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Datei: sep_set_lems.pvs Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/Extraction.thy
    Author:     Stefan Berghofer, TU Muenchen
*)

section \<open>Program extraction for HOL\<close>

theory Extraction
imports Option
begin

subsection \<open>Setup\<close>

setup \<open>
  Extraction.add_types
      [("bool", ([], NONE))] #>
  Extraction.set_preprocessor (fn thy =>
    Proofterm.rewrite_proof_notypes
      ([], Rewrite_HOL_Proof.elim_cong :: Proof_Rewrite_Rules.rprocs true) o
    Proofterm.rewrite_proof thy
      (Rewrite_HOL_Proof.rews,
       Proof_Rewrite_Rules.rprocs true @ [Proof_Rewrite_Rules.expand_of_class thy]) o
    Proof_Rewrite_Rules.elim_vars (curry Const \<^const_name>\<open>default\<close>))
\<close>

lemmas [extraction_expand] =
  meta_spec atomize_eq atomize_all atomize_imp atomize_conj
  allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2
  notE' impE' impE iffE imp_cong simp_thms eq_True eq_False
  induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
  induct_atomize induct_atomize' induct_rulify induct_rulify'
  induct_rulify_fallback induct_trueI
  True_implies_equals implies_True_equals TrueE
  False_implies_equals implies_False_swap

lemmas [extraction_expand_def] =
  HOL.induct_forall_def HOL.induct_implies_def HOL.induct_equal_def HOL.induct_conj_def
  HOL.induct_true_def HOL.induct_false_def

datatype (plugins only: code extraction) sumbool = Left | Right

subsection \<open>Type of extracted program\<close>

extract_type
  "typeof (Trueprop P) \ typeof P"

  "typeof P \ Type (TYPE(Null)) \ typeof Q \ Type (TYPE('Q)) \
     typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('Q))"

  "typeof Q \ Type (TYPE(Null)) \ typeof (P \ Q) \ Type (TYPE(Null))"

  "typeof P \ Type (TYPE('P)) \ typeof Q \ Type (TYPE('Q)) \
     typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('P \<Rightarrow> 'Q))"

  "(\x. typeof (P x)) \ (\x. Type (TYPE(Null))) \
     typeof (\<forall>x. P x) \<equiv> Type (TYPE(Null))"

  "(\x. typeof (P x)) \ (\x. Type (TYPE('P))) \
     typeof (\<forall>x::'a. P x) \<equiv> Type (TYPE('a \<Rightarrow> 'P))"

  "(\x. typeof (P x)) \ (\x. Type (TYPE(Null))) \
     typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a))"

  "(\x. typeof (P x)) \ (\x. Type (TYPE('P))) \
     typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a \<times> 'P))"

  "typeof P \ Type (TYPE(Null)) \ typeof Q \ Type (TYPE(Null)) \
     typeof (P \<or> Q) \<equiv> Type (TYPE(sumbool))"

  "typeof P \ Type (TYPE(Null)) \ typeof Q \ Type (TYPE('Q)) \
     typeof (P \<or> Q) \<equiv> Type (TYPE('Q option))"

  "typeof P \ Type (TYPE('P)) \ typeof Q \ Type (TYPE(Null)) \
     typeof (P \<or> Q) \<equiv> Type (TYPE('P option))"

  "typeof P \ Type (TYPE('P)) \ typeof Q \ Type (TYPE('Q)) \
     typeof (P \<or> Q) \<equiv> Type (TYPE('P + 'Q))"

  "typeof P \ Type (TYPE(Null)) \ typeof Q \ Type (TYPE('Q)) \
     typeof (P \<and> Q) \<equiv> Type (TYPE('Q))"

  "typeof P \ Type (TYPE('P)) \ typeof Q \ Type (TYPE(Null)) \
     typeof (P \<and> Q) \<equiv> Type (TYPE('P))"

  "typeof P \ Type (TYPE('P)) \ typeof Q \ Type (TYPE('Q)) \
     typeof (P \<and> Q) \<equiv> Type (TYPE('P \<times> 'Q))"

  "typeof (P = Q) \ typeof ((P \ Q) \ (Q \ P))"

  "typeof (x \ P) \ typeof P"

subsection \<open>Realizability\<close>

realizability
  "(realizes t (Trueprop P)) \ (Trueprop (realizes t P))"

  "(typeof P) \ (Type (TYPE(Null))) \
     (realizes t (P \<longrightarrow> Q)) \<equiv> (realizes Null P \<longrightarrow> realizes t Q)"

  "(typeof P) \ (Type (TYPE('P))) \
   (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
     (realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x::'P. realizes x P \<longrightarrow> realizes Null Q)"

