src/HOL/Extraction.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 62922 96691631c1eb
child 69593 3dda49e08b9d
permissions -rw-r--r--
executable domain membership checks
     1 (*  Title:      HOL/Extraction.thy
     2     Author:     Stefan Berghofer, TU Muenchen
     3 *)
     4 
     5 section \<open>Program extraction for HOL\<close>
     6 
     7 theory Extraction
     8 imports Option
     9 begin
    10 
    11 ML_file "Tools/rewrite_hol_proof.ML"
    12 
    13 subsection \<open>Setup\<close>
    14 
    15 setup \<open>
    16   Extraction.add_types
    17       [("bool", ([], NONE))] #>
    18   Extraction.set_preprocessor (fn thy =>
    19     let val ctxt = Proof_Context.init_global thy in
    20       Proofterm.rewrite_proof_notypes
    21         ([], RewriteHOLProof.elim_cong :: ProofRewriteRules.rprocs true) o
    22       Proofterm.rewrite_proof thy
    23         (RewriteHOLProof.rews,
    24          ProofRewriteRules.rprocs true @ [ProofRewriteRules.expand_of_class ctxt]) o
    25       ProofRewriteRules.elim_vars (curry Const @{const_name default})
    26     end)
    27 \<close>
    28 
    29 lemmas [extraction_expand] =
    30   meta_spec atomize_eq atomize_all atomize_imp atomize_conj
    31   allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2
    32   notE' impE' impE iffE imp_cong simp_thms eq_True eq_False
    33   induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
    34   induct_atomize induct_atomize' induct_rulify induct_rulify'
    35   induct_rulify_fallback induct_trueI
    36   True_implies_equals implies_True_equals TrueE
    37   False_implies_equals implies_False_swap
    38 
    39 lemmas [extraction_expand_def] =
    40   HOL.induct_forall_def HOL.induct_implies_def HOL.induct_equal_def HOL.induct_conj_def
    41   HOL.induct_true_def HOL.induct_false_def
    42 
    43 datatype (plugins only: code extraction) sumbool = Left | Right
    44 
    45 subsection \<open>Type of extracted program\<close>
    46 
    47 extract_type
    48   "typeof (Trueprop P) \<equiv> typeof P"
    49 
    50   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    51      typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('Q))"
    52 
    53   "typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE(Null))"
    54 
    55   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    56      typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('P \<Rightarrow> 'Q))"
    57 
    58   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
    59      typeof (\<forall>x. P x) \<equiv> Type (TYPE(Null))"
    60 
    61   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
    62      typeof (\<forall>x::'a. P x) \<equiv> Type (TYPE('a \<Rightarrow> 'P))"
    63 
    64   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
    65      typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a))"
    66 
    67   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
    68      typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a \<times> 'P))"
    69 
    70   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    71      typeof (P \<or> Q) \<equiv> Type (TYPE(sumbool))"
    72 
    73   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    74      typeof (P \<or> Q) \<equiv> Type (TYPE('Q option))"
    75 
    76   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    77      typeof (P \<or> Q) \<equiv> Type (TYPE('P option))"
    78 
    79   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    80      typeof (P \<or> Q) \<equiv> Type (TYPE('P + 'Q))"
    81 
    82   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    83      typeof (P \<and> Q) \<equiv> Type (TYPE('Q))"
    84 
    85   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
    86      typeof (P \<and> Q) \<equiv> Type (TYPE('P))"
    87 
    88   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
    89      typeof (P \<and> Q) \<equiv> Type (TYPE('P \<times> 'Q))"
    90 
    91   "typeof (P = Q) \<equiv> typeof ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))"
    92 
    93   "typeof (x \<in> P) \<equiv> typeof P"
    94 
    95 subsection \<open>Realizability\<close>
    96 
    97 realizability
    98   "(realizes t (Trueprop P)) \<equiv> (Trueprop (realizes t P))"
    99 
   100   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   101      (realizes t (P \<longrightarrow> Q)) \<equiv> (realizes Null P \<longrightarrow> realizes t Q)"
   102 
   103   "(typeof P) \<equiv> (Type (TYPE('P))) \<Longrightarrow>
