A \<and> is_empty B"
huffman@36008
   599
  by (auto simp add: is_empty_def)
haftmann@32578
   600
haftmann@40616
   601
definition singleton :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a pred \<Rightarrow> 'a" where
bulwahn@33111
   602
  "singleton dfault A = (if \<exists>!x. eval A x then THE x. eval A x else dfault ())"
haftmann@32578
   603
haftmann@32578
   604
lemma singleton_eqI:
bulwahn@33110
   605
  "\<exists>!x. eval A x \<Longrightarrow> eval A x \<Longrightarrow> singleton dfault A = x"
haftmann@32578
   606
  by (auto simp add: singleton_def)
haftmann@32578
   607
haftmann@32578
   608
lemma eval_singletonI:
bulwahn@33110
   609
  "\<exists>!x. eval A x \<Longrightarrow> eval A (singleton dfault A)"
haftmann@32578
   610
proof -
haftmann@32578
   611
  assume assm: "\<exists>!x. eval A x"
haftmann@32578
   612
  then obtain x where "eval A x" ..
bulwahn@33110
   613
  moreover with assm have "singleton dfault A = x" by (rule singleton_eqI)
haftmann@32578
   614
  ultimately show ?thesis by simp 
haftmann@32578
   615
qed
haftmann@32578
   616
haftmann@32578
   617
lemma single_singleton:
bulwahn@33110
   618
  "\<exists>!x. eval A x \<Longrightarrow> single (singleton dfault A) = A"
haftmann@32578
   619
proof -
haftmann@32578
   620
  assume assm: "\<exists>!x. eval A x"
bulwahn@33110
   621
  then have "eval A (singleton dfault A)"
haftmann@32578
   622
    by (rule eval_singletonI)
bulwahn@33110
   623
  moreover from assm have "\<And>x. eval A x \<Longrightarrow> singleton dfault A = x"
haftmann@32578
   624
    by (rule singleton_eqI)
bulwahn@33110
   625
  ultimately have "eval (single (singleton dfault A)) = eval A"
nipkow@39302
   626
    by (simp (no_asm_use) add: single_def fun_eq_iff) blast
haftmann@40616
   627
  then have "\<And>x. eval (single (singleton dfault A)) x = eval A x"
haftmann@40616
   628
    by simp
haftmann@40616
   629
  then show ?thesis by (rule pred_eqI)
haftmann@32578
   630
qed
haftmann@32578
   631
haftmann@32578
   632
lemma singleton_undefinedI:
bulwahn@33111
   633
  "\<not> (\<exists>!x. eval A x) \<Longrightarrow> singleton dfault A = dfault ()"
haftmann@32578
   634
  by (simp add: singleton_def)
haftmann@32578
   635
haftmann@32578
   636
lemma singleton_bot:
bulwahn@33111
   637
  "singleton dfault \<bottom> = dfault ()"
haftmann@32578
   638
  by (auto simp add: bot_pred_def intro: singleton_undefinedI)
haftmann@32578
   639
haftmann@32578
   640
lemma singleton_single:
bulwahn@33110
   641
  "singleton dfault (single x) = x"
haftmann@32578
   642
  by (auto simp add: intro: singleton_eqI singleI elim: singleE)
haftmann@32578
   643
haftmann@32578
   644
lemma singleton_sup_single_single:
bulwahn@33111
   645
  "singleton dfault (single x \<squnion> single y) = (if x = y then x else dfault ())"
haftmann@32578
   646
proof (cases "x = y")
haftmann@32578
   647
  case True then show ?thesis by (simp add: singleton_single)
haftmann@32578
   648
next
haftmann@32578
   649
  case False
haftmann@32578
   650
  have "eval (single x \<squnion> single y) x"
haftmann@32578
   651
    and "eval (single x \<squnion> single y) y"
haftmann@32578
   652
  by (auto intro: supI1 supI2 singleI)
haftmann@32578
   653
  with False have "\<not> (\<exists>!z. eval (single x \<squnion> single y) z)"
haftmann@32578
   654
    by blast
bulwahn@33111
   655
  then have "singleton dfault (single x \<squnion> single y) = dfault ()"
haftmann@32578
   656
    by (rule singleton_undefinedI)
haftmann@32578
   657
  with False show ?thesis by simp
haftmann@32578