  "(realizes t (P \ Q)) \ (\x. realizes x P \ realizes (t x) Q)"

  "(\x. typeof (P x)) \ (\x. Type (TYPE(Null))) \
     (realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes Null (P x))"

  "(realizes t (\x. P x)) \ (\x. realizes (t x) (P x))"

  "(\x. typeof (P x)) \ (\x. Type (TYPE(Null))) \
     (realizes t (\<exists>x. P x)) \<equiv> (realizes Null (P t))"

  "(realizes t (\x. P x)) \ (realizes (snd t) (P (fst t)))"

  "(typeof P) \ (Type (TYPE(Null))) \
   (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
     (realizes t (P \<or> Q)) \<equiv>
     (case t of Left \<Rightarrow> realizes Null P | Right \<Rightarrow> realizes Null Q)"

  "(typeof P) \ (Type (TYPE(Null))) \
     (realizes t (P \<or> Q)) \<equiv>
     (case t of None \<Rightarrow> realizes Null P | Some q \<Rightarrow> realizes q Q)"

  "(typeof Q) \ (Type (TYPE(Null))) \
     (realizes t (P \<or> Q)) \<equiv>
     (case t of None \<Rightarrow> realizes Null Q | Some p \<Rightarrow> realizes p P)"

  "(realizes t (P \ Q)) \
   (case t of Inl p \<Rightarrow> realizes p P | Inr q \<Rightarrow> realizes q Q)"

  "(typeof P) \ (Type (TYPE(Null))) \
     (realizes t (P \<and> Q)) \<equiv> (realizes Null P \<and> realizes t Q)"

  "(typeof Q) \ (Type (TYPE(Null))) \
     (realizes t (P \<and> Q)) \<equiv> (realizes t P \<and> realizes Null Q)"

  "(realizes t (P \ Q)) \ (realizes (fst t) P \ realizes (snd t) Q)"

  "typeof P \ Type (TYPE(Null)) \
     realizes t (\<not> P) \<equiv> \<not> realizes Null P"

  "typeof P \ Type (TYPE('P)) \
     realizes t (\<not> P) \<equiv> (\<forall>x::'P. \<not> realizes x P)"

  "typeof (P::bool) \ Type (TYPE(Null)) \
   typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
     realizes t (P = Q) \<equiv> realizes Null P = realizes Null Q"

  "(realizes t (P = Q)) \ (realizes t ((P \ Q) \ (Q \ P)))"

subsection \<open>Computational content of basic inference rules\<close>

theorem disjE_realizer:
  assumes r: "case x of Inl p \ P p | Inr q \ Q q"
  and r1: "\p. P p \ R (f p)" and r2: "\q. Q q \ R (g q)"
  shows "R (case x of Inl p \ f p | Inr q \ g q)"
proof (cases x)
  case Inl
  with r show ?thesis by simp (rule r1)
next
  case Inr
  with r show ?thesis by simp (rule r2)
qed

theorem disjE_realizer2:
  assumes r: "case x of None \ P | Some q \ Q q"
  and r1: "P \ R f" and r2: "\q. Q q \ R (g q)"
  shows "R (case x of None \ f | Some q \ g q)"
proof (cases x)
  case None
  with r show ?thesis by simp (rule r1)
next
  case Some
  with r show ?thesis by simp (rule r2)
qed

theorem disjE_realizer3:
  assumes r: "case x of Left \ P | Right \ Q"
  and r1: "P \ R f" and r2: "Q \ R g"
  shows "R (case x of Left \ f | Right \ g)"
proof (cases x)
  case Left
  with r show ?thesis by simp (rule r1)
next
  case Right
  with r show ?thesis by simp (rule r2)
qed

theorem conjI_realizer:
  "P p \ Q q \ P (fst (p, q)) \ Q (snd (p, q))"
  by simp

theorem exI_realizer:
  "P y x \ P (snd (x, y)) (fst (x, y))" by simp

theorem exE_realizer: "P (snd p) (fst p) \
  (\<And>x y. P y x \<Longrightarrow> Q (f x y)) \<Longrightarrow> Q (let (x, y) = p in f x y)"
  by (cases p) (simp add: Let_def)

theorem exE_realizer': "P (snd p) (fst p) \
  (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp

realizers
  impI (P, Q): "\pq. pq"
    "\<^bold>\(c: _) (d: _) P Q pq (h: _). allI \ _ \ c \ (\<^bold>\x. impI \ _ \ _ \ (h \ x))"

  impI (P): "Null"
    "\<^bold>\(c: _) P Q (h: _). allI \ _ \ c \ (\<^bold>\x. impI \ _ \ _ \ (h \ x))"

