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src/HOL/Extraction.thy
changeset 37233 b78f31ca4675
parent 34913 d8cb720c9c53
child 48891 c0eafbd55de3
equal deleted inserted replaced
37232:c10fb22a5e0c 37233:b78f31ca4675 
    16       [("bool", ([], NONE))] #>
 
    16       [("bool", ([], NONE))] #>
 
    17   Extraction.set_preprocessor (fn thy =>
 
    17   Extraction.set_preprocessor (fn thy =>
 
    18       Proofterm.rewrite_proof_notypes
 
    18       Proofterm.rewrite_proof_notypes
 
    19         ([], RewriteHOLProof.elim_cong :: ProofRewriteRules.rprocs true) o
 
    19         ([], RewriteHOLProof.elim_cong :: ProofRewriteRules.rprocs true) o
 
    20       Proofterm.rewrite_proof thy
 
    20       Proofterm.rewrite_proof thy
 
    21         (RewriteHOLProof.rews, ProofRewriteRules.rprocs true) o
 
    21         (RewriteHOLProof.rews,
 
       
    22          ProofRewriteRules.rprocs true @ [ProofRewriteRules.expand_of_class thy]) o
 
    22       ProofRewriteRules.elim_vars (curry Const @{const_name default}))
 
    23       ProofRewriteRules.elim_vars (curry Const @{const_name default}))
 
    23 *}
 
    24 *}
 
    24 
    25 
    25 lemmas [extraction_expand] =
 
    26 lemmas [extraction_expand] =
 
    26   meta_spec atomize_eq atomize_all atomize_imp atomize_conj
 
    27   meta_spec atomize_eq atomize_all atomize_imp atomize_conj
 
   197   by (cases p) (simp add: Let_def)
 
   198   by (cases p) (simp add: Let_def)
 
   198 
   199 
   199 theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
 
   200 theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
 
   200   (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
 
   201   (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
 
   201 
   202 
   202 setup {*
 
       
   203   Sign.add_const_constraint (@{const_name "default"}, SOME @{typ "'a::type"})
 
       
   204 *}
 
       
   205 
       
   206 realizers
 
   203 realizers
 
   207   impI (P, Q): "\<lambda>pq. pq"
 
   204   impI (P, Q): "\<lambda>pq. pq"
 
   208     "\<Lambda> P Q pq (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
 
   205     "\<Lambda> (c: _) (d: _) P Q pq (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
 
   209 
   206 
   210   impI (P): "Null"
 
   207   impI (P): "Null"
 
   211     "\<Lambda> P Q (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
 
   208     "\<Lambda> (c: _) P Q (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
 
   212 
   209 
   213   impI (Q): "\<lambda>q. q" "\<Lambda> P Q q. impI \<cdot> _ \<cdot> _"
 
   210   impI (Q): "\<lambda>q. q" "\<Lambda> (c: _) P Q q. impI \<cdot> _ \<cdot> _"
 
   214 
   211 
   215   impI: "Null" "impI"
 
   212   impI: "Null" "impI"
 
   216 
   213 
   217   mp (P, Q): "\<lambda>pq. pq"
 
   214   mp (P, Q): "\<lambda>pq. pq"
 
   218     "\<Lambda> P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
 
   215     "\<Lambda> (c: _) (d: _) P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
 
   219 
   216 
   220   mp (P): "Null"
 
   217   mp (P): "Null"
 
   221     "\<Lambda> P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
 
   218     "\<Lambda> (c: _) P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
 
   222 
   219 
   223   mp (Q): "\<lambda>q. q" "\<Lambda> P Q q. mp \<cdot> _ \<cdot> _"
 
   220   mp (Q): "\<lambda>q. q" "\<Lambda> (c: _) P Q q. mp \<cdot> _ \<cdot> _"
 
   224 
   221 
   225   mp: "Null" "mp"
 
   222   mp: "Null" "mp"
 
   226 
   223 
   227   allI (P): "\<lambda>p. p" "\<Lambda> P p. allI \<cdot> _"
 
   224   allI (P): "\<lambda>p. p" "\<Lambda> (c: _) P (d: _) p. allI \<cdot> _ \<bullet> d"
 
   228 
   225 
   229   allI: "Null" "allI"
 
   226   allI: "Null" "allI"
 
   230 
   227 
   231   spec (P): "\<lambda>x p. p x" "\<Lambda> P x p. spec \<cdot> _ \<cdot> x"
 
   228   spec (P): "\<lambda>x p. p x" "\<Lambda> (c: _) P x (d: _) p. spec \<cdot> _ \<cdot> x \<bullet> d"
 
   232 
   229 
   233   spec: "Null" "spec"
 
   230   spec: "Null" "spec"
 
   234 
   231 
   235   exI (P): "\<lambda>x p. (x, p)" "\<Lambda> P x p. exI_realizer \<cdot> P \<cdot> p \<cdot> x"
 
   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"
 