   658
qed
haftmann@32578
   659
haftmann@32578
   660
lemma singleton_sup_aux:
bulwahn@33110
   661
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
bulwahn@33110
   662
    else if B = \<bottom> then singleton dfault A
bulwahn@33110
   663
    else singleton dfault
bulwahn@33110
   664
      (single (singleton dfault A) \<squnion> single (singleton dfault B)))"
haftmann@32578
   665
proof (cases "(\<exists>!x. eval A x) \<and> (\<exists>!y. eval B y)")
haftmann@32578
   666
  case True then show ?thesis by (simp add: single_singleton)
haftmann@32578
   667
next
haftmann@32578
   668
  case False
haftmann@32578
   669
  from False have A_or_B:
bulwahn@33111
   670
    "singleton dfault A = dfault () \<or> singleton dfault B = dfault ()"
haftmann@32578
   671
    by (auto intro!: singleton_undefinedI)
bulwahn@33110
   672
  then have rhs: "singleton dfault
bulwahn@33111
   673
    (single (singleton dfault A) \<squnion> single (singleton dfault B)) = dfault ()"
haftmann@32578
   674
    by (auto simp add: singleton_sup_single_single singleton_single)
haftmann@32578
   675
  from False have not_unique:
haftmann@32578
   676
    "\<not> (\<exists>!x. eval A x) \<or> \<not> (\<exists>!y. eval B y)" by simp
haftmann@32578
   677
  show ?thesis proof (cases "A \<noteq> \<bottom> \<and> B \<noteq> \<bottom>")
haftmann@32578
   678
    case True
haftmann@32578
   679
    then obtain a b where a: "eval A a" and b: "eval B b"
haftmann@32578
   680
      by (blast elim: not_bot)
haftmann@32578
   681
    with True not_unique have "\<not> (\<exists>!x. eval (A \<squnion> B) x)"
haftmann@32578
   682
      by (auto simp add: sup_pred_def bot_pred_def)
bulwahn@33111
   683
    then have "singleton dfault (A \<squnion> B) = dfault ()" by (rule singleton_undefinedI)
haftmann@32578
   684
    with True rhs show ?thesis by simp
haftmann@32578
   685
  next
haftmann@32578
   686
    case False then show ?thesis by auto
haftmann@32578
   687
  qed
haftmann@32578
   688
qed
haftmann@32578
   689
haftmann@32578
   690
lemma singleton_sup:
bulwahn@33110
   691
  "singleton dfault (A \<squnion> B) = (if A = \<bottom> then singleton dfault B
bulwahn@33110
   692
    else if B = \<bottom> then singleton dfault A
bulwahn@33111
   693
    else if singleton dfault A = singleton dfault B then singleton dfault A else dfault ())"
bulwahn@33110
   694
using singleton_sup_aux [of dfault A B] by (simp only: singleton_sup_single_single)
haftmann@32578
   695
haftmann@30328
   696
haftmann@30328
   697
subsubsection {* Derived operations *}
haftmann@30328
   698
haftmann@30328
   699
definition if_pred :: "bool \<Rightarrow> unit pred" where
haftmann@30328
   700
  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
haftmann@30328
   701
bulwahn@33754
   702
definition holds :: "unit pred \<Rightarrow> bool" where
bulwahn@33754
   703
  holds_eq: "holds P = eval P ()"
bulwahn@33754
   704
haftmann@30328
   705
definition not_pred :: "unit pred \<Rightarrow> unit pred" where
haftmann@30328
   706
  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
haftmann@30328
   707
haftmann@30328
   708
lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
haftmann@30328
   709
  unfolding if_pred_eq by (auto intro: singleI)
haftmann@30328
   710
haftmann@30328
   711
lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30328
   712
  unfolding if_pred_eq by (cases b) (auto elim: botE)
haftmann@30328
   713
haftmann@30328
   714
lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
haftmann@30328
   715
  unfolding not_pred_eq eval_pred by (auto intro: singleI)