  impI (Q): "\q. q" "\<^bold>\(c: _) P Q q. impI \ _ \ _"

  impI: "Null" "impI"

  mp (P, Q): "\pq. pq"
    "\<^bold>\(c: _) (d: _) P Q pq (h: _) p. mp \ _ \ _ \ (spec \ _ \ p \ c \ h)"

  mp (P): "Null"
    "\<^bold>\(c: _) P Q (h: _) p. mp \ _ \ _ \ (spec \ _ \ p \ c \ h)"

  mp (Q): "\q. q" "\<^bold>\(c: _) P Q q. mp \ _ \ _"

  mp: "Null" "mp"

  allI (P): "\p. p" "\<^bold>\(c: _) P (d: _) p. allI \ _ \ d"

  allI: "Null" "allI"

  spec (P): "\x p. p x" "\<^bold>\(c: _) P x (d: _) p. spec \ _ \ x \ d"

  spec: "Null" "spec"

  exI (P): "\x p. (x, p)" "\<^bold>\(c: _) P x (d: _) p. exI_realizer \ P \ p \ x \ c \ d"

  exI: "\x. x" "\<^bold>\P x (c: _) (h: _). h"

  exE (P, Q): "\p pq. let (x, y) = p in pq x y"
    "\<^bold>\(c: _) (d: _) P Q (e: _) p (h: _) pq. exE_realizer \ P \ p \ Q \ pq \ c \ e \ d \ h"

  exE (P): "Null"
    "\<^bold>\(c: _) P Q (d: _) p. exE_realizer' \ _ \ _ \ _ \ c \ d"

  exE (Q): "\x pq. pq x"
    "\<^bold>\(c: _) P Q (d: _) x (h1: _) pq (h2: _). h2 \ x \ h1"

  exE: "Null"
    "\<^bold>\P Q (c: _) x (h1: _) (h2: _). h2 \ x \ h1"

  conjI (P, Q): "Pair"
    "\<^bold>\(c: _) (d: _) P Q p (h: _) q. conjI_realizer \ P \ p \ Q \ q \ c \ d \ h"

  conjI (P): "\p. p"
    "\<^bold>\(c: _) P Q p. conjI \ _ \ _"

  conjI (Q): "\q. q"
    "\<^bold>\(c: _) P Q (h: _) q. conjI \ _ \ _ \ h"

  conjI: "Null" "conjI"

  conjunct1 (P, Q): "fst"
    "\<^bold>\(c: _) (d: _) P Q pq. conjunct1 \ _ \ _"

  conjunct1 (P): "\p. p"
    "\<^bold>\(c: _) P Q p. conjunct1 \ _ \ _"

  conjunct1 (Q): "Null"
    "\<^bold>\(c: _) P Q q. conjunct1 \ _ \ _"

  conjunct1: "Null" "conjunct1"

  conjunct2 (P, Q): "snd"
    "\<^bold>\(c: _) (d: _) P Q pq. conjunct2 \ _ \ _"

  conjunct2 (P): "Null"
    "\<^bold>\(c: _) P Q p. conjunct2 \ _ \ _"

  conjunct2 (Q): "\p. p"
    "\<^bold>\(c: _) P Q p. conjunct2 \ _ \ _"

  conjunct2: "Null" "conjunct2"

  disjI1 (P, Q): "Inl"
    "\<^bold>\(c: _) (d: _) P Q p. iffD2 \ _ \ _ \ (sum.case_1 \ P \ _ \ p \ arity_type_bool \ c \ d)"

  disjI1 (P): "Some"
    "\<^bold>\(c: _) P Q p. iffD2 \ _ \ _ \ (option.case_2 \ _ \ P \ p \ arity_type_bool \ c)"

  disjI1 (Q): "None"
    "\<^bold>\(c: _) P Q. iffD2 \ _ \ _ \ (option.case_1 \ _ \ _ \ arity_type_bool \ c)"

  disjI1: "Left"
    "\<^bold>\P Q. iffD2 \ _ \ _ \ (sumbool.case_1 \ _ \ _ \ arity_type_bool)"

  disjI2 (P, Q): "Inr"
    "\<^bold>\(d: _) (c: _) Q P q. iffD2 \ _ \ _ \ (sum.case_2 \ _ \ Q \ q \ arity_type_bool \ c \ d)"

  disjI2 (P): "None"
    "\<^bold>\(c: _) Q P. iffD2 \ _ \ _ \ (option.case_1 \ _ \ _ \ arity_type_bool \ c)"

  disjI2 (Q): "Some"
    "\<^bold>\(c: _) Q P q. iffD2 \ _ \ _ \ (option.case_2 \ _ \ Q \ q \ arity_type_bool \ c)"

