src/HOL/Extraction.thy
author hoelzl
Thu Sep 02 10:14:32 2010 +0200 (2010-09-02)
changeset 39072 1030b1a166ef
parent 37233 b78f31ca4675
child 48891 c0eafbd55de3
permissions -rw-r--r--
Add lessThan_Suc_eq_insert_0
     1 (*  Title:      HOL/Extraction.thy
     2     Author:     Stefan Berghofer, TU Muenchen
     3 *)
     4 
     5 header {* Program extraction for HOL *}
     6 
     7 theory Extraction
     8 imports Option
     9 uses "Tools/rewrite_hol_proof.ML"
    10 begin
    11 
    12 subsection {* Setup *}
    13 
    14 setup {*
    15   Extraction.add_types
    16       [("bool", ([], NONE))] #>
    17   Extraction.set_preprocessor (fn thy =>
    18       Proofterm.rewrite_proof_notypes
    19         ([], RewriteHOLProof.elim_cong :: ProofRewriteRules.rprocs true) o
    20       Proofterm.rewrite_proof thy
    21         (RewriteHOLProof.rews,
    22          ProofRewriteRules.rprocs true @ [ProofRewriteRules.expand_of_class thy]) o
    23       ProofRewriteRules.elim_vars (curry Const @{const_name default}))
    24 *}
    25 
    26 lemmas [extraction_expand] =
    27   meta_spec atomize_eq atomize_all atomize_imp atomize_conj
    28   allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2
    29   notE' impE' impE iffE imp_cong simp_thms eq_True eq_False
    30   induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
    31   induct_atomize induct_atomize' induct_rulify induct_rulify'
    32   induct_rulify_fallback induct_trueI
    33   True_implies_equals TrueE
    34 
    35 lemmas [extraction_expand_def] =
    36   induct_forall_def induct_implies_def induct_equal_def induct_conj_def
    37   induct_true_def induct_false_def
    38 
    39 datatype sumbool = Left | Right
    40 
    41 subsection {* Type of extracted program *}
    42 
    43 extract_type
    44   "typeof (Trueprop P) \<equiv> typeof P"
    45 
    46   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    47      typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('Q))"
    48 
    49   "typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE(Null))"
    50 
    51   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    52      typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('P \<Rightarrow> 'Q))"
    53 
    54   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
    55      typeof (\<forall>x. P x) \<equiv> Type (TYPE(Null))"
    56 
    57   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
    58      typeof (\<forall>x::'a. P x) \<equiv> Type (TYPE('a \<Rightarrow> 'P))"
    59 
    60   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
    61      typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a))"
    62 
    63   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
    64      typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a \<times> 'P))"
    65 
    66   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    67      typeof (P \<or> Q) \<equiv> Type (TYPE(sumbool))"
    68 
    69   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    70      typeof (P \<or> Q) \<equiv> Type (TYPE('Q option))"
    71 
    72   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    73      typeof (P \<or> Q) \<equiv> Type (TYPE('P option))"
    74 
    75   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    76      typeof (P \<or> Q) \<equiv> Type (TYPE('P + 'Q))"
    77 
    78   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    79      typeof (P \<and> Q) \<equiv> Type (TYPE('Q))"
    80 
    81   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    82      typeof (P \<and> Q) \<equiv> Type (TYPE('P))"
    83 
    84   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    85      typeof (P \<and> Q) \<equiv> Type (TYPE('P \<times> 'Q))"
    86 
    87   "typeof (P = Q) \<equiv> typeof ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))"
    88 
    89   "typeof (x \<in> P) \<equiv> typeof P"
    90 
    91 subsection {* Realizability *}
    92 
    93 realizability
    94   "(realizes t (Trueprop P)) \<equiv> (Trueprop (realizes t P))"
    95 
    96   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
    97      (realizes t (P \<longrightarrow> Q)) \<equiv> (realizes Null P \<longrightarrow> realizes t Q)"
    98 
    99   "(typeof P) \<equiv> (Type (TYPE('P))) \<Longrightarrow>
   100    (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   101      (realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x::'P. realizes x P \<longrightarrow> realizes Null Q)"
