author | berghofe |
Thu, 10 Oct 2002 14:18:01 +0200 | |
changeset 13635 | c41e88151b54 |
parent 13599 | cfdf7e4cd0d2 |
child 13725 | 12404b452034 |
permissions | -rw-r--r-- |
13403 | 1 |
(* Title: HOL/Extraction.thy |
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ID: $Id$ |
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Author: Stefan Berghofer, TU Muenchen |
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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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header {* Program extraction for HOL *} |
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theory Extraction = Datatype |
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files |
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"Tools/rewrite_hol_proof.ML": |
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subsection {* Setup *} |
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ML_setup {* |
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Context.>> (fn thy => thy |> |
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Extraction.set_preprocessor (fn sg => |
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Proofterm.rewrite_proof_notypes |
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([], ("HOL/elim_cong", RewriteHOLProof.elim_cong) :: |
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ProofRewriteRules.rprocs true) o |
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Proofterm.rewrite_proof (Sign.tsig_of sg) |
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(RewriteHOLProof.rews, ProofRewriteRules.rprocs true) o |
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ProofRewriteRules.elim_vars (curry Const "arbitrary"))) |
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*} |
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lemmas [extraction_expand] = |
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71118807d303
Removed (now unneeded) declaration of realizers for induction on natural numbers.
berghofe
parents:
13452
diff
changeset
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atomize_eq atomize_all atomize_imp |
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allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2 |
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notE' impE' impE iffE imp_cong simp_thms |
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induct_forall_eq induct_implies_eq induct_equal_eq |
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induct_forall_def induct_implies_def |
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induct_atomize induct_rulify1 induct_rulify2 |
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datatype sumbool = Left | Right |
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subsection {* Type of extracted program *} |
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extract_type |
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"typeof (Trueprop P) \<equiv> typeof P" |
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"typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow> |
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typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('Q))" |
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"typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE(Null))" |
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"typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow> |
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typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('P \<Rightarrow> 'Q))" |
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"(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow> |
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typeof (\<forall>x. P x) \<equiv> Type (TYPE(Null))" |
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"(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow> |
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typeof (\<forall>x::'a. P x) \<equiv> Type (TYPE('a \<Rightarrow> 'P))" |
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"(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow> |
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typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a))" |
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"(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow> |
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typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a \<times> 'P))" |
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"typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> |
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typeof (P \<or> Q) \<equiv> Type (TYPE(sumbool))" |
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"typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow> |
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typeof (P \<or> Q) \<equiv> Type (TYPE('Q option))" |
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"typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> |
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typeof (P \<or> Q) \<equiv> Type (TYPE('P option))" |
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"typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow> |
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typeof (P \<or> Q) \<equiv> Type (TYPE('P + 'Q))" |
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"typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow> |
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typeof (P \<and> Q) \<equiv> Type (TYPE('Q))" |
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"typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> |
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typeof (P \<and> Q) \<equiv> Type (TYPE('P))" |
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"typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow> |
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typeof (P \<and> Q) \<equiv> Type (TYPE('P \<times> 'Q))" |
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"typeof (P = Q) \<equiv> typeof ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))" |
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"typeof (x \<in> P) \<equiv> typeof P" |
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subsection {* Realizability *} |
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realizability |
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"(realizes t (Trueprop P)) \<equiv> (Trueprop (realizes t P))" |
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"(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<longrightarrow> Q)) \<equiv> (realizes Null P \<longrightarrow> realizes t Q)" |
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"(typeof P) \<equiv> (Type (TYPE('P))) \<Longrightarrow> |
