src/HOL/Extraction.thy
author blanchet
Fri Feb 21 00:09:56 2014 +0100 (2014-02-21)
changeset 55642 63beb38e9258
parent 55404 5cb95b79a51f
child 58112 8081087096ad
permissions -rw-r--r--
adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
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(*  Title:      HOL/Extraction.thy
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    Author:     Stefan Berghofer, TU Muenchen
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*)
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header {* Program extraction for HOL *}
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theory Extraction
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imports Datatype Option
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begin
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ML_file "Tools/rewrite_hol_proof.ML"
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subsection {* Setup *}
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setup {*
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  Extraction.add_types
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      [("bool", ([], NONE))] #>
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  Extraction.set_preprocessor (fn thy =>
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      Proofterm.rewrite_proof_notypes
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        ([], RewriteHOLProof.elim_cong :: ProofRewriteRules.rprocs true) o
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      Proofterm.rewrite_proof thy
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        (RewriteHOLProof.rews,
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         ProofRewriteRules.rprocs true @ [ProofRewriteRules.expand_of_class thy]) o
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      ProofRewriteRules.elim_vars (curry Const @{const_name default}))
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*}
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lemmas [extraction_expand] =
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  meta_spec atomize_eq atomize_all atomize_imp atomize_conj
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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 eq_True eq_False
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  induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
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  induct_atomize induct_atomize' induct_rulify induct_rulify'
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  induct_rulify_fallback induct_trueI
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  True_implies_equals TrueE
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lemmas [extraction_expand_def] =
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  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
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  induct_true_def induct_false_def
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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 y x \<Longrightarrow> P (snd (x, y)) (fst (x, y))" by simp
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theorem exE_realizer: "P (snd p) (fst p) \<Longrightarrow>
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  (\<And>x y. P y x \<Longrightarrow> Q (f x y)) \<Longrightarrow> Q (let (x, y) = p in f x y)"
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  by (cases p) (simp add: Let_def)
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theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
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  (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
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realizers
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  impI (P, Q): "\<lambda>pq. pq"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
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  impI (P): "Null"
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    "\<^bold>\<lambda>(c: _) P Q (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
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  impI (Q): "\<lambda>q. q" "\<^bold>\<lambda>(c: _) P Q q. impI \<cdot> _ \<cdot> _"
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  impI: "Null" "impI"
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  mp (P, Q): "\<lambda>pq. pq"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
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  mp (P): "Null"
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    "\<^bold>\<lambda>(c: _) P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
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  mp (Q): "\<lambda>q. q" "\<^bold>\<lambda>(c: _) P Q q. mp \<cdot> _ \<cdot> _"
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  mp: "Null" "mp"
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  allI (P): "\<lambda>p. p" "\<^bold>\<lambda>(c: _) P (d: _) p. allI \<cdot> _ \<bullet> d"
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  allI: "Null" "allI"
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  spec (P): "\<lambda>x p. p x" "\<^bold>\<lambda>(c: _) P x (d: _) p. spec \<cdot> _ \<cdot> x \<bullet> d"
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  spec: "Null" "spec"
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  exI (P): "\<lambda>x p. (x, p)" "\<^bold>\<lambda>(c: _) P x (d: _) p. exI_realizer \<cdot> P \<cdot> p \<cdot> x \<bullet> c \<bullet> d"
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  exI: "\<lambda>x. x" "\<^bold>\<lambda>P x (c: _) (h: _). h"
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  exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q (e: _) p (h: _) pq. exE_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> pq \<bullet> c \<bullet> e \<bullet> d \<bullet> h"
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  exE (P): "Null"
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    "\<^bold>\<lambda>(c: _) P Q (d: _) p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _ \<bullet> c \<bullet> d"
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  exE (Q): "\<lambda>x pq. pq x"
