src/HOL/Extraction.thy
author wenzelm
Fri Mar 28 19:43:54 2008 +0100 (2008-03-28)
changeset 26462 dac4e2bce00d
parent 25424 170f4cc34697
child 27982 2aaa4a5569a6
permissions -rw-r--r--
avoid rebinding of existing facts;
     1 (*  Title:      HOL/Extraction.thy
     2     ID:         $Id$
     3     Author:     Stefan Berghofer, TU Muenchen
     4 *)
     5 
     6 header {* Program extraction for HOL *}
     7 
     8 theory Extraction
     9 imports Datatype
    10 uses "Tools/rewrite_hol_proof.ML"
    11 begin
    12 
    13 subsection {* Setup *}
    14 
    15 setup {*
    16 let
    17 fun realizes_set_proc (Const ("realizes", Type ("fun", [Type ("Null", []), _])) $ r $
    18       (Const ("op :", _) $ x $ S)) = (case strip_comb S of
    19         (Var (ixn, U), ts) => SOME (list_comb (Var (ixn, binder_types U @
    20            [HOLogic.dest_setT (body_type U)] ---> HOLogic.boolT), ts @ [x]))
    21       | (Free (s, U), ts) => SOME (list_comb (Free (s, binder_types U @
    22            [HOLogic.dest_setT (body_type U)] ---> HOLogic.boolT), ts @ [x]))
    23       | _ => NONE)
    24   | realizes_set_proc (Const ("realizes", Type ("fun", [T, _])) $ r $
    25       (Const ("op :", _) $ x $ S)) = (case strip_comb S of
    26         (Var (ixn, U), ts) => SOME (list_comb (Var (ixn, T :: binder_types U @
    27            [HOLogic.dest_setT (body_type U)] ---> HOLogic.boolT), r :: ts @ [x]))
    28       | (Free (s, U), ts) => SOME (list_comb (Free (s, T :: binder_types U @
    29            [HOLogic.dest_setT (body_type U)] ---> HOLogic.boolT), r :: ts @ [x]))
    30       | _ => NONE)
    31   | realizes_set_proc _ = NONE;
    32 
    33 fun mk_realizes_set r rT s (setT as Type ("set", [elT])) =
    34   Abs ("x", elT, Const ("realizes", rT --> HOLogic.boolT --> HOLogic.boolT) $
    35     incr_boundvars 1 r $ (Const ("op :", elT --> setT --> HOLogic.boolT) $
    36       Bound 0 $ incr_boundvars 1 s));
    37 in
    38   Extraction.add_types
    39       [("bool", ([], NONE)),
    40        ("set", ([realizes_set_proc], SOME mk_realizes_set))] #>
    41   Extraction.set_preprocessor (fn thy =>
    42       Proofterm.rewrite_proof_notypes
    43         ([], ("HOL/elim_cong", RewriteHOLProof.elim_cong) ::
    44           ProofRewriteRules.rprocs true) o
    45       Proofterm.rewrite_proof thy
    46         (RewriteHOLProof.rews, ProofRewriteRules.rprocs true) o
    47       ProofRewriteRules.elim_vars (curry Const "arbitrary"))
    48 end
    49 *}
    50 
    51 lemmas [extraction_expand] =
    52   meta_spec atomize_eq atomize_all atomize_imp atomize_conj
    53   allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2
    54   notE' impE' impE iffE imp_cong simp_thms eq_True eq_False
    55   induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
    56   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
    57   induct_atomize induct_rulify induct_rulify_fallback
    58   True_implies_equals TrueE
    59 
    60 datatype sumbool = Left | Right
    61 
    62 subsection {* Type of extracted program *}
    63 
    64 extract_type
    65   "typeof (Trueprop P) \<equiv> typeof P"
    66 
    67   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    68      typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('Q))"
    69 
    70   "typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE(Null))"
    71 
    72   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    73      typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('P \<Rightarrow> 'Q))"
    74 
    75   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
    76      typeof (\<forall>x. P x) \<equiv> Type (TYPE(Null))"
    77 
    78   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
    79      typeof (\<forall>x::'a. P x) \<equiv> Type (TYPE('a \<Rightarrow> 'P))"
    80 
    81   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
    82      typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a))"
    83 
    84   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
    85      typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a \<times> 'P))"
    86 
    87   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    88      typeof (P \<or> Q) \<equiv> Type (TYPE(sumbool))"
    89 
    90   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    91      typeof (P \<or> Q) \<equiv> Type (TYPE('Q option))"
    92 
    93   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    94      typeof (P \<or> Q) \<equiv> Type (TYPE('P option))"
    95 
    96   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    97      typeof (P \<or> Q) \<equiv> Type (TYPE('P + 'Q))"
    98 
    99   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