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(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x::'P. realizes x P \<longrightarrow> realizes Null Q)" |
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"(realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x. realizes x P \<longrightarrow> realizes (t x) Q)" |
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"(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes Null (P x))" |
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"(realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes (t x) (P x))" |
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"(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (\<exists>x. P x)) \<equiv> (realizes Null (P t))" |
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"(realizes t (\<exists>x. P x)) \<equiv> (realizes (snd t) (P (fst t)))" |
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"(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<or> Q)) \<equiv> |
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(case t of Left \<Rightarrow> realizes Null P | Right \<Rightarrow> realizes Null Q)" |
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"(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<or> Q)) \<equiv> |
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(case t of None \<Rightarrow> realizes Null P | Some q \<Rightarrow> realizes q Q)" |
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"(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<or> Q)) \<equiv> |
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(case t of None \<Rightarrow> realizes Null Q | Some p \<Rightarrow> realizes p P)" |
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"(realizes t (P \<or> Q)) \<equiv> |
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(case t of Inl p \<Rightarrow> realizes p P | Inr q \<Rightarrow> realizes q Q)" |
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"(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<and> Q)) \<equiv> (realizes Null P \<and> realizes t Q)" |
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"(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow> |
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(realizes t (P \<and> Q)) \<equiv> (realizes t P \<and> realizes Null Q)" |
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"(realizes t (P \<and> Q)) \<equiv> (realizes (fst t) P \<and> realizes (snd t) Q)" |
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"typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> |
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realizes t (\<not> P) \<equiv> \<not> realizes Null P" |
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"typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> |
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realizes t (\<not> P) \<equiv> (\<forall>x::'P. \<not> realizes x P)" |
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"typeof (P::bool) \<equiv> Type (TYPE(Null)) \<Longrightarrow> |
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typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> |
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realizes t (P = Q) \<equiv> realizes Null P = realizes Null Q" |
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"(realizes t (P = Q)) \<equiv> (realizes t ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)))" |
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subsection {* Computational content of basic inference rules *} |
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theorem disjE_realizer: |
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assumes r: "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q" |
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and r1: "\<And>p. P p \<Longrightarrow> R (f p)" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)" |
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shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)" |
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proof (cases x) |
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case Inl |
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with r show ?thesis by simp (rule r1) |
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next |
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case Inr |
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with r show ?thesis by simp (rule r2) |
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qed |
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theorem disjE_realizer2: |
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assumes r: "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q" |
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and r1: "P \<Longrightarrow> R f" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)" |
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shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)" |
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proof (cases x) |
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case None |
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with r show ?thesis by simp (rule r1) |
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next |
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case Some |
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with r show ?thesis by simp (rule r2) |
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qed |
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theorem disjE_realizer3: |
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assumes r: "case x of Left \<Rightarrow> P | Right \<Rightarrow> Q" |
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and r1: "P \<Longrightarrow> R f" and r2: "Q \<Longrightarrow> R g" |
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shows "R (case x of Left \<Rightarrow> f | Right \<Rightarrow> g)" |
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proof (cases x) |
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case Left |
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with r show ?thesis by simp (rule r1) |
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next |
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case Right |
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with r show ?thesis by simp (rule r2) |
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qed |
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theorem conjI_realizer: |
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"P p \<Longrightarrow> Q q \<Longrightarrow> P (fst (p, q)) \<and> Q (snd (p, q))" |
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by simp |
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theorem exI_realizer: |
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"P x y \<Longrightarrow> P (fst (x, y)) (snd (x, y))" by simp |
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realizers |
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impI (P, Q): "\<lambda>P Q pq. pq" |