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    "\<^bold>\<lambda>(c: _) P Q (d: _) x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
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  exE: "Null"
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    "\<^bold>\<lambda>P Q (c: _) x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
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  conjI (P, Q): "Pair"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q p (h: _) q. conjI_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> q \<bullet> c \<bullet> d \<bullet> h"
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  conjI (P): "\<lambda>p. p"
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    "\<^bold>\<lambda>(c: _) P Q p. conjI \<cdot> _ \<cdot> _"
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  conjI (Q): "\<lambda>q. q"
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    "\<^bold>\<lambda>(c: _) P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
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  conjI: "Null" "conjI"
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  conjunct1 (P, Q): "fst"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q pq. conjunct1 \<cdot> _ \<cdot> _"
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  conjunct1 (P): "\<lambda>p. p"
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    "\<^bold>\<lambda>(c: _) P Q p. conjunct1 \<cdot> _ \<cdot> _"
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  conjunct1 (Q): "Null"
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    "\<^bold>\<lambda>(c: _) P Q q. conjunct1 \<cdot> _ \<cdot> _"
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  conjunct1: "Null" "conjunct1"
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  conjunct2 (P, Q): "snd"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q pq. conjunct2 \<cdot> _ \<cdot> _"
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  conjunct2 (P): "Null"
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    "\<^bold>\<lambda>(c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
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  conjunct2 (Q): "\<lambda>p. p"
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    "\<^bold>\<lambda>(c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
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  conjunct2: "Null" "conjunct2"
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  disjI1 (P, Q): "Inl"
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    "\<^bold>\<lambda>(c: _) (d: _) P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.case_1 \<cdot> P \<cdot> _ \<cdot> p \<bullet> arity_type_bool \<bullet> c \<bullet> d)"
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  disjI1 (P): "Some"
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   286
    "\<^bold>\<lambda>(c: _) P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_2 \<cdot> _ \<cdot> P \<cdot> p \<bullet> arity_type_bool \<bullet> c)"
berghofe@13403
   287
berghofe@13725
   288
  disjI1 (Q): "None"
blanchet@55642
   289
    "\<^bold>\<lambda>(c: _) P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
berghofe@13403
   290
berghofe@13725
   291
  disjI1: "Left"
blanchet@55642
   292
    "\<^bold>\<lambda>P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.case_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
berghofe@13403
   293
berghofe@13725
   294
  disjI2 (P, Q): "Inr"
blanchet@55642
   295
    "\<^bold>\<lambda>(d: _) (c: _) Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.case_2 \<cdot> _ \<cdot> Q \<cdot> q \<bullet> arity_type_bool \<bullet> c \<bullet> d)"
berghofe@13403
   296
berghofe@13725
   297
  disjI2 (P): "None"
blanchet@55642
   298
    "\<^bold>\<lambda>(c: _) Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
berghofe@13403
   299
berghofe@13725
   300
  disjI2 (Q): "Some"
blanchet@55642
   301
    "\<^bold>\<lambda>(c: _) Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_2 \<cdot> _ \<cdot> Q \<cdot> q \<bullet> arity_type_bool \<bullet> c)"
berghofe@13403
   302
berghofe@13725
   303
  disjI2: "Right"
blanchet@55642
   304
    "\<^bold>\<lambda>Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.case_2 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
berghofe@13403
   305
berghofe@13725
   306
  disjE (P, Q, R): "\<lambda>pq pr qr.
berghofe@13403
   307
     (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
wenzelm@52486
   308
    "\<^bold>\<lambda>(c: _) (d: _) (e: _) P Q R pq (h1: _) pr (h2: _) qr.
berghofe@37233
   309
       disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> e \<bullet> h1 \<bullet> h2"
berghofe@13403
   310
berghofe@13725
   311
  disjE (Q, R): "\<lambda>pq pr qr.
berghofe@13403
   312
     (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
wenzelm@52486
   313
    "\<^bold>\<lambda>(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr.
berghofe@37233
   314
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h2"
berghofe@13403
   315
berghofe@13725
   316
  disjE (P, R): "\<lambda>pq pr qr.
berghofe@13403
   317
     (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
wenzelm@52486
   318
    "\<^bold>\<lambda>(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr (h3: _).
berghofe@37233
   319
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h3 \<bullet> h2"
berghofe@13403
   320
berghofe@13725
   321
  disjE (R): "\<lambda>pq pr qr.
berghofe@13403
   322
     (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
wenzelm@52486
   323
    "\<^bold>\<lambda>(c: _) P Q R pq (h1: _) pr (h2: _) qr.
berghofe@37233
   324
       disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> h1 \<bullet> h2"
berghofe@13403
   325
berghofe@13403
   326
  disjE (P, Q): "Null"
wenzelm@52486
   327
    "\<^bold>\<lambda>(c: _) (d: _) P Q R pq. disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> d \<bullet> arity_type_bool"
berghofe@13403
   328
berghofe@13403
   329
  disjE (Q): "Null"
wenzelm@52486
   330
    "\<^bold>\<lambda>(c: _) P Q R pq. disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> arity_type_bool"
berghofe@13403
   331
berghofe@13403
   332
  disjE (P): "Null"
wenzelm@52486
   333
    "\<^bold>\<lambda>(c: _) P Q R pq (h1: _) (h2: _) (h3: _).