haftmann@30328
   716
haftmann@30328
   717
lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
haftmann@30328
   718
  unfolding not_pred_eq by (auto intro: singleI)
haftmann@30328
   719
haftmann@30328
   720
lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   721
  unfolding not_pred_eq
haftmann@30328
   722
  by (auto split: split_if_asm elim: botE)
haftmann@30328
   723
haftmann@30328
   724
lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
haftmann@30328
   725
  unfolding not_pred_eq
haftmann@30328
   726
  by (auto split: split_if_asm elim: botE)
bulwahn@33754
   727
lemma "f () = False \<or> f () = True"
bulwahn@33754
   728
by simp
haftmann@30328
   729
blanchet@37549
   730
lemma closure_of_bool_cases [no_atp]:
haftmann@44007
   731
  fixes f :: "unit \<Rightarrow> bool"
haftmann@44007
   732
  assumes "f = (\<lambda>u. False) \<Longrightarrow> P f"
haftmann@44007
   733
  assumes "f = (\<lambda>u. True) \<Longrightarrow> P f"
haftmann@44007
   734
  shows "P f"
bulwahn@33754
   735
proof -
haftmann@44007
   736
  have "f = (\<lambda>u. False) \<or> f = (\<lambda>u. True)"
bulwahn@33754
   737
    apply (cases "f ()")
bulwahn@33754
   738
    apply (rule disjI2)
bulwahn@33754
   739
    apply (rule ext)
bulwahn@33754
   740
    apply (simp add: unit_eq)
bulwahn@33754
   741
    apply (rule disjI1)
bulwahn@33754
   742
    apply (rule ext)
bulwahn@33754
   743
    apply (simp add: unit_eq)
bulwahn@33754
   744
    done
wenzelm@41550
   745
  from this assms show ?thesis by blast
bulwahn@33754
   746
qed
bulwahn@33754
   747
bulwahn@33754
   748
lemma unit_pred_cases:
haftmann@44007
   749
  assumes "P \<bottom>"
haftmann@44007
   750
  assumes "P (single ())"
haftmann@44007
   751
  shows "P Q"
haftmann@44415
   752
using assms unfolding bot_pred_def bot_fun_def bot_bool_def empty_def single_def proof (cases Q)
haftmann@44007
   753
  fix f
haftmann@44007
   754
  assume "P (Pred (\<lambda>u. False))" "P (Pred (\<lambda>u. () = u))"
haftmann@44007
   755
  then have "P (Pred f)" 
haftmann@44007
   756
    by (cases _ f rule: closure_of_bool_cases) simp_all
haftmann@44007
   757
  moreover assume "Q = Pred f"
haftmann@44007
   758
  ultimately show "P Q" by simp
haftmann@44007
   759
qed
haftmann@44007
   760
  
bulwahn@33754
   761
lemma holds_if_pred:
bulwahn@33754
   762
  "holds (if_pred b) = b"
bulwahn@33754
   763
unfolding if_pred_eq holds_eq
bulwahn@33754
   764
by (cases b) (auto intro: singleI elim: botE)
bulwahn@33754
   765
bulwahn@33754
   766
lemma if_pred_holds:
bulwahn@33754
   767
  "if_pred (holds P) = P"
bulwahn@33754
   768
unfolding if_pred_eq holds_eq
bulwahn@33754
   769
by (rule unit_pred_cases) (auto intro: singleI elim: botE)
bulwahn@33754
   770
bulwahn@33754
   771
lemma is_empty_holds:
bulwahn@33754
   772
  "is_empty P \<longleftrightarrow> \<not> holds P"
bulwahn@33754
   773
unfolding is_empty_def holds_eq
bulwahn@33754
   774
by (rule unit_pred_cases) (auto elim: botE intro: singleI)
haftmann@30328
   775
haftmann@41311
   776
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a pred \<Rightarrow> 'b pred" where
haftmann@41311
   777
  "map f P = P \<guillemotright>= (single o f)"
haftmann@41311
   778
haftmann@41311
   779
lemma eval_map [simp]:
haftmann@44363
   780
  "eval (map f P) = (\<Squnion>x\<in>{x. eval P x}. (\<lambda>y. f x = y))"
haftmann@44415
   781
  by (auto simp add: map_def comp_def)
haftmann@41311
   782
haftmann@41505
   783
enriched_type map: map
haftmann@44363
   784
  by (rule ext, rule pred_eqI, auto)+
