(* Title: HOL/Extraction.thy Author: Stefan Berghofer, TU Muenchen *) section ‹Program extraction for HOL› theory Extraction imports Option begin subsection ‹Setup› setup ‹ 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>‹default›)) › 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 ‹Type of extracted program› extract_type "typeof (Trueprop P) ≡ typeof P" "typeof P ≡ Type (TYPE(Null)) ⟹ typeof Q ≡ Type (TYPE('Q)) ⟹ typeof (P ⟶ Q) ≡ 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 ⟶ Q) ≡ Type (TYPE('P ⇒ 'Q))" "(λx. typeof (P x)) ≡ (λx. Type (TYPE(Null))) ⟹ typeof (∀x. P x) ≡ Type (TYPE(Null))" "(λx. typeof (P x)) ≡ (λx. Type (TYPE('P))) ⟹ typeof (∀x::'a. P x) ≡ Type (TYPE('a ⇒ 'P))" "(λx. typeof (P x)) ≡ (λx. Type (TYPE(Null))) ⟹ typeof (∃x::'a. P x) ≡ Type (TYPE('a))" "(λx. typeof (P x)) ≡ (λx. Type (TYPE('P))) ⟹ typeof (∃x::'a. P x) ≡ Type (TYPE('a × 'P))" "typeof P ≡ Type (TYPE(Null)) ⟹ typeof Q ≡ Type (TYPE(Null)) ⟹ typeof (P ∨ Q) ≡ Type (TYPE(sumbool))" "typeof P ≡ Type (TYPE(Null)) ⟹ typeof Q ≡ Type (TYPE('Q)) ⟹ typeof (P ∨ Q) ≡ Type (TYPE('Q option))" "typeof P ≡ Type (TYPE('P)) ⟹ typeof Q ≡ Type (TYPE(Null)) ⟹ typeof (P ∨ Q) ≡ Type (TYPE('P option))" "typeof P ≡ Type (TYPE('P)) ⟹ typeof Q ≡ Type (TYPE('Q)) ⟹ typeof (P ∨ Q) ≡ Type (TYPE('P + 'Q))" "typeof P ≡ Type (TYPE(Null)) ⟹ typeof Q ≡ Type (TYPE('Q)) ⟹ typeof (P ∧ Q) ≡ Type (TYPE('Q))" "typeof P ≡ Type (TYPE('P)) ⟹ typeof Q ≡ Type (TYPE(Null)) ⟹ typeof (P ∧ Q) ≡ Type (TYPE('P))" "typeof P ≡ Type (TYPE('P)) ⟹ typeof Q ≡ Type (TYPE('Q)) ⟹ typeof (P ∧ Q) ≡ Type (TYPE('P × 'Q))" "typeof (P = Q) ≡ typeof ((P ⟶ Q) ∧ (Q ⟶ P))" "typeof (x ∈ P) ≡ typeof P" subsection ‹Realizability› realizability "(realizes t (Trueprop P)) ≡ (Trueprop (realizes t P))" "(typeof P) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ⟶ Q)) ≡ (realizes Null P ⟶ realizes t Q)" "(typeof P) ≡ (Type (TYPE('P))) ⟹ (typeof Q) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ⟶ Q)) ≡ (∀x::'P. realizes x P ⟶ realizes Null Q)" "(realizes t (P ⟶ Q)) ≡ (∀x. realizes x P ⟶ realizes (t x) Q)" "(λx. typeof (P x)) ≡ (λx. Type (TYPE(Null))) ⟹ (realizes t (∀x. P x)) ≡ (∀x. realizes Null (P x))" "(realizes t (∀x. P x)) ≡ (∀x. realizes (t x) (P x))" "(λx. typeof (P x)) ≡ (λx. Type (TYPE(Null))) ⟹ (realizes t (∃x. P x)) ≡ (realizes Null (P t))" "(realizes t (∃x. P x)) ≡ (realizes (snd t) (P (fst t)))" "(typeof P) ≡ (Type (TYPE(Null))) ⟹ (typeof Q) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ∨ Q)) ≡ (case t of Left ⇒ realizes Null P | Right ⇒ realizes Null Q)" "(typeof P) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ∨ Q)) ≡ (case t of None ⇒ realizes Null P | Some q ⇒ realizes q Q)" "(typeof Q) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ∨ Q)) ≡ (case t of None ⇒ realizes Null Q | Some p ⇒ realizes p P)" "(realizes t (P ∨ Q)) ≡ (case t of Inl p ⇒ realizes p P | Inr q ⇒ realizes q Q)" "(typeof P) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ∧ Q)) ≡ (realizes Null P ∧ realizes t Q)" "(typeof Q) ≡ (Type (TYPE(Null))) ⟹ (realizes t (P ∧ Q)) ≡ (realizes t P ∧ realizes Null Q)" "(realizes t (P ∧ Q)) ≡ (realizes (fst t) P ∧ realizes (snd t) Q)" "typeof P ≡ Type (TYPE(Null)) ⟹ realizes t (¬ P) ≡ ¬ realizes Null P" "typeof P ≡ Type (TYPE('P)) ⟹ realizes t (¬ P) ≡ (∀x::'P. ¬ realizes x P)" "typeof (P::bool) ≡ Type (TYPE(Null)) ⟹ typeof Q ≡ Type (TYPE(Null)) ⟹ realizes t (P = Q) ≡ realizes Null P = realizes Null Q" "(realizes t (P = Q)) ≡ (realizes t ((P ⟶ Q) ∧ (Q ⟶ P)))" subsection ‹Computational content of basic inference rules› 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) ⟹ (⋀x y. P y x ⟹ Q (f x y)) ⟹ Q (let (x, y) = p in f x y)" by (cases p) (simp add: Let_def) theorem exE_realizer': "P (snd p) (fst p) ⟹ (⋀x y. P y x ⟹ Q) ⟹ Q" by (cases p) simp realizers impI (P, Q): "λpq. pq" "❙λ(c: _) (d: _) P Q pq (h: _). allI ⋅ _ ∙ c ∙ (❙λx. impI ⋅ _ ⋅ _ ∙ (h ⋅ x))" impI (P): "Null" "❙λ(c: _) P Q (h: _). allI ⋅ _ ∙ c ∙ (❙λx. impI ⋅ _ ⋅ _ ∙ (h ⋅ x))" impI (Q): "λq. q" "❙λ(c: _) P Q q. impI ⋅ _ ⋅ _" impI: "Null" "impI" mp (P, Q): "λpq. pq" "❙λ(c: _) (d: _) P Q pq (h: _) p. mp ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ p ∙ c ∙ h)" mp (P): "Null" "❙λ(c: _) P Q (h: _) p. mp ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ p ∙ c ∙ h)" mp (Q): "λq. q" "❙λ(c: _) P Q q. mp ⋅ _ ⋅ _" mp: "Null" "mp" allI (P): "λp. p" "❙λ(c: _) P (d: _) p. allI ⋅ _ ∙ d" allI: "Null" "allI" spec (P): "λx p. p x" "❙λ(c: _) P x (d: _) p. spec ⋅ _ ⋅ x ∙ d" spec: "Null" "spec" exI (P): "λx p. (x, p)" "❙λ(c: _) P x (d: _) p. exI_realizer ⋅ P ⋅ p ⋅ x ∙ c ∙ d" exI: "λx. x" "❙λP x (c: _) (h: _). h" exE (P, Q): "λp pq. let (x, y) = p in pq x y" "❙λ(c: _) (d: _) P Q (e: _) p (h: _) pq. exE_realizer ⋅ P ⋅ p ⋅ Q ⋅ pq ∙ c ∙ e ∙ d ∙ h" exE (P): "Null" "❙λ(c: _) P Q (d: _) p. exE_realizer' ⋅ _ ⋅ _ ⋅ _ ∙ c ∙ d" exE (Q): "λx pq. pq x" "❙λ(c: _) P Q (d: _) x (h1: _) pq (h2: _). h2 ⋅ x ∙ h1" exE: "Null" "❙λP Q (c: _) x (h1: _) (h2: _). h2 ⋅ x ∙ h1" conjI (P, Q): "Pair" "❙λ(c: _) (d: _) P Q p (h: _) q. conjI_realizer ⋅ P ⋅ p ⋅ Q ⋅ q ∙ c ∙ d ∙ h" conjI (P): "λp. p" "❙λ(c: _) P Q p. conjI ⋅ _ ⋅ _" conjI (Q): "λq. q" "❙λ(c: _) P Q (h: _) q. conjI ⋅ _ ⋅ _ ∙ h" conjI: "Null" "conjI" conjunct1 (P, Q): "fst" "❙λ(c: _) (d: _) P Q pq. conjunct1 ⋅ _ ⋅ _" conjunct1 (P): "λp. p" "❙λ(c: _) P Q p. conjunct1 ⋅ _ ⋅ _" conjunct1 (Q): "Null" "❙λ(c: _) P Q q. conjunct1 ⋅ _ ⋅ _" conjunct1: "Null" "conjunct1" conjunct2 (P, Q): "snd" "❙λ(c: _) (d: _) P Q pq. conjunct2 ⋅ _ ⋅ _" conjunct2 (P): "Null" "❙λ(c: _) P Q p. conjunct2 ⋅ _ ⋅ _" conjunct2 (Q): "λp. p" "❙λ(c: _) P Q p. conjunct2 ⋅ _ ⋅ _" conjunct2: "Null" "conjunct2" disjI1 (P, Q): "Inl" "❙λ(c: _) (d: _) P Q p. iffD2 ⋅ _ ⋅ _ ∙ (sum.case_1 ⋅ P ⋅ _ ⋅ p ∙ arity_type_bool ∙ c ∙ d)" disjI1 (P): "Some" "❙λ(c: _) P Q p. iffD2 ⋅ _ ⋅ _ ∙ (option.case_2 ⋅ _ ⋅ P ⋅ p ∙ arity_type_bool ∙ c)" disjI1 (Q): "None" "❙λ(c: _) P Q. iffD2 ⋅ _ ⋅ _ ∙ (option.case_1 ⋅ _ ⋅ _ ∙ arity_type_bool ∙ c)" disjI1: "Left" "❙λP Q. iffD2 ⋅ _ ⋅ _ ∙ (sumbool.case_1 ⋅ _ ⋅ _ ∙ arity_type_bool)" disjI2 (P, Q): "Inr" "❙λ(d: _) (c: _) Q P q. iffD2 ⋅ _ ⋅ _ ∙ (sum.case_2 ⋅ _ ⋅ Q ⋅ q ∙ arity_type_bool ∙ c ∙ d)" disjI2 (P): "None" "❙λ(c: _) Q P. iffD2 ⋅ _ ⋅ _ ∙ (option.case_1 ⋅ _ ⋅ _ ∙ arity_type_bool ∙ c)" disjI2 (Q): "Some" "❙λ(c: _) Q P q. iffD2 ⋅ _ ⋅ _ ∙ (option.case_2 ⋅ _ ⋅ Q ⋅ q ∙ arity_type_bool ∙ c)" disjI2: "Right" "❙λQ P. iffD2 ⋅ _ ⋅ _ ∙ (sumbool.case_2 ⋅ _ ⋅ _ ∙ arity_type_bool)" disjE (P, Q, R): "λpq pr qr. (case pq of Inl p ⇒ pr p | Inr q ⇒ qr q)" "❙λ(c: _) (d: _) (e: _) P Q R pq (h1: _) pr (h2: _) qr. disjE_realizer ⋅ _ ⋅ _ ⋅ pq ⋅ R ⋅ pr ⋅ qr ∙ c ∙ d ∙ e ∙ h1 ∙ h2" disjE (Q, R): "λpq pr qr. (case pq of None ⇒ pr | Some q ⇒ qr q)" "❙λ(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr. disjE_realizer2 ⋅ _ ⋅ _ ⋅ pq ⋅ R ⋅ pr ⋅ qr ∙ c ∙ d ∙ h1 ∙ h2" disjE (P, R): "λpq pr qr. (case pq of None ⇒ qr | Some p ⇒ pr p)" "❙λ(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr (h3: _). disjE_realizer2 ⋅ _ ⋅ _ ⋅ pq ⋅ R ⋅ qr ⋅ pr ∙ c ∙ d ∙ h1 ∙ h3 ∙ h2" disjE (R): "λpq pr qr. (case pq of Left ⇒ pr | Right ⇒ qr)" "❙λ(c: _) P Q R pq (h1: _) pr (h2: _) qr. disjE_realizer3 ⋅ _ ⋅ _ ⋅ pq ⋅ R ⋅ pr ⋅ qr ∙ c ∙ h1 ∙ h2" disjE (P, Q): "Null" "❙λ(c: _) (d: _) P Q R pq. disjE_realizer ⋅ _ ⋅ _ ⋅ pq ⋅ (λx. R) ⋅ _ ⋅ _ ∙ c ∙ d ∙ arity_type_bool" disjE (Q): "Null" "❙λ(c: _) P Q R pq. disjE_realizer2 ⋅ _ ⋅ _ ⋅ pq ⋅ (λx. R) ⋅ _ ⋅ _ ∙ c ∙ arity_type_bool" disjE (P): "Null" "❙λ(c: _) P Q R pq (h1: _) (h2: _) (h3: _). disjE_realizer2 ⋅ _ ⋅ _ ⋅ pq ⋅ (λx. R) ⋅ _ ⋅ _ ∙ c ∙ arity_type_bool ∙ h1 ∙ h3 ∙ h2" disjE: "Null" "❙λP Q R pq. disjE_realizer3 ⋅ _ ⋅ _ ⋅ pq ⋅ (λx. R) ⋅ _ ⋅ _ ∙ arity_type_bool" FalseE (P): "default" "❙λ(c: _) P. FalseE ⋅ _" FalseE: "Null" "FalseE" notI (P): "Null" "❙λ(c: _) P (h: _). allI ⋅ _ ∙ c ∙ (❙λx. notI ⋅ _ ∙ (h ⋅ x))" notI: "Null" "notI" notE (P, R): "λp. default" "❙λ(c: _) (d: _) P R (h: _) p. notE ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ p ∙ c ∙ h)" notE (P): "Null" "❙λ(c: _) P R (h: _) p. notE ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ p ∙ c ∙ h)" notE (R): "default" "❙λ(c: _) P R. notE ⋅ _ ⋅ _" notE: "Null" "notE" subst (P): "λs t ps. ps" "❙λ(c: _) s t P (d: _) (h: _) ps. subst ⋅ s ⋅ t ⋅ P ps ∙ d ∙ h" subst: "Null" "subst" iffD1 (P, Q): "fst" "❙λ(d: _) (c: _) Q P pq (h: _) p. mp ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ p ∙ d ∙ (conjunct1 ⋅ _ ⋅ _ ∙ h))" iffD1 (P): "λp. p" "❙λ(c: _) Q P p (h: _). mp ⋅ _ ⋅ _ ∙ (conjunct1 ⋅ _ ⋅ _ ∙ h)" iffD1 (Q): "Null" "❙λ(c: _) Q P q1 (h: _) q2. mp ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ q2 ∙ c ∙ (conjunct1 ⋅ _ ⋅ _ ∙ h))" iffD1: "Null" "iffD1" iffD2 (P, Q): "snd" "❙λ(c: _) (d: _) P Q pq (h: _) q. mp ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ q ∙ d ∙ (conjunct2 ⋅ _ ⋅ _ ∙ h))" iffD2 (P): "λp. p" "❙λ(c: _) P Q p (h: _). mp ⋅ _ ⋅ _ ∙ (conjunct2 ⋅ _ ⋅ _ ∙ h)" iffD2 (Q): "Null" "❙λ(c: _) P Q q1 (h: _) q2. mp ⋅ _ ⋅ _ ∙ (spec ⋅ _ ⋅ q2 ∙ c ∙ (conjunct2 ⋅ _ ⋅ _ ∙ h))" iffD2: "Null" "iffD2" iffI (P, Q): "Pair" "❙λ(c: _) (d: _) P Q pq (h1 : _) qp (h2 : _). conjI_realizer ⋅ (λpq. ∀x. P x ⟶ Q (pq x)) ⋅ pq ⋅ (λqp. ∀x. Q x ⟶ P (qp x)) ⋅ qp ∙ (arity_type_fun ∙ c ∙ d) ∙ (arity_type_fun ∙ d ∙ c) ∙ (allI ⋅ _ ∙ c ∙ (❙λx. impI ⋅ _ ⋅ _ ∙ (h1 ⋅ x))) ∙ (allI ⋅ _ ∙ d ∙ (❙λx. impI ⋅ _ ⋅ _ ∙ (h2 ⋅ x)))" iffI (P): "λp. p" "❙λ(c: _) P Q (h1 : _) p (h2 : _). conjI ⋅ _ ⋅ _ ∙ (allI ⋅ _ ∙ c ∙ (❙λx. impI ⋅ _ ⋅ _ ∙ (h1 ⋅ x))) ∙ (impI ⋅ _ ⋅ _ ∙ h2)" iffI (Q): "λq. q" "❙λ(c: _) P Q q (h1 : _) (h2 : _). conjI ⋅ _ ⋅ _ ∙ (impI ⋅ _ ⋅ _ ∙ h1) ∙ (allI ⋅ _ ∙ c ∙ (❙λx. impI ⋅ _ ⋅ _ ∙ (h2 ⋅ x)))" iffI: "Null" "iffI" end