   104    (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   105      (realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x::'P. realizes x P \<longrightarrow> realizes Null Q)"
   106 
   107   "(realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x. realizes x P \<longrightarrow> realizes (t x) Q)"
   108 
   109   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
   110      (realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes Null (P x))"
   111 
   112   "(realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes (t x) (P x))"
   113 
   114   "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
   115      (realizes t (\<exists>x. P x)) \<equiv> (realizes Null (P t))"
   116 
   117   "(realizes t (\<exists>x. P x)) \<equiv> (realizes (snd t) (P (fst t)))"
   118 
   119   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   120    (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   121      (realizes t (P \<or> Q)) \<equiv>
   122      (case t of Left \<Rightarrow> realizes Null P | Right \<Rightarrow> realizes Null Q)"
   123 
   124   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   125      (realizes t (P \<or> Q)) \<equiv>
   126      (case t of None \<Rightarrow> realizes Null P | Some q \<Rightarrow> realizes q Q)"
   127 
   128   "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   129      (realizes t (P \<or> Q)) \<equiv>
   130      (case t of None \<Rightarrow> realizes Null Q | Some p \<Rightarrow> realizes p P)"
   131 
   132   "(realizes t (P \<or> Q)) \<equiv>
   133    (case t of Inl p \<Rightarrow> realizes p P | Inr q \<Rightarrow> realizes q Q)"
   134 
   135   "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   136      (realizes t (P \<and> Q)) \<equiv> (realizes Null P \<and> realizes t Q)"
   137 
   138   "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
   139      (realizes t (P \<and> Q)) \<equiv> (realizes t P \<and> realizes Null Q)"
   140 
   141   "(realizes t (P \<and> Q)) \<equiv> (realizes (fst t) P \<and> realizes (snd t) Q)"
   142 
   143   "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   144      realizes t (\<not> P) \<equiv> \<not> realizes Null P"
   145 
   146   "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow>
   147      realizes t (\<not> P) \<equiv> (\<forall>x::'P. \<not> realizes x P)"
   148 
   149   "typeof (P::bool) \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   150    typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
   151      realizes t (P = Q) \<equiv> realizes Null P = realizes Null Q"
   152 
   153   "(realizes t (P = Q)) \<equiv> (realizes t ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)))"
   154 
   155 subsection \<open>Computational content of basic inference rules\<close>
   156 
   157 theorem disjE_realizer:
   158   assumes r: "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q"
   159   and r1: "\<And>p. P p \<Longrightarrow> R (f p)" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
   160   shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)"
   161 proof (cases x)
   162   case Inl
   163   with r show ?thesis by simp (rule r1)
   164 next
   165   case Inr
   166   with r show ?thesis by simp (rule r2)
   167 qed
   168 
   169 theorem disjE_realizer2:
   170   assumes r: "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q"
   171   and r1: "P \<Longrightarrow> R f" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
   172   shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)"
   173 proof (cases x)
   174   case None
   175   with r show ?thesis by simp (rule r1)
   176 next
   177   case Some
   178   with r show ?thesis by simp (rule r2)
   179 qed
   180 
   181 theorem disjE_realizer3:
   182   assumes r: "case x of Left \<Rightarrow> P | Right \<Rightarrow> Q"
   183   and r1: "P \<Longrightarrow> R f" and r2: "Q \<Longrightarrow> R g"
   184   shows "R (case x of Left \<Rightarrow> f | Right \<Rightarrow> g)"
   185 proof (cases x)
   186   case Left
   187   with r show ?thesis by simp (rule r1)
   188 next
   189   case Right
   190   with r show ?thesis by simp (rule r2)
   191 qed
   192 
   193 theorem conjI_realizer:
   194   "P p \<Longrightarrow> Q q \<Longrightarrow> P (fst (p, q)) \<and> Q (snd (p, q))"
   195   by simp
   196 
   197 theorem exI_realizer:
   198   "P y x \<Longrightarrow> P (snd (x, y)) (fst (x, y))" by simp
   199 
   200 theorem exE_realizer: "P (snd p) (fst p) \<Longrightarrow>