  disjI2: "Right"
    "\<^bold>\Q P. iffD2 \ _ \ _ \ (sumbool.case_2 \ _ \ _ \ arity_type_bool)"

  disjE (P, Q, R): "\pq pr qr.
     (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
    "\<^bold>\(c: _) (d: _) (e: _) P Q R pq (h1: _) pr (h2: _) qr.
       disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> e \<bullet> h1 \<bullet> h2"

  disjE (Q, R): "\pq pr qr.
     (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
    "\<^bold>\(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr.
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h2"

  disjE (P, R): "\pq pr qr.
     (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
    "\<^bold>\(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr (h3: _).
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h3 \<bullet> h2"

  disjE (R): "\pq pr qr.
     (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
    "\<^bold>\(c: _) P Q R pq (h1: _) pr (h2: _) qr.
       disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> h1 \<bullet> h2"

  disjE (P, Q): "Null"
    "\<^bold>\(c: _) (d: _) P Q R pq. disjE_realizer \ _ \ _ \ pq \ (\x. R) \ _ \ _ \ c \ d \ arity_type_bool"

  disjE (Q): "Null"
    "\<^bold>\(c: _) P Q R pq. disjE_realizer2 \ _ \ _ \ pq \ (\x. R) \ _ \ _ \ c \ arity_type_bool"

  disjE (P): "Null"
    "\<^bold>\(c: _) P Q R pq (h1: _) (h2: _) (h3: _).
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> arity_type_bool \<bullet> h1 \<bullet> h3 \<bullet> h2"

  disjE: "Null"
    "\<^bold>\P Q R pq. disjE_realizer3 \ _ \ _ \ pq \ (\x. R) \ _ \ _ \ arity_type_bool"

  FalseE (P): "default"
    "\<^bold>\(c: _) P. FalseE \ _"

  FalseE: "Null" "FalseE"

  notI (P): "Null"
    "\<^bold>\(c: _) P (h: _). allI \ _ \ c \ (\<^bold>\x. notI \ _ \ (h \ x))"

  notI: "Null" "notI"

  notE (P, R): "\p. default"
    "\<^bold>\(c: _) (d: _) P R (h: _) p. notE \ _ \ _ \ (spec \ _ \ p \ c \ h)"

  notE (P): "Null"
    "\<^bold>\(c: _) P R (h: _) p. notE \ _ \ _ \ (spec \ _ \ p \ c \ h)"

  notE (R): "default"
    "\<^bold>\(c: _) P R. notE \ _ \ _"

  notE: "Null" "notE"

  subst (P): "\s t ps. ps"
    "\<^bold>\(c: _) s t P (d: _) (h: _) ps. subst \ s \ t \ P ps \ d \ h"

  subst: "Null" "subst"

  iffD1 (P, Q): "fst"
    "\<^bold>\(d: _) (c: _) Q P pq (h: _) p.
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> d \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"

  iffD1 (P): "\p. p"
    "\<^bold>\(c: _) Q P p (h: _). mp \ _ \ _ \ (conjunct1 \ _ \ _ \ h)"

  iffD1 (Q): "Null"
    "\<^bold>\(c: _) Q P q1 (h: _) q2.
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"

  iffD1: "Null" "iffD1"

  iffD2 (P, Q): "snd"
    "\<^bold>\(c: _) (d: _) P Q pq (h: _) q.
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> d \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"

  iffD2 (P): "\p. p"
    "\<^bold>\(c: _) P Q p (h: _). mp \ _ \ _ \ (conjunct2 \ _ \ _ \ h)"

  iffD2 (Q): "Null"
    "\<^bold>\(c: _) P Q q1 (h: _) q2.
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"

  iffD2: "Null" "iffD2"

  iffI (P, Q): "Pair"
    "\<^bold>\(c: _) (d: _) P Q pq (h1 : _) qp (h2 : _). conjI_realizer \
       (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
       (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
       (arity_type_fun \<bullet> c \<bullet> d) \<bullet>
       (arity_type_fun \<bullet> d \<bullet> c) \<bullet>
       (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
       (allI \<cdot> _ \<bullet> d \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"

  iffI (P): "\p. p"
    "\<^bold>\(c: _) P Q (h1 : _) p (h2 : _). conjI \ _ \ _ \
       (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
       (impI \<cdot> _ \<cdot> _ \<bullet> h2)"

  iffI (Q): "\q. q"
    "\<^bold>\(c: _) P Q q (h1 : _) (h2 : _). conjI \ _ \ _ \
       (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
       (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"

  iffI: "Null" "iffI"

end

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