haftmann@41311
   785
haftmann@41311
   786
haftmann@30328
   787
subsubsection {* Implementation *}
haftmann@30328
   788
haftmann@30328
   789
datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
haftmann@30328
   790
haftmann@30328
   791
primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
haftmann@44414
   792
  "pred_of_seq Empty = \<bottom>"
haftmann@44414
   793
| "pred_of_seq (Insert x P) = single x \<squnion> P"
haftmann@44414
   794
| "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
haftmann@30328
   795
haftmann@30328
   796
definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
haftmann@30328
   797
  "Seq f = pred_of_seq (f ())"
haftmann@30328
   798
haftmann@30328
   799
code_datatype Seq
haftmann@30328
   800
haftmann@30328
   801
primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
haftmann@30328
   802
  "member Empty x \<longleftrightarrow> False"
haftmann@44414
   803
| "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
haftmann@44414
   804
| "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
haftmann@30328
   805
haftmann@30328
   806
lemma eval_member:
haftmann@30328
   807
  "member xq = eval (pred_of_seq xq)"
haftmann@30328
   808
proof (induct xq)
haftmann@30328
   809
  case Empty show ?case
nipkow@39302
   810
  by (auto simp add: fun_eq_iff elim: botE)
haftmann@30328
   811
next
haftmann@30328
   812
  case Insert show ?case
nipkow@39302
   813
  by (auto simp add: fun_eq_iff elim: supE singleE intro: supI1 supI2 singleI)
haftmann@30328
   814
next
haftmann@30328
   815
  case Join then show ?case
nipkow@39302
   816
  by (auto simp add: fun_eq_iff elim: supE intro: supI1 supI2)
haftmann@30328
   817
qed
haftmann@30328
   818
haftmann@30328
   819
lemma eval_code [code]: "eval (Seq f) = member (f ())"
haftmann@30328
   820
  unfolding Seq_def by (rule sym, rule eval_member)
haftmann@30328
   821
haftmann@30328
   822
lemma single_code [code]:
haftmann@30328
   823
  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
haftmann@30328
   824
  unfolding Seq_def by simp
haftmann@30328
   825
haftmann@41080
   826
primrec "apply" :: "('a \<Rightarrow> 'b pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
haftmann@44415
   827
  "apply f Empty = Empty"
haftmann@44415
   828
| "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
haftmann@44415
   829
| "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
haftmann@30328
   830
haftmann@30328
   831
lemma apply_bind:
haftmann@30328
   832
  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
haftmann@30328
   833
proof (induct xq)
haftmann@30328
   834
  case Empty show ?case
haftmann@30328
   835
    by (simp add: bottom_bind)
haftmann@30328
   836
next
haftmann@30328
   837
  case Insert show ?case
haftmann@30328
   838
    by (simp add: single_bind sup_bind)
haftmann@30328
   839
next
haftmann@30328
   840
  case Join then show ?case
haftmann@30328
   841
    by (simp add: sup_bind)
haftmann@30328
   842
qed
haftmann@30328
   843
  
haftmann@30328
   844
lemma bind_code [code]:
haftmann@30328
   845
  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
haftmann@30328
   846
  unfolding Seq_def by (rule sym, rule apply_bind)
haftmann@30328
   847
haftmann@30328
   848
lemma bot_set_code [code]:
haftmann@30328
   849
  "\<bottom> = Seq (\<lambda>u. Empty)"
haftmann@30328
   850
  unfolding Seq_def by simp
haftmann@30328
   851
haftmann@30376
   852
primrec adjunct :: "'a pred \<Rightarrow> 'a seq \<Rightarrow> 'a seq" where
haftmann@44415
   853
  "adjunct P Empty = Join P Empty"