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"\<Lambda>P Q pq (h: _). allI \<cdot> _ \<bullet> (\<Lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))" |
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impI (P): "Null" |
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"\<Lambda>P Q (h: _). allI \<cdot> _ \<bullet> (\<Lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))" |
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impI (Q): "\<lambda>P Q q. q" "\<Lambda>P Q q. impI \<cdot> _ \<cdot> _" |
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impI: "Null" "\<Lambda>P Q. impI \<cdot> _ \<cdot> _" |
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mp (P, Q): "\<lambda>P Q pq. pq" |
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"\<Lambda>P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)" |
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mp (P): "Null" |
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"\<Lambda>P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)" |
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mp (Q): "\<lambda>P Q q. q" "\<Lambda>P Q q. mp \<cdot> _ \<cdot> _" |
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mp: "Null" "\<Lambda>P Q. mp \<cdot> _ \<cdot> _" |
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allI (P): "\<lambda>P p. p" "\<Lambda>P p. allI \<cdot> _" |
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allI: "Null" "\<Lambda>P. allI \<cdot> _" |
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spec (P): "\<lambda>P x p. p x" "\<Lambda>P x p. spec \<cdot> _ \<cdot> x" |
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spec: "Null" "\<Lambda>P x. spec \<cdot> _ \<cdot> x" |
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exI (P): "\<lambda>P x p. (x, p)" "\<Lambda>P. exI_realizer \<cdot> _" |
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exI: "\<lambda>P x. x" "\<Lambda>P x (h: _). h" |
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exE (P, Q): "\<lambda>P Q p pq. pq (fst p) (snd p)" |
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"\<Lambda>P Q p (h1: _) pq (h2: _). h2 \<cdot> (fst p) \<cdot> (snd p) \<bullet> h1" |
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exE (P): "Null" |
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"\<Lambda>P Q p (h1: _) (h2: _). h2 \<cdot> (fst p) \<cdot> (snd p) \<bullet> h1" |
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exE (Q): "\<lambda>P Q x pq. pq x" |
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"\<Lambda>P Q x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1" |
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exE: "Null" |
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"\<Lambda>P Q x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1" |
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conjI (P, Q): "\<lambda>P Q p q. (p, q)" |
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"\<Lambda>P Q p (h: _) q. conjI_realizer \<cdot> |
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(\<lambda>p. realizes p P) \<cdot> p \<cdot> (\<lambda>q. realizes q Q) \<cdot> q \<bullet> h" |
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conjI (P): "\<lambda>P Q p. p" |
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"\<Lambda>P Q p. conjI \<cdot> _ \<cdot> _" |
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conjI (Q): "\<lambda>P Q q. q" |
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"\<Lambda>P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h" |
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conjI: "Null" |
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"\<Lambda>P Q. conjI \<cdot> _ \<cdot> _" |
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conjunct1 (P, Q): "\<lambda>P Q. fst" |
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"\<Lambda>P Q pq. conjunct1 \<cdot> _ \<cdot> _" |
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conjunct1 (P): "\<lambda>P Q p. p" |
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"\<Lambda>P Q p. conjunct1 \<cdot> _ \<cdot> _" |
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conjunct1 (Q): "Null" |
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"\<Lambda>P Q q. conjunct1 \<cdot> _ \<cdot> _" |
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conjunct1: "Null" |
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"\<Lambda>P Q. conjunct1 \<cdot> _ \<cdot> _" |
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conjunct2 (P, Q): "\<lambda>P Q. snd" |
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"\<Lambda>P Q pq. conjunct2 \<cdot> _ \<cdot> _" |
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conjunct2 (P): "Null" |
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"\<Lambda>P Q p. conjunct2 \<cdot> _ \<cdot> _" |
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conjunct2 (Q): "\<lambda>P Q p. p" |
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"\<Lambda>P Q p. conjunct2 \<cdot> _ \<cdot> _" |
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conjunct2: "Null" |
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"\<Lambda>P Q. conjunct2 \<cdot> _ \<cdot> _" |
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disjI1 (P, Q): "\<lambda>P Q. Inl" |
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"\<Lambda>P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_1 \<cdot> (\<lambda>p. realizes p P) \<cdot> _ \<cdot> p)" |
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disjI1 (P): "\<lambda>P Q. Some" |
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"\<Lambda>P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> (\<lambda>p. realizes p P) \<cdot> p)" |
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disjI1 (Q): "\<lambda>P Q. None" |
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"\<Lambda>P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)" |
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disjI1: "\<lambda>P Q. Left" |
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"\<Lambda>P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_1 \<cdot> _ \<cdot> _)" |
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disjI2 (P, Q): "\<lambda>Q P. Inr" |
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"\<Lambda>Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_2 \<cdot> _ \<cdot> (\<lambda>q. realizes q Q) \<cdot> q)" |
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disjI2 (P): "\<lambda>Q P. None" |
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"\<Lambda>Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)" |
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disjI2 (Q): "\<lambda>Q P. Some" |
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"\<Lambda>Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> (\<lambda>q. realizes q Q) \<cdot> q)" |
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disjI2: "\<lambda>Q P. Right" |