   100      typeof (P \<and> Q) \<equiv> Type (TYPE('Q))"
   101 
   102   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   103      typeof (P \<and> Q) \<equiv> Type (TYPE('P))"
   104 
   105   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
   106      typeof (P \<and> Q) \<equiv> Type (TYPE('P \<times> 'Q))"
   107 
   108   "typeof (P = Q) \<equiv> typeof ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))"
   109 
   110   "typeof (x \<in> P) \<equiv> typeof P"
   111 
   112 subsection {* Realizability *}
   113 
   114 realizability
   115   "(realizes t (Trueprop P)) \<equiv> (Trueprop (realizes t P))"
   116 
   117   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   118      (realizes t (P \<longrightarrow> Q)) \<equiv> (realizes Null P \<longrightarrow> realizes t Q)"
   119 
   120   "(typeof P) \<equiv> (Type (TYPE('P))) \<Longrightarrow>
   121    (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   122      (realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x::'P. realizes x P \<longrightarrow> realizes Null Q)"
   123 
   124   "(realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x. realizes x P \<longrightarrow> realizes (t x) Q)"
   125 
   126   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
   127      (realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes Null (P x))"
   128 
   129   "(realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes (t x) (P x))"
   130 
   131   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
   132      (realizes t (\<exists>x. P x)) \<equiv> (realizes Null (P t))"
   133 
   134   "(realizes t (\<exists>x. P x)) \<equiv> (realizes (snd t) (P (fst t)))"
   135 
   136   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   137    (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   138      (realizes t (P \<or> Q)) \<equiv>
   139      (case t of Left \<Rightarrow> realizes Null P | Right \<Rightarrow> realizes Null Q)"
   140 
   141   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   142      (realizes t (P \<or> Q)) \<equiv>
   143      (case t of None \<Rightarrow> realizes Null P | Some q \<Rightarrow> realizes q Q)"
   144 
   145   "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   146      (realizes t (P \<or> Q)) \<equiv>
   147      (case t of None \<Rightarrow> realizes Null Q | Some p \<Rightarrow> realizes p P)"
   148 
   149   "(realizes t (P \<or> Q)) \<equiv>
   150    (case t of Inl p \<Rightarrow> realizes p P | Inr q \<Rightarrow> realizes q Q)"
   151 
   152   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   153      (realizes t (P \<and> Q)) \<equiv> (realizes Null P \<and> realizes t Q)"
   154 
   155   "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   156      (realizes t (P \<and> Q)) \<equiv> (realizes t P \<and> realizes Null Q)"
   157 
   158   "(realizes t (P \<and> Q)) \<equiv> (realizes (fst t) P \<and> realizes (snd t) Q)"
   159 
   160   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   161      realizes t (\<not> P) \<equiv> \<not> realizes Null P"
   162 
   163   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow>
   164      realizes t (\<not> P) \<equiv> (\<forall>x::'P. \<not> realizes x P)"
   165 
   166   "typeof (P::bool) \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   167    typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   168      realizes t (P = Q) \<equiv> realizes Null P = realizes Null Q"
   169 
   170   "(realizes t (P = Q)) \<equiv> (realizes t ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)))"
   171 
   172 subsection {* Computational content of basic inference rules *}
   173 
   174 theorem disjE_realizer:
   175   assumes r: "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q"
   176   and r1: "\<And>p. P p \<Longrightarrow> R (f p)" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
   177   shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)"
   178 proof (cases x)
   179   case Inl
   180   with r show ?thesis by simp (rule r1)
   181 next
   182   case Inr
   183   with r show ?thesis by simp (rule r2)
   184 qed
   185 
   186 theorem disjE_realizer2:
   187   assumes r: "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q"
   188   and r1: "P \<Longrightarrow> R f" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
   189   shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)"
   190 proof (cases x)
   191   case None
   192   with r show ?thesis by simp (rule r1)
   193 next
   194   case Some
   195   with r show ?thesis by simp (rule r2)
   196 qed
   197 
   198 theorem disjE_realizer3:
   199   assumes r: "case x of Left \<Rightarrow> P | Right \<Rightarrow> Q"
   200   and r1: "P \<Longrightarrow> R f" and r2: "Q \<Longrightarrow> R g"