haftmann@44415
   854
| "adjunct P (Insert x Q) = Insert x (Q \<squnion> P)"
haftmann@44415
   855
| "adjunct P (Join Q xq) = Join Q (adjunct P xq)"
haftmann@30376
   856
haftmann@30376
   857
lemma adjunct_sup:
haftmann@30376
   858
  "pred_of_seq (adjunct P xq) = P \<squnion> pred_of_seq xq"
haftmann@30376
   859
  by (induct xq) (simp_all add: sup_assoc sup_commute sup_left_commute)
haftmann@30376
   860
haftmann@30328
   861
lemma sup_code [code]:
haftmann@30328
   862
  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
haftmann@30328
   863
    of Empty \<Rightarrow> g ()
haftmann@30328
   864
     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
haftmann@30376
   865
     | Join P xq \<Rightarrow> adjunct (Seq g) (Join P xq))"
haftmann@30328
   866
proof (cases "f ()")
haftmann@30328
   867
  case Empty
haftmann@30328
   868
  thus ?thesis
haftmann@34007
   869
    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"])
haftmann@30328
   870
next
haftmann@30328
   871
  case Insert
haftmann@30328
   872
  thus ?thesis
haftmann@30328
   873
    unfolding Seq_def by (simp add: sup_assoc)
haftmann@30328
   874
next
haftmann@30328
   875
  case Join
haftmann@30328
   876
  thus ?thesis
haftmann@30376
   877
    unfolding Seq_def
haftmann@30376
   878
    by (simp add: adjunct_sup sup_assoc sup_commute sup_left_commute)
haftmann@30328
   879
qed
haftmann@30328
   880
haftmann@30430
   881
primrec contained :: "'a seq \<Rightarrow> 'a pred \<Rightarrow> bool" where
haftmann@44415
   882
  "contained Empty Q \<longleftrightarrow> True"
haftmann@44415
   883
| "contained (Insert x P) Q \<longleftrightarrow> eval Q x \<and> P \<le> Q"
haftmann@44415
   884
| "contained (Join P xq) Q \<longleftrightarrow> P \<le> Q \<and> contained xq Q"
haftmann@30430
   885
haftmann@30430
   886
lemma single_less_eq_eval:
haftmann@30430
   887
  "single x \<le> P \<longleftrightarrow> eval P x"
haftmann@44415
   888
  by (auto simp add: less_eq_pred_def le_fun_def)
haftmann@30430
   889
haftmann@30430
   890
lemma contained_less_eq:
haftmann@30430
   891
  "contained xq Q \<longleftrightarrow> pred_of_seq xq \<le> Q"
haftmann@30430
   892
  by (induct xq) (simp_all add: single_less_eq_eval)
haftmann@30430
   893
haftmann@30430
   894
lemma less_eq_pred_code [code]:
haftmann@30430
   895
  "Seq f \<le> Q = (case f ()
haftmann@30430
   896
   of Empty \<Rightarrow> True
haftmann@30430
   897
    | Insert x P \<Rightarrow> eval Q x \<and> P \<le> Q
haftmann@30430
   898
    | Join P xq \<Rightarrow> P \<le> Q \<and> contained xq Q)"
haftmann@30430
   899
  by (cases "f ()")
haftmann@30430
   900
    (simp_all add: Seq_def single_less_eq_eval contained_less_eq)
haftmann@30430
   901
haftmann@30430
   902
lemma eq_pred_code [code]:
haftmann@31133
   903
  fixes P Q :: "'a pred"
haftmann@38857
   904
  shows "HOL.equal P Q \<longleftrightarrow> P \<le> Q \<and> Q \<le> P"
haftmann@38857
   905
  by (auto simp add: equal)
haftmann@38857
   906
haftmann@38857
   907
lemma [code nbe]:
haftmann@38857
   908
  "HOL.equal (x :: 'a pred) x \<longleftrightarrow> True"
haftmann@38857
   909
  by (fact equal_refl)
haftmann@30430
   910
haftmann@30430
   911
lemma [code]:
haftmann@30430
   912
  "pred_case f P = f (eval P)"
haftmann@30430
   913
  by (cases P) simp
haftmann@30430
   914
haftmann@30430
   915
lemma [code]:
haftmann@30430
   916
  "pred_rec f P = f (eval P)"
haftmann@30430
   917
  by (cases P) simp
haftmann@30328
   918
bulwahn@31105
   919
inductive eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "eq x x"