   102 
   103   "(realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x. realizes x P \<longrightarrow> realizes (t x) Q)"
   104 
   105   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
   106      (realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes Null (P x))"
   107 
   108   "(realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes (t x) (P x))"
   109 
   110   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
   111      (realizes t (\<exists>x. P x)) \<equiv> (realizes Null (P t))"
   112 
   113   "(realizes t (\<exists>x. P x)) \<equiv> (realizes (snd t) (P (fst t)))"
   114 
   115   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   116    (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   117      (realizes t (P \<or> Q)) \<equiv>
   118      (case t of Left \<Rightarrow> realizes Null P | Right \<Rightarrow> realizes Null Q)"
   119 
   120   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   121      (realizes t (P \<or> Q)) \<equiv>
   122      (case t of None \<Rightarrow> realizes Null P | Some q \<Rightarrow> realizes q Q)"
   123 
   124   "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   125      (realizes t (P \<or> Q)) \<equiv>
   126      (case t of None \<Rightarrow> realizes Null Q | Some p \<Rightarrow> realizes p P)"
   127 
   128   "(realizes t (P \<or> Q)) \<equiv>
   129    (case t of Inl p \<Rightarrow> realizes p P | Inr q \<Rightarrow> realizes q Q)"
   130 
   131   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   132      (realizes t (P \<and> Q)) \<equiv> (realizes Null P \<and> realizes t Q)"
   133 
   134   "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   135      (realizes t (P \<and> Q)) \<equiv> (realizes t P \<and> realizes Null Q)"
   136 
   137   "(realizes t (P \<and> Q)) \<equiv> (realizes (fst t) P \<and> realizes (snd t) Q)"
   138 
   139   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   140      realizes t (\<not> P) \<equiv> \<not> realizes Null P"
   141 
   142   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow>
   143      realizes t (\<not> P) \<equiv> (\<forall>x::'P. \<not> realizes x P)"
   144 
   145   "typeof (P::bool) \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   146    typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   147      realizes t (P = Q) \<equiv> realizes Null P = realizes Null Q"
   148 
   149   "(realizes t (P = Q)) \<equiv> (realizes t ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)))"
   150 
   151 subsection {* Computational content of basic inference rules *}
   152 
   153 theorem disjE_realizer:
   154   assumes r: "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q"
   155   and r1: "\<And>p. P p \<Longrightarrow> R (f p)" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
   156   shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)"
   157 proof (cases x)
   158   case Inl
   159   with r show ?thesis by simp (rule r1)
   160 next
   161   case Inr
   162   with r show ?thesis by simp (rule r2)
   163 qed
   164 
   165 theorem disjE_realizer2:
   166   assumes r: "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q"
   167   and r1: "P \<Longrightarrow> R f" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
   168   shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)"
   169 proof (cases x)
   170   case None
   171   with r show ?thesis by simp (rule r1)
   172 next
   173   case Some
   174   with r show ?thesis by simp (rule r2)
   175 qed
   176 
   177 theorem disjE_realizer3:
   178   assumes r: "case x of Left \<Rightarrow> P | Right \<Rightarrow> Q"
   179   and r1: "P \<Longrightarrow> R f" and r2: "Q \<Longrightarrow> R g"
   180   shows "R (case x of Left \<Rightarrow> f | Right \<Rightarrow> g)"
   181 proof (cases x)
   182   case Left
   183   with r show ?thesis by simp (rule r1)
   184 next
   185   case Right
   186   with r show ?thesis by simp (rule r2)
   187 qed
   188 
   189 theorem conjI_realizer:
   190   "P p \<Longrightarrow> Q q \<Longrightarrow> P (fst (p, q)) \<and> Q (snd (p, q))"
   191   by simp
   192 
   193 theorem exI_realizer:
   194   "P y x \<Longrightarrow> P (snd (x, y)) (fst (x, y))" by simp
   195 
   196 theorem exE_realizer: "P (snd p) (fst p) \<Longrightarrow>
   197   (\<And>x y. P y x \<Longrightarrow> Q (f x y)) \<Longrightarrow> Q (let (x, y) = p in f x y)"
   198   by (cases p) (simp add: Let_def)
   199 
   200 theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