   201   shows "R (case x of Left \<Rightarrow> f | Right \<Rightarrow> g)"
   202 proof (cases x)
   203   case Left
   204   with r show ?thesis by simp (rule r1)
   205 next
   206   case Right
   207   with r show ?thesis by simp (rule r2)
   208 qed
   209 
   210 theorem conjI_realizer:
   211   "P p \<Longrightarrow> Q q \<Longrightarrow> P (fst (p, q)) \<and> Q (snd (p, q))"
   212   by simp
   213 
   214 theorem exI_realizer:
   215   "P y x \<Longrightarrow> P (snd (x, y)) (fst (x, y))" by simp
   216 
   217 theorem exE_realizer: "P (snd p) (fst p) \<Longrightarrow>
   218   (\<And>x y. P y x \<Longrightarrow> Q (f x y)) \<Longrightarrow> Q (let (x, y) = p in f x y)"
   219   by (cases p) (simp add: Let_def)
   220 
   221 theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
   222   (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
   223 
   224 realizers
   225   impI (P, Q): "\<lambda>pq. pq"
   226     "\<Lambda> P Q pq (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
   227 
   228   impI (P): "Null"
   229     "\<Lambda> P Q (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
   230 
   231   impI (Q): "\<lambda>q. q" "\<Lambda> P Q q. impI \<cdot> _ \<cdot> _"
   232 
   233   impI: "Null" "impI"
   234 
   235   mp (P, Q): "\<lambda>pq. pq"
   236     "\<Lambda> P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
   237 
   238   mp (P): "Null"
   239     "\<Lambda> P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
   240 
   241   mp (Q): "\<lambda>q. q" "\<Lambda> P Q q. mp \<cdot> _ \<cdot> _"
   242 
   243   mp: "Null" "mp"
   244 
   245   allI (P): "\<lambda>p. p" "\<Lambda> P p. allI \<cdot> _"
   246 
   247   allI: "Null" "allI"
   248 
   249   spec (P): "\<lambda>x p. p x" "\<Lambda> P x p. spec \<cdot> _ \<cdot> x"
   250 
   251   spec: "Null" "spec"
   252 
   253   exI (P): "\<lambda>x p. (x, p)" "\<Lambda> P x p. exI_realizer \<cdot> P \<cdot> p \<cdot> x"
   254 
   255   exI: "\<lambda>x. x" "\<Lambda> P x (h: _). h"
   256 
   257   exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
   258     "\<Lambda> P Q p (h: _) pq. exE_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> pq \<bullet> h"
   259 
   260   exE (P): "Null"
   261     "\<Lambda> P Q p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _"
   262 
   263   exE (Q): "\<lambda>x pq. pq x"
   264     "\<Lambda> P Q x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
   265 
   266   exE: "Null"
   267     "\<Lambda> P Q x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
   268 
   269   conjI (P, Q): "Pair"
   270     "\<Lambda> P Q p (h: _) q. conjI_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> q \<bullet> h"
   271 
   272   conjI (P): "\<lambda>p. p"
   273     "\<Lambda> P Q p. conjI \<cdot> _ \<cdot> _"
   274 
   275   conjI (Q): "\<lambda>q. q"
   276     "\<Lambda> P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
   277 
   278   conjI: "Null" "conjI"
   279 
   280   conjunct1 (P, Q): "fst"
   281     "\<Lambda> P Q pq. conjunct1 \<cdot> _ \<cdot> _"
   282 
   283   conjunct1 (P): "\<lambda>p. p"
   284     "\<Lambda> P Q p. conjunct1 \<cdot> _ \<cdot> _"
   285 
   286   conjunct1 (Q): "Null"
   287     "\<Lambda> P Q q. conjunct1 \<cdot> _ \<cdot> _"
   288 
   289   conjunct1: "Null" "conjunct1"
   290 
   291   conjunct2 (P, Q): "snd"
   292     "\<Lambda> P Q pq. conjunct2 \<cdot> _ \<cdot> _"
   293 
   294   conjunct2 (P): "Null"
   295     "\<Lambda> P Q p. conjunct2 \<cdot> _ \<cdot> _"
   296 
   297   conjunct2 (Q): "\<lambda>p. p"
   298     "\<Lambda> P Q p. conjunct2 \<cdot> _ \<cdot> _"
   299 
   300   conjunct2: "Null" "conjunct2"
   301 
   302   disjI1 (P, Q): "Inl"
   303     "\<Lambda> P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_1 \<cdot> P \<cdot> _ \<cdot> p)"
   304 
   305   disjI1 (P): "Some"
   306     "\<Lambda> P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> P \<cdot> p)"
   307 
   308   disjI1 (Q): "None"
   309     "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)"
   310 
   311   disjI1: "Left"
   312     "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_1 \<cdot> _ \<cdot> _)"
   313 
   314   disjI2 (P, Q): "Inr"
   315     "\<Lambda> Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_2 \<cdot> _ \<cdot> Q \<cdot> q)"
   316 
   317   disjI2 (P): "None"
   318     "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)"
   319 
   320   disjI2 (Q): "Some"
   321     "\<Lambda> Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> Q \<cdot> q)"
   322 
   323   disjI2: "Right"
   324     "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_2 \<cdot> _ \<cdot> _)"
   325 
   326   disjE (P, Q, R): "\<lambda>pq pr qr.