bulwahn@31105
   920
bulwahn@31105
   921
lemma eq_is_eq: "eq x y \<equiv> (x = y)"
haftmann@31108
   922
  by (rule eq_reflection) (auto intro: eq.intros elim: eq.cases)
haftmann@30948
   923
haftmann@32578
   924
primrec null :: "'a seq \<Rightarrow> bool" where
haftmann@44415
   925
  "null Empty \<longleftrightarrow> True"
haftmann@44415
   926
| "null (Insert x P) \<longleftrightarrow> False"
haftmann@44415
   927
| "null (Join P xq) \<longleftrightarrow> is_empty P \<and> null xq"
haftmann@32578
   928
haftmann@32578
   929
lemma null_is_empty:
haftmann@32578
   930
  "null xq \<longleftrightarrow> is_empty (pred_of_seq xq)"
haftmann@32578
   931
  by (induct xq) (simp_all add: is_empty_bot not_is_empty_single is_empty_sup)
haftmann@32578
   932
haftmann@32578
   933
lemma is_empty_code [code]:
haftmann@32578
   934
  "is_empty (Seq f) \<longleftrightarrow> null (f ())"
haftmann@32578
   935
  by (simp add: null_is_empty Seq_def)
haftmann@32578
   936
bulwahn@33111
   937
primrec the_only :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a seq \<Rightarrow> 'a" where
bulwahn@33111
   938
  [code del]: "the_only dfault Empty = dfault ()"
haftmann@44415
   939
| "the_only dfault (Insert x P) = (if is_empty P then x else let y = singleton dfault P in if x = y then x else dfault ())"
haftmann@44415
   940
| "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
   941
       else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111
   942
       if x = y then x else dfault ())"
haftmann@32578
   943
haftmann@32578
   944
lemma the_only_singleton:
bulwahn@33110
   945
  "the_only dfault xq = singleton dfault (pred_of_seq xq)"
haftmann@32578
   946
  by (induct xq)
haftmann@32578
   947
    (auto simp add: singleton_bot singleton_single is_empty_def
haftmann@32578
   948
    null_is_empty Let_def singleton_sup)
haftmann@32578
   949
haftmann@32578
   950
lemma singleton_code [code]:
bulwahn@33110
   951
  "singleton dfault (Seq f) = (case f ()
bulwahn@33111
   952
   of Empty \<Rightarrow> dfault ()
haftmann@32578
   953
    | Insert x P \<Rightarrow> if is_empty P then x
bulwahn@33110
   954
        else let y = singleton dfault P in
bulwahn@33111
   955
          if x = y then x else dfault ()
bulwahn@33110
   956
    | Join P xq \<Rightarrow> if is_empty P then the_only dfault xq
bulwahn@33110
   957
        else if null xq then singleton dfault P
bulwahn@33110
   958
        else let x = singleton dfault P; y = the_only dfault xq in
bulwahn@33111
   959
          if x = y then x else dfault ())"
haftmann@32578
   960
  by (cases "f ()")
haftmann@32578
   961
   (auto simp add: Seq_def the_only_singleton is_empty_def
haftmann@32578
   962
      null_is_empty singleton_bot singleton_single singleton_sup Let_def)
haftmann@32578
   963
haftmann@44414
   964
definition the :: "'a pred \<Rightarrow> 'a" where
haftmann@37767
   965
  "the A = (THE x. eval A x)"
bulwahn@33111
   966
haftmann@40674
   967
lemma the_eqI:
haftmann@41080
   968
  "(THE x. eval P x) = x \<Longrightarrow> the P = x"
haftmann@40674
   969
  by (simp add: the_def)
haftmann@40674
   970
haftmann@44414
   971
definition not_unique :: "'a pred \<Rightarrow> 'a" where
haftmann@44414
   972
  [code del]: "not_unique A = (THE x. eval A x)"
haftmann@44414
   973
haftmann@44414
   974
code_abort not_unique
haftmann@44414
   975
haftmann@40674
   976
lemma the_eq [code]: "the A = singleton (\<lambda>x. not_unique A) A"
haftmann@40674
   977
  by (rule the_eqI) (simp add: singleton_def not_unique_def)
bulwahn@33110
   978