   201   (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
   202 
   203 realizers
   204   impI (P, Q): "\<lambda>pq. pq"
   205     "\<Lambda> (c: _) (d: _) P Q pq (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
   206 
   207   impI (P): "Null"
   208     "\<Lambda> (c: _) P Q (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
   209 
   210   impI (Q): "\<lambda>q. q" "\<Lambda> (c: _) P Q q. impI \<cdot> _ \<cdot> _"
   211 
   212   impI: "Null" "impI"
   213 
   214   mp (P, Q): "\<lambda>pq. pq"
   215     "\<Lambda> (c: _) (d: _) P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   216 
   217   mp (P): "Null"
   218     "\<Lambda> (c: _) P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   219 
   220   mp (Q): "\<lambda>q. q" "\<Lambda> (c: _) P Q q. mp \<cdot> _ \<cdot> _"
   221 
   222   mp: "Null" "mp"
   223 
   224   allI (P): "\<lambda>p. p" "\<Lambda> (c: _) P (d: _) p. allI \<cdot> _ \<bullet> d"
   225 
   226   allI: "Null" "allI"
   227 
   228   spec (P): "\<lambda>x p. p x" "\<Lambda> (c: _) P x (d: _) p. spec \<cdot> _ \<cdot> x \<bullet> d"
   229 
   230   spec: "Null" "spec"
   231 
   232   exI (P): "\<lambda>x p. (x, p)" "\<Lambda> (c: _) P x (d: _) p. exI_realizer \<cdot> P \<cdot> p \<cdot> x \<bullet> c \<bullet> d"
   233 
   234   exI: "\<lambda>x. x" "\<Lambda> P x (c: _) (h: _). h"
   235 
   236   exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
   237     "\<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"
   238 
   239   exE (P): "Null"
   240     "\<Lambda> (c: _) P Q (d: _) p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _ \<bullet> c \<bullet> d"
   241 
   242   exE (Q): "\<lambda>x pq. pq x"
   243     "\<Lambda> (c: _) P Q (d: _) x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
   244 
   245   exE: "Null"
   246     "\<Lambda> P Q (c: _) x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
   247 
   248   conjI (P, Q): "Pair"
   249     "\<Lambda> (c: _) (d: _) P Q p (h: _) q. conjI_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> q \<bullet> c \<bullet> d \<bullet> h"
   250 
   251   conjI (P): "\<lambda>p. p"
   252     "\<Lambda> (c: _) P Q p. conjI \<cdot> _ \<cdot> _"
   253 
   254   conjI (Q): "\<lambda>q. q"
   255     "\<Lambda> (c: _) P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
   256 
   257   conjI: "Null" "conjI"
   258 
   259   conjunct1 (P, Q): "fst"
   260     "\<Lambda> (c: _) (d: _) P Q pq. conjunct1 \<cdot> _ \<cdot> _"
   261 
   262   conjunct1 (P): "\<lambda>p. p"
   263     "\<Lambda> (c: _) P Q p. conjunct1 \<cdot> _ \<cdot> _"
   264 
   265   conjunct1 (Q): "Null"
   266     "\<Lambda> (c: _) P Q q. conjunct1 \<cdot> _ \<cdot> _"
   267 
   268   conjunct1: "Null" "conjunct1"
   269 
   270   conjunct2 (P, Q): "snd"
   271     "\<Lambda> (c: _) (d: _) P Q pq. conjunct2 \<cdot> _ \<cdot> _"
   272 
   273   conjunct2 (P): "Null"
   274     "\<Lambda> (c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
   275 
   276   conjunct2 (Q): "\<lambda>p. p"
   277     "\<Lambda> (c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
   278 
   279   conjunct2: "Null" "conjunct2"
   280 
   281   disjI1 (P, Q): "Inl"
   282     "\<Lambda> (c: _) (d: _) P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_1 \<cdot> P \<cdot> _ \<cdot> p \<bullet> arity_type_bool \<bullet> c \<bullet> d)"
   283 
   284   disjI1 (P): "Some"
   285     "\<Lambda> (c: _) P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> P \<cdot> p \<bullet> arity_type_bool \<bullet> c)"
   286 
   287   disjI1 (Q): "None"
   288     "\<Lambda> (c: _) P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
   289 
   290   disjI1: "Left"
   291     "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
   292 
   293   disjI2 (P, Q): "Inr"
   294     "\<Lambda> (d: _) (c: _) Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_2 \<cdot> _ \<cdot> Q \<cdot> q \<bullet> arity_type_bool \<bullet> c \<bullet> d)"
   295 
   296   disjI2 (P): "None"
   297     "\<Lambda> (c: _) Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
   298 
   299   disjI2 (Q): "Some"
   300     "\<Lambda> (c: _) Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> Q \<cdot> q \<bullet> arity_type_bool \<bullet> c)"
   301 
   302   disjI2: "Right"
   303     "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_2 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
   304 
   305   disjE (P, Q, R): "\<lambda>pq pr qr.