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"\<Lambda>Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_2 \<cdot> _ \<cdot> _)" |
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disjE (P, Q, R): "\<lambda>P Q R pq pr qr. |
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(case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)" |
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"\<Lambda>P Q R pq (h1: _) pr (h2: _) qr. |
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disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes r R) \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2" |
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disjE (Q, R): "\<lambda>P Q R pq pr qr. |
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(case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)" |
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"\<Lambda>P Q R pq (h1: _) pr (h2: _) qr. |
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disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes r R) \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2" |
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disjE (P, R): "\<lambda>P Q R pq pr qr. |
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(case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)" |
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"\<Lambda>P Q R pq (h1: _) pr (h2: _) qr (h3: _). |
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disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes r R) \<cdot> qr \<cdot> pr \<bullet> h1 \<bullet> h3 \<bullet> h2" |
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disjE (R): "\<lambda>P Q R pq pr qr. |
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(case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)" |
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"\<Lambda>P Q R pq (h1: _) pr (h2: _) qr. |
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disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes r R) \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2" |
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disjE (P, Q): "Null" |
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"\<Lambda>P Q R pq. disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes Null R) \<cdot> _ \<cdot> _" |
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disjE (Q): "Null" |
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"\<Lambda>P Q R pq. disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes Null R) \<cdot> _ \<cdot> _" |
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disjE (P): "Null" |
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"\<Lambda>P Q R pq (h1: _) (h2: _) (h3: _). |
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disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes Null R) \<cdot> _ \<cdot> _ \<bullet> h1 \<bullet> h3 \<bullet> h2" |
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disjE: "Null" |
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"\<Lambda>P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>r. realizes Null R) \<cdot> _ \<cdot> _" |
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FalseE (P): "\<lambda>P. arbitrary" |
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"\<Lambda>P. FalseE \<cdot> _" |
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FalseE: "Null" |
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"\<Lambda>P. FalseE \<cdot> _" |
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notI (P): "Null" |
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"\<Lambda>P (h: _). allI \<cdot> _ \<bullet> (\<Lambda>x. notI \<cdot> _ \<bullet> (h \<cdot> x))" |
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notI: "Null" |
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"\<Lambda>P. notI \<cdot> _" |
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notE (P, R): "\<lambda>P R p. arbitrary" |
|
343 |
"\<Lambda>P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)" |
|
344 |
||
345 |
notE (P): "Null" |
|
346 |
"\<Lambda>P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)" |
|
347 |
||
348 |
notE (R): "\<lambda>P R. arbitrary" |
|
349 |
"\<Lambda>P R. notE \<cdot> _ \<cdot> _" |
|
350 |
||
351 |
notE: "Null" |
|
352 |
"\<Lambda>P R. notE \<cdot> _ \<cdot> _" |
|
353 |
||
354 |
subst (P): "\<lambda>s t P ps. ps" |
|
355 |
"\<Lambda>s t P (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> (\<lambda>x. realizes ps (P x)) \<bullet> h" |
|
356 |
||
357 |
subst: "Null" |
|
358 |
"\<Lambda>s t P. subst \<cdot> s \<cdot> t \<cdot> (\<lambda>x. realizes Null (P x))" |
|
359 |
||
360 |
iffD1 (P, Q): "\<lambda>Q P. fst" |
|
361 |
"\<Lambda>Q P pq (h: _) p. |
|
362 |
mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))" |
|
363 |
||
364 |
iffD1 (P): "\<lambda>Q P p. p" |
|
365 |
"\<Lambda>Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)" |
|
366 |
||
367 |
iffD1 (Q): "Null" |
|
368 |
"\<Lambda>Q P q1 (h: _) q2. |
|
369 |
mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))" |
|
370 |
||
371 |
iffD1: "Null" |
|
372 |
"\<Lambda>Q P. iffD1 \<cdot> _ \<cdot> _" |
|
373 |
||
374 |
iffD2 (P, Q): "\<lambda>P Q. snd" |
|
375 |
"\<Lambda>P Q pq (h: _) q. |
|
376 |
mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))" |
|
377 |
||
378 |
iffD2 (P): "\<lambda>P Q p. p" |
|
379 |
"\<Lambda>P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)" |
|
380 |
||
381 |
iffD2 (Q): "Null" |
|
382 |
"\<Lambda>P Q q1 (h: _) q2. |
|
383 |
mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))" |
|
384 |
||
385 |
iffD2: "Null" |
|
386 |
"\<Lambda>P Q. iffD2 \<cdot> _ \<cdot> _" |
|
387 |
||
388 |
iffI (P, Q): "\<lambda>P Q pq qp. (pq, qp)" |
|
389 |
"\<Lambda>P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot> |
|
390 |
(\<lambda>pq. \<forall>x. realizes x P \<longrightarrow> realizes (pq x) Q) \<cdot> pq \<cdot> |
|
391 |
(\<lambda>qp. \<forall>x. realizes x Q \<longrightarrow> realizes (qp x) P) \<cdot> qp \<bullet> |
|
392 |
(allI \<cdot> _ \<bullet> (\<Lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet> |
|
393 |
(allI \<cdot> _ \<bullet> (\<Lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))" |
|
394 |
||
395 |
iffI (P): "\<lambda>P Q p. p" |
|
396 |
"\<Lambda>P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet> |
|
397 |
(allI \<cdot> _ \<bullet> (\<Lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet> |
|
398 |
(impI \<cdot> _ \<cdot> _ \<bullet> h2)" |
|
399 |
||
400 |
iffI (Q): "\<lambda>P Q q. q" |
|
401 |
"\<Lambda>P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet> |
|
402 |
(impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet> |
|
403 |
(allI \<cdot> _ \<bullet> (\<Lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))" |
|
404 |
||
405 |
iffI: "Null" |
|
406 |
"\<Lambda>P Q. iffI \<cdot> _ \<cdot> _" |
|
407 |
||
408 |
classical: "Null" |
|
409 |
"\<Lambda>P. classical \<cdot> _" |
|
410 |
||
411 |
end |