   201   (\<And>x y. P y x \<Longrightarrow> Q (f x y)) \<Longrightarrow> Q (let (x, y) = p in f x y)"
   202   by (cases p) (simp add: Let_def)
   203 
   204 theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
   205   (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
   206 
   207 realizers
   208   impI (P, Q): "\<lambda>pq. pq"
   209     "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
   210 
   211   impI (P): "Null"
   212     "\<^bold>\<lambda>(c: _) P Q (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
   213 
   214   impI (Q): "\<lambda>q. q" "\<^bold>\<lambda>(c: _) P Q q. impI \<cdot> _ \<cdot> _"
   215 
   216   impI: "Null" "impI"
   217 
   218   mp (P, Q): "\<lambda>pq. pq"
   219     "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   220 
   221   mp (P): "Null"
   222     "\<^bold>\<lambda>(c: _) P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   223 
   224   mp (Q): "\<lambda>q. q" "\<^bold>\<lambda>(c: _) P Q q. mp \<cdot> _ \<cdot> _"
   225 
   226   mp: "Null" "mp"
   227 
   228   allI (P): "\<lambda>p. p" "\<^bold>\<lambda>(c: _) P (d: _) p. allI \<cdot> _ \<bullet> d"
   229 
   230   allI: "Null" "allI"
   231 
   232   spec (P): "\<lambda>x p. p x" "\<^bold>\<lambda>(c: _) P x (d: _) p. spec \<cdot> _ \<cdot> x \<bullet> d"
   233 
   234   spec: "Null" "spec"
   235 
   236   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"
   237 
   238   exI: "\<lambda>x. x" "\<^bold>\<lambda>P x (c: _) (h: _). h"
   239 
   240   exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
   241     "\<^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"
   242 
   243   exE (P): "Null"
   244     "\<^bold>\<lambda>(c: _) P Q (d: _) p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _ \<bullet> c \<bullet> d"
   245 
   246   exE (Q): "\<lambda>x pq. pq x"
   247     "\<^bold>\<lambda>(c: _) P Q (d: _) x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
   248 
   249   exE: "Null"
   250     "\<^bold>\<lambda>P Q (c: _) x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
   251 
   252   conjI (P, Q): "Pair"
   253     "\<^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"
   254 
   255   conjI (P): "\<lambda>p. p"
   256     "\<^bold>\<lambda>(c: _) P Q p. conjI \<cdot> _ \<cdot> _"
   257 
   258   conjI (Q): "\<lambda>q. q"
   259     "\<^bold>\<lambda>(c: _) P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
   260 
   261   conjI: "Null" "conjI"
   262 
   263   conjunct1 (P, Q): "fst"
   264     "\<^bold>\<lambda>(c: _) (d: _) P Q pq. conjunct1 \<cdot> _ \<cdot> _"
   265 
   266   conjunct1 (P): "\<lambda>p. p"
   267     "\<^bold>\<lambda>(c: _) P Q p. conjunct1 \<cdot> _ \<cdot> _"
   268 
   269   conjunct1 (Q): "Null"
   270     "\<^bold>\<lambda>(c: _) P Q q. conjunct1 \<cdot> _ \<cdot> _"
   271 
   272   conjunct1: "Null" "conjunct1"
   273 
   274   conjunct2 (P, Q): "snd"
   275     "\<^bold>\<lambda>(c: _) (d: _) P Q pq. conjunct2 \<cdot> _ \<cdot> _"
   276 
   277   conjunct2 (P): "Null"
   278     "\<^bold>\<lambda>(c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
   279 
   280   conjunct2 (Q): "\<lambda>p. p"
   281     "\<^bold>\<lambda>(c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
   282 
   283   conjunct2: "Null" "conjunct2"
   284 
   285   disjI1 (P, Q): "Inl"
   286     "\<^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)"
   287 
   288   disjI1 (P): "Some"
   289     "\<^bold>\<lambda>(c: _) P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_2 \<cdot> _ \<cdot> P \<cdot> p \<bullet> arity_type_bool \<bullet> c)"
   290 
   291   disjI1 (Q): "None"
   292     "\<^bold>\<lambda>(c: _) P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
   293 
   294   disjI1: "Left"
   295     "\<^bold>\<lambda>P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.case_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
   296 
   297   disjI2 (P, Q): "Inr"
   298     "\<^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)"
   299 
   300   disjI2 (P): "None"
   301     "\<^bold>\<lambda>(c: _) Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
   302 
   303   disjI2 (Q): "Some"
   304     "\<^bold>\<lambda>(c: _) Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.case_2 \<cdot> _ \<cdot> Q \<cdot> q \<bullet> arity_type_bool \<bullet> c)"
   305 
   306   disjI2: "Right"
   307     "\<^bold>\<lambda>Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.case_2 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
   308 
   309   disjE (P, Q, R): "\<lambda>pq pr qr.