   306      (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
   307     "\<Lambda> (c: _) (d: _) (e: _) P Q R pq (h1: _) pr (h2: _) qr.
   308        disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> e \<bullet> h1 \<bullet> h2"
   309 
   310   disjE (Q, R): "\<lambda>pq pr qr.
   311      (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
   312     "\<Lambda> (c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr.
   313        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h2"
   314 
   315   disjE (P, R): "\<lambda>pq pr qr.
   316      (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
   317     "\<Lambda> (c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr (h3: _).
   318        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h3 \<bullet> h2"
   319 
   320   disjE (R): "\<lambda>pq pr qr.
   321      (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
   322     "\<Lambda> (c: _) P Q R pq (h1: _) pr (h2: _) qr.
   323        disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> h1 \<bullet> h2"
   324 
   325   disjE (P, Q): "Null"
   326     "\<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"
   327 
   328   disjE (Q): "Null"
   329     "\<Lambda> (c: _) P Q R pq. disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> arity_type_bool"
   330 
   331   disjE (P): "Null"
   332     "\<Lambda> (c: _) P Q R pq (h1: _) (h2: _) (h3: _).
   333        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> arity_type_bool \<bullet> h1 \<bullet> h3 \<bullet> h2"
   334 
   335   disjE: "Null"
   336     "\<Lambda> P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> arity_type_bool"
   337 
   338   FalseE (P): "default"
   339     "\<Lambda> (c: _) P. FalseE \<cdot> _"
   340 
   341   FalseE: "Null" "FalseE"
   342 
   343   notI (P): "Null"
   344     "\<Lambda> (c: _) P (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
   345 
   346   notI: "Null" "notI"
   347 
   348   notE (P, R): "\<lambda>p. default"
   349     "\<Lambda> (c: _) (d: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   350 
   351   notE (P): "Null"
   352     "\<Lambda> (c: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   353 
   354   notE (R): "default"
   355     "\<Lambda> (c: _) P R. notE \<cdot> _ \<cdot> _"
   356 
   357   notE: "Null" "notE"
   358 
   359   subst (P): "\<lambda>s t ps. ps"
   360     "\<Lambda> (c: _) s t P (d: _) (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> d \<bullet> h"
   361 
   362   subst: "Null" "subst"
   363 
   364   iffD1 (P, Q): "fst"
   365     "\<Lambda> (d: _) (c: _) Q P pq (h: _) p.
   366        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> d \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
   367 
   368   iffD1 (P): "\<lambda>p. p"
   369     "\<Lambda> (c: _) Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
   370 
   371   iffD1 (Q): "Null"
   372     "\<Lambda> (c: _) Q P q1 (h: _) q2.
   373        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
   374 
   375   iffD1: "Null" "iffD1"
   376 
   377   iffD2 (P, Q): "snd"
   378     "\<Lambda> (c: _) (d: _) P Q pq (h: _) q.
   379        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> d \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
   380 
   381   iffD2 (P): "\<lambda>p. p"
   382     "\<Lambda> (c: _) P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
   383 
   384   iffD2 (Q): "Null"
   385     "\<Lambda> (c: _) P Q q1 (h: _) q2.
   386        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
   387 
   388   iffD2: "Null" "iffD2"
   389 
   390   iffI (P, Q): "Pair"
   391     "\<Lambda> (c: _) (d: _) P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
   392        (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
   393        (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
   394        (arity_type_fun \<bullet> c \<bullet> d) \<bullet>
   395        (arity_type_fun \<bullet> d \<bullet> c) \<bullet>
   396        (allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
   397        (allI \<cdot> _ \<bullet> d \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
   398 
   399   iffI (P): "\<lambda>p. p"
   400     "\<Lambda> (c: _) P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
   401        (allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
   402        (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
   403 
   404   iffI (Q): "\<lambda>q. q"
   405     "\<Lambda> (c: _) P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
   406        (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
   407        (allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
   408 
   409   iffI: "Null" "iffI"
   410 
   411 end