   236 
   233 
   237   exI: "\<lambda>x. x" "\<Lambda> P x (h: _). h"
 
   234   exI: "\<lambda>x. x" "\<Lambda> P x (c: _) (h: _). h"
 
   238 
   235 
   239   exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
 
   236   exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
 
   240     "\<Lambda> P Q p (h: _) pq. exE_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> pq \<bullet> h"
 
   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"
 
   241 
   238 
   242   exE (P): "Null"
 
   239   exE (P): "Null"
 
   243     "\<Lambda> P Q p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _"
 
   240     "\<Lambda> (c: _) P Q (d: _) p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _ \<bullet> c \<bullet> d"
 
   244 
   241 
   245   exE (Q): "\<lambda>x pq. pq x"
 
   242   exE (Q): "\<lambda>x pq. pq x"
 
   246     "\<Lambda> P Q x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
 
   243     "\<Lambda> (c: _) P Q (d: _) x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
 
   247 
   244 
   248   exE: "Null"
 
   245   exE: "Null"
 
   249     "\<Lambda> P Q x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
 
   246     "\<Lambda> P Q (c: _) x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
 
   250 
   247 
   251   conjI (P, Q): "Pair"
 
   248   conjI (P, Q): "Pair"
 
   252     "\<Lambda> P Q p (h: _) q. conjI_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> q \<bullet> h"
 
   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"
 
   253 
   250 
   254   conjI (P): "\<lambda>p. p"
 
   251   conjI (P): "\<lambda>p. p"
 
   255     "\<Lambda> P Q p. conjI \<cdot> _ \<cdot> _"
 
   252     "\<Lambda> (c: _) P Q p. conjI \<cdot> _ \<cdot> _"
 
   256 
   253 
   257   conjI (Q): "\<lambda>q. q"
 
   254   conjI (Q): "\<lambda>q. q"
 
   258     "\<Lambda> P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
 
   255     "\<Lambda> (c: _) P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
 
   259 
   256 
   260   conjI: "Null" "conjI"
 
   257   conjI: "Null" "conjI"
 
   261 
   258 
   262   conjunct1 (P, Q): "fst"
 
   259   conjunct1 (P, Q): "fst"
 
   263     "\<Lambda> P Q pq. conjunct1 \<cdot> _ \<cdot> _"
 
   260     "\<Lambda> (c: _) (d: _) P Q pq. conjunct1 \<cdot> _ \<cdot> _"
 
   264 
   261 
   265   conjunct1 (P): "\<lambda>p. p"
 
   262   conjunct1 (P): "\<lambda>p. p"
 
   266     "\<Lambda> P Q p. conjunct1 \<cdot> _ \<cdot> _"
 
   263     "\<Lambda> (c: _) P Q p. conjunct1 \<cdot> _ \<cdot> _"
 
   267 
   264 
   268   conjunct1 (Q): "Null"
 
   265   conjunct1 (Q): "Null"
 
   269     "\<Lambda> P Q q. conjunct1 \<cdot> _ \<cdot> _"
 
   266     "\<Lambda> (c: _) P Q q. conjunct1 \<cdot> _ \<cdot> _"
 
   270 
   267 
   271   conjunct1: "Null" "conjunct1"
 
   268   conjunct1: "Null" "conjunct1"
 
   272 
   269 
   273   conjunct2 (P, Q): "snd"
 
   270   conjunct2 (P, Q): "snd"
 
   274     "\<Lambda> P Q pq. conjunct2 \<cdot> _ \<cdot> _"
 
   271     "\<Lambda> (c: _) (d: _) P Q pq. conjunct2 \<cdot> _ \<cdot> _"
 
   275 
   272 
   276   conjunct2 (P): "Null"
 
   273   conjunct2 (P): "Null"
 
   277     "\<Lambda> P Q p. conjunct2 \<cdot> _ \<cdot> _"
 
   274     "\<Lambda> (c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
 
   278 
   275 
   279   conjunct2 (Q): "\<lambda>p. p"
 
   276   conjunct2 (Q): "\<lambda>p. p"
 
   280     "\<Lambda> P Q p. conjunct2 \<cdot> _ \<cdot> _"
 
   277     "\<Lambda> (c: _) P Q p. conjunct2 \<cdot> _ \<cdot> _"
 
   281 
   278 
   282   conjunct2: "Null" "conjunct2"
 
   279   conjunct2: "Null" "conjunct2"
 
   283 
   280 
   284   disjI1 (P, Q): "Inl"
 
   281   disjI1 (P, Q): "Inl"
 
   285     "\<Lambda> P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_1 \<cdot> P \<cdot> _ \<cdot> p)"
 
   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)"
 
   286 
   283 
   287   disjI1 (P): "Some"
 
   284   disjI1 (P): "Some"
 
   288     "\<Lambda> P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> P \<cdot> p)"
 
   285     "\<Lambda> (c: _) P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> P \<cdot> p \<bullet> arity_type_bool \<bullet> c)"
 