haftmann@36531
   979
code_reflect Predicate
haftmann@36513
   980
  datatypes pred = Seq and seq = Empty | Insert | Join
haftmann@36513
   981
  functions map
haftmann@36513
   982
haftmann@30948
   983
ML {*
haftmann@30948
   984
signature PREDICATE =
haftmann@30948
   985
sig
haftmann@30948
   986
  datatype 'a pred = Seq of (unit -> 'a seq)
haftmann@30948
   987
  and 'a seq = Empty | Insert of 'a * 'a pred | Join of 'a pred * 'a seq
haftmann@30959
   988
  val yield: 'a pred -> ('a * 'a pred) option
haftmann@30959
   989
  val yieldn: int -> 'a pred -> 'a list * 'a pred
haftmann@31222
   990
  val map: ('a -> 'b) -> 'a pred -> 'b pred
haftmann@30948
   991
end;
haftmann@30948
   992
haftmann@30948
   993
structure Predicate : PREDICATE =
haftmann@30948
   994
struct
haftmann@30948
   995
haftmann@36513
   996
datatype pred = datatype Predicate.pred
haftmann@36513
   997
datatype seq = datatype Predicate.seq
haftmann@36513
   998
haftmann@36513
   999
fun map f = Predicate.map f;
haftmann@30959
  1000
haftmann@36513
  1001
fun yield (Seq f) = next (f ())
haftmann@36513
  1002
and next Empty = NONE
haftmann@36513
  1003
  | next (Insert (x, P)) = SOME (x, P)
haftmann@36513
  1004
  | next (Join (P, xq)) = (case yield P
haftmann@30959
  1005
     of NONE => next xq
haftmann@36513
  1006
      | SOME (x, Q) => SOME (x, Seq (fn _ => Join (Q, xq))));
haftmann@30959
  1007
haftmann@30959
  1008
fun anamorph f k x = (if k = 0 then ([], x)
haftmann@30959
  1009
  else case f x
haftmann@30959
  1010
   of NONE => ([], x)
haftmann@30959
  1011
    | SOME (v, y) => let
haftmann@30959
  1012
        val (vs, z) = anamorph f (k - 1) y
haftmann@33607
  1013
      in (v :: vs, z) end);
haftmann@30959
  1014
haftmann@30959
  1015
fun yieldn P = anamorph yield P;
haftmann@30948
  1016
haftmann@30948
  1017
end;
haftmann@30948
  1018
*}
haftmann@30948
  1019
haftmann@44363
  1020
lemma eval_mem [simp]:
haftmann@44363
  1021
  "x \<in> eval P \<longleftrightarrow> eval P x"
haftmann@44363
  1022
  by (simp add: mem_def)
haftmann@44363
  1023
haftmann@44363
  1024
lemma eq_mem [simp]:
haftmann@44363
  1025
  "x \<in> (op =) y \<longleftrightarrow> x = y"
haftmann@44363
  1026
  by (auto simp add: mem_def)
haftmann@44363
  1027
haftmann@30328
  1028
no_notation
haftmann@41082
  1029
  bot ("\<bottom>") and
haftmann@41082
  1030
  top ("\<top>") and
haftmann@30328
  1031
  inf (infixl "\<sqinter>" 70) and
haftmann@30328
  1032
  sup (infixl "\<squnion>" 65) and
haftmann@30328
  1033
  Inf ("\<Sqinter>_" [900] 900) and
haftmann@30328
  1034
  Sup ("\<Squnion>_" [900] 900) and
haftmann@30328
  1035
  bind (infixl "\<guillemotright>=" 70)
haftmann@30328
  1036
haftmann@41080
  1037
no_syntax (xsymbols)
haftmann@41082
  1038
  "_INF1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Sqinter>_./ _)" [0, 10] 10)
haftmann@41082
  1039
  "_INF"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Sqinter>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1040
  "_SUP1"     :: "pttrns \<Rightarrow> 'b \<Rightarrow> 'b"           ("(3\<Squnion>_./ _)" [0, 10] 10)
haftmann@41080
  1041
  "_SUP"      :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"  ("(3\<Squnion>_\<in>_./ _)" [0, 0, 10] 10)
haftmann@41080
  1042
wenzelm@36176
  1043
hide_type (open) pred seq
wenzelm@36176
  1044
hide_const (open) Pred eval single bind is_empty singleton if_pred not_pred holds
bulwahn@33111
  1045
  Empty Insert Join Seq member pred_of_seq "apply" adjunct null the_only eq map not_unique the
haftmann@30328
  1046
haftmann@30328
  1047
end