   327      (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
   328     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
   329        disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
   330 
   331   disjE (Q, R): "\<lambda>pq pr qr.
   332      (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
   333     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
   334        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
   335 
   336   disjE (P, R): "\<lambda>pq pr qr.
   337      (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
   338     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr (h3: _).
   339        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> h1 \<bullet> h3 \<bullet> h2"
   340 
   341   disjE (R): "\<lambda>pq pr qr.
   342      (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
   343     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
   344        disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
   345 
   346   disjE (P, Q): "Null"
   347     "\<Lambda> P Q R pq. disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
   348 
   349   disjE (Q): "Null"
   350     "\<Lambda> P Q R pq. disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
   351 
   352   disjE (P): "Null"
   353     "\<Lambda> P Q R pq (h1: _) (h2: _) (h3: _).
   354        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> h1 \<bullet> h3 \<bullet> h2"
   355 
   356   disjE: "Null"
   357     "\<Lambda> P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
   358 
   359   FalseE (P): "arbitrary"
   360     "\<Lambda> P. FalseE \<cdot> _"
   361 
   362   FalseE: "Null" "FalseE"
   363 
   364   notI (P): "Null"
   365     "\<Lambda> P (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
   366 
   367   notI: "Null" "notI"
   368 
   369   notE (P, R): "\<lambda>p. arbitrary"
   370     "\<Lambda> P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
   371 
   372   notE (P): "Null"
   373     "\<Lambda> P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
   374 
   375   notE (R): "arbitrary"
   376     "\<Lambda> P R. notE \<cdot> _ \<cdot> _"
   377 
   378   notE: "Null" "notE"
   379 
   380   subst (P): "\<lambda>s t ps. ps"
   381     "\<Lambda> s t P (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> h"
   382 
   383   subst: "Null" "subst"
   384 
   385   iffD1 (P, Q): "fst"
   386     "\<Lambda> Q P pq (h: _) p.
   387        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
   388 
   389   iffD1 (P): "\<lambda>p. p"
   390     "\<Lambda> Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
   391 
   392   iffD1 (Q): "Null"
   393     "\<Lambda> Q P q1 (h: _) q2.
   394        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
   395 
   396   iffD1: "Null" "iffD1"
   397 
   398   iffD2 (P, Q): "snd"
   399     "\<Lambda> P Q pq (h: _) q.
   400        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
   401 
   402   iffD2 (P): "\<lambda>p. p"
   403     "\<Lambda> P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
   404 
   405   iffD2 (Q): "Null"
   406     "\<Lambda> P Q q1 (h: _) q2.
   407        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
   408 
   409   iffD2: "Null" "iffD2"
   410 
   411   iffI (P, Q): "Pair"
   412     "\<Lambda> P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
   413        (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
   414        (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
   415        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
   416        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
   417 
   418   iffI (P): "\<lambda>p. p"
   419     "\<Lambda> P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
   420        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
   421        (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
   422 
   423   iffI (Q): "\<lambda>q. q"
   424     "\<Lambda> P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
   425        (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
   426        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
   427 
   428   iffI: "Null" "iffI"
   429 
   430 (*
   431   classical: "Null"
   432     "\<Lambda> P. classical \<cdot> _"
   433 *)
   434 
   435 end