berghofe@37233
   334
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> arity_type_bool \<bullet> h1 \<bullet> h3 \<bullet> h2"
berghofe@13403
   335
berghofe@13403
   336
  disjE: "Null"
wenzelm@52486
   337
    "\<^bold>\<lambda>P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> arity_type_bool"
berghofe@13403
   338
haftmann@27982
   339
  FalseE (P): "default"
wenzelm@52486
   340
    "\<^bold>\<lambda>(c: _) P. FalseE \<cdot> _"
berghofe@13403
   341
berghofe@13725
   342
  FalseE: "Null" "FalseE"
berghofe@13403
   343
berghofe@13403
   344
  notI (P): "Null"
wenzelm@52486
   345
    "\<^bold>\<lambda>(c: _) P (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
berghofe@13403
   346
berghofe@13725
   347
  notI: "Null" "notI"
berghofe@13403
   348
haftmann@27982
   349
  notE (P, R): "\<lambda>p. default"
wenzelm@52486
   350
    "\<^bold>\<lambda>(c: _) (d: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
berghofe@13403
   351
berghofe@13403
   352
  notE (P): "Null"
wenzelm@52486
   353
    "\<^bold>\<lambda>(c: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
berghofe@13403
   354
haftmann@27982
   355
  notE (R): "default"
wenzelm@52486
   356
    "\<^bold>\<lambda>(c: _) P R. notE \<cdot> _ \<cdot> _"
berghofe@13403
   357
berghofe@13725
   358
  notE: "Null" "notE"
berghofe@13403
   359
berghofe@13725
   360
  subst (P): "\<lambda>s t ps. ps"
wenzelm@52486
   361
    "\<^bold>\<lambda>(c: _) s t P (d: _) (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> d \<bullet> h"
berghofe@13403
   362
berghofe@13725
   363
  subst: "Null" "subst"
berghofe@13725
   364
berghofe@13725
   365
  iffD1 (P, Q): "fst"
wenzelm@52486
   366
    "\<^bold>\<lambda>(d: _) (c: _) Q P pq (h: _) p.
berghofe@37233
   367
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> d \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   368
berghofe@13725
   369
  iffD1 (P): "\<lambda>p. p"
wenzelm@52486
   370
    "\<^bold>\<lambda>(c: _) Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
berghofe@13403
   371
berghofe@13403
   372
  iffD1 (Q): "Null"
wenzelm@52486
   373
    "\<^bold>\<lambda>(c: _) Q P q1 (h: _) q2.
berghofe@37233
   374
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   375
berghofe@13725
   376
  iffD1: "Null" "iffD1"
berghofe@13403
   377
berghofe@13725
   378
  iffD2 (P, Q): "snd"
wenzelm@52486
   379
    "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h: _) q.
berghofe@37233
   380
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> d \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   381
berghofe@13725
   382
  iffD2 (P): "\<lambda>p. p"
wenzelm@52486
   383
    "\<^bold>\<lambda>(c: _) P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
berghofe@13403
   384
berghofe@13403
   385
  iffD2 (Q): "Null"
wenzelm@52486
   386
    "\<^bold>\<lambda>(c: _) P Q q1 (h: _) q2.
berghofe@37233
   387
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   388
berghofe@13725
   389
  iffD2: "Null" "iffD2"
berghofe@13403
   390
berghofe@13725
   391
  iffI (P, Q): "Pair"
wenzelm@52486
   392
    "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
berghofe@13725
   393
       (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
berghofe@13725
   394
       (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
berghofe@37233
   395
       (arity_type_fun \<bullet> c \<bullet> d) \<bullet>
berghofe@37233
   396
       (arity_type_fun \<bullet> d \<bullet> c) \<bullet>
wenzelm@52486
   397
       (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
wenzelm@52486
   398
       (allI \<cdot> _ \<bullet> d \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
berghofe@13403
   399
berghofe@13725
   400
  iffI (P): "\<lambda>p. p"
wenzelm@52486
   401
    "\<^bold>\<lambda>(c: _) P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
wenzelm@52486
   402
       (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
berghofe@13403
   403
       (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
berghofe@13403
   404
berghofe@13725
   405
  iffI (Q): "\<lambda>q. q"
wenzelm@52486
   406
    "\<^bold>\<lambda>(c: _) P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
berghofe@13403
   407
       (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
wenzelm@52486
   408
       (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
berghofe@13403
   409
berghofe@13725
   410
  iffI: "Null" "iffI"
berghofe@13403
   411
berghofe@13403
   412
end