   310      (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
   311     "\<^bold>\<lambda>(c: _) (d: _) (e: _) P Q R pq (h1: _) pr (h2: _) qr.
   312        disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> e \<bullet> h1 \<bullet> h2"
   313 
   314   disjE (Q, R): "\<lambda>pq pr qr.
   315      (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
   316     "\<^bold>\<lambda>(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr.
   317        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h2"
   318 
   319   disjE (P, R): "\<lambda>pq pr qr.
   320      (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
   321     "\<^bold>\<lambda>(c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr (h3: _).
   322        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h3 \<bullet> h2"
   323 
   324   disjE (R): "\<lambda>pq pr qr.
   325      (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
   326     "\<^bold>\<lambda>(c: _) P Q R pq (h1: _) pr (h2: _) qr.
   327        disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> h1 \<bullet> h2"
   328 
   329   disjE (P, Q): "Null"
   330     "\<^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"
   331 
   332   disjE (Q): "Null"
   333     "\<^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"
   334 
   335   disjE (P): "Null"
   336     "\<^bold>\<lambda>(c: _) P Q R pq (h1: _) (h2: _) (h3: _).
   337        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> c \<bullet> arity_type_bool \<bullet> h1 \<bullet> h3 \<bullet> h2"
   338 
   339   disjE: "Null"
   340     "\<^bold>\<lambda>P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> arity_type_bool"
   341 
   342   FalseE (P): "default"
   343     "\<^bold>\<lambda>(c: _) P. FalseE \<cdot> _"
   344 
   345   FalseE: "Null" "FalseE"
   346 
   347   notI (P): "Null"
   348     "\<^bold>\<lambda>(c: _) P (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
   349 
   350   notI: "Null" "notI"
   351 
   352   notE (P, R): "\<lambda>p. default"
   353     "\<^bold>\<lambda>(c: _) (d: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   354 
   355   notE (P): "Null"
   356     "\<^bold>\<lambda>(c: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
   357 
   358   notE (R): "default"
   359     "\<^bold>\<lambda>(c: _) P R. notE \<cdot> _ \<cdot> _"
   360 
   361   notE: "Null" "notE"
   362 
   363   subst (P): "\<lambda>s t ps. ps"
   364     "\<^bold>\<lambda>(c: _) s t P (d: _) (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> d \<bullet> h"
   365 
   366   subst: "Null" "subst"
   367 
   368   iffD1 (P, Q): "fst"
   369     "\<^bold>\<lambda>(d: _) (c: _) Q P pq (h: _) p.
   370        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> d \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
   371 
   372   iffD1 (P): "\<lambda>p. p"
   373     "\<^bold>\<lambda>(c: _) Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
   374 
   375   iffD1 (Q): "Null"
   376     "\<^bold>\<lambda>(c: _) Q P q1 (h: _) q2.
   377        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
   378 
   379   iffD1: "Null" "iffD1"
   380 
   381   iffD2 (P, Q): "snd"
   382     "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h: _) q.
   383        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> d \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
   384 
   385   iffD2 (P): "\<lambda>p. p"
   386     "\<^bold>\<lambda>(c: _) P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
   387 
   388   iffD2 (Q): "Null"
   389     "\<^bold>\<lambda>(c: _) P Q q1 (h: _) q2.
   390        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
   391 
   392   iffD2: "Null" "iffD2"
   393 
   394   iffI (P, Q): "Pair"
   395     "\<^bold>\<lambda>(c: _) (d: _) P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
   396        (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
   397        (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
   398        (arity_type_fun \<bullet> c \<bullet> d) \<bullet>
   399        (arity_type_fun \<bullet> d \<bullet> c) \<bullet>
   400        (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
   401        (allI \<cdot> _ \<bullet> d \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
   402 
   403   iffI (P): "\<lambda>p. p"
   404     "\<^bold>\<lambda>(c: _) P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
   405        (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
   406        (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
   407 
   408   iffI (Q): "\<lambda>q. q"
   409     "\<^bold>\<lambda>(c: _) P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
   410        (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
   411        (allI \<cdot> _ \<bullet> c \<bullet> (\<^bold>\<lambda>x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
   412 
   413   iffI: "Null" "iffI"
   414 
   415 end