   289 
   286 
   290   disjI1 (Q): "None"
 
   287   disjI1 (Q): "None"
 
   291     "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)"
 
   288     "\<Lambda> (c: _) P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
 
   292 
   289 
   293   disjI1: "Left"
 
   290   disjI1: "Left"
 
   294     "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_1 \<cdot> _ \<cdot> _)"
 
   291     "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
 
   295 
   292 
   296   disjI2 (P, Q): "Inr"
 
   293   disjI2 (P, Q): "Inr"
 
   297     "\<Lambda> Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_2 \<cdot> _ \<cdot> Q \<cdot> q)"
 
   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)"
 
   298 
   295 
   299   disjI2 (P): "None"
 
   296   disjI2 (P): "None"
 
   300     "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)"
 
   297     "\<Lambda> (c: _) Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool \<bullet> c)"
 
   301 
   298 
   302   disjI2 (Q): "Some"
 
   299   disjI2 (Q): "Some"
 
   303     "\<Lambda> Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> Q \<cdot> q)"
 
   300     "\<Lambda> (c: _) Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> Q \<cdot> q \<bullet> arity_type_bool \<bullet> c)"
 
   304 
   301 
   305   disjI2: "Right"
 
   302   disjI2: "Right"
 
   306     "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_2 \<cdot> _ \<cdot> _)"
 
   303     "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_2 \<cdot> _ \<cdot> _ \<bullet> arity_type_bool)"
 
   307 
   304 
   308   disjE (P, Q, R): "\<lambda>pq pr qr.
 
   305   disjE (P, Q, R): "\<lambda>pq pr qr.
 
   309      (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
 
   306      (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
 
   310     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
 
   307     "\<Lambda> (c: _) (d: _) (e: _) P Q R pq (h1: _) pr (h2: _) qr.
 
   311        disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
 
   308        disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> e \<bullet> h1 \<bullet> h2"
 
   312 
   309 
   313   disjE (Q, R): "\<lambda>pq pr qr.
 
   310   disjE (Q, R): "\<lambda>pq pr qr.
 
   314      (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
 
   311      (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
 
   315     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
 
   312     "\<Lambda> (c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr.
 
   316        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
 
   313        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h2"
 
   317 
   314 
   318   disjE (P, R): "\<lambda>pq pr qr.
 
   315   disjE (P, R): "\<lambda>pq pr qr.
 
   319      (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
 
   316      (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
 
   320     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr (h3: _).
 
   317     "\<Lambda> (c: _) (d: _) P Q R pq (h1: _) pr (h2: _) qr (h3: _).
 
   321        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> h1 \<bullet> h3 \<bullet> h2"
 
   318        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> c \<bullet> d \<bullet> h1 \<bullet> h3 \<bullet> h2"
 
   322 
   319 
   323   disjE (R): "\<lambda>pq pr qr.
 
   320   disjE (R): "\<lambda>pq pr qr.
 
   324      (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
 
   321      (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
 
   325     "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
 
   322     "\<Lambda> (c: _) P Q R pq (h1: _) pr (h2: _) qr.
 
   326        disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
 
   323        disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> c \<bullet> h1 \<bullet> h2"
 
   327 
   324 
   328   disjE (P, Q): "Null"
 
   325   disjE (P, Q): "Null"
 
   329     "\<Lambda> P Q R pq. disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
 
   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"
 
   330 
   327 
   331   disjE (Q): "Null"
 
   328   disjE (Q): "Null"
 
   332     "\<Lambda> P Q R pq. disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
 
   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"
 
   333 
   330 
   334   disjE (P): "Null"
 
   331   disjE (P): "Null"
 
   335     "\<Lambda> P Q R pq (h1: _) (h2: _) (h3: _).
 
   332     "\<Lambda> (c: _) P Q R pq (h1: _) (h2: _) (h3: _).
 
   336        disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> h1 \<bullet> h3 \<bullet> h2"
 
   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"
 
   337 
   334 
   338   disjE: "Null"
 
   335   disjE: "Null"
 
   339     "\<Lambda> P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
 
   336     "\<Lambda> P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> arity_type_bool"
 
   340 
   337 
   341   FalseE (P): "default"
 
   338   FalseE (P): "default"
 
   342     "\<Lambda> P. FalseE \<cdot> _"
 
   339     "\<Lambda> (c: _) P. FalseE \<cdot> _"
 
   343 
   340 
   344   FalseE: "Null" "FalseE"
 
   341   FalseE: "Null" "FalseE"
 
   345 
   342 
   346   notI (P): "Null"
 
   343   notI (P): "Null"
 
   347     "\<Lambda> P (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
 
   344     "\<Lambda> (c: _) P (h: _). allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
 
   348 
   345 
   349   notI: "Null" "notI"
 
   346   notI: "Null" "notI"
 
   350 
   347 
   351   notE (P, R): "\<lambda>p. default"
 
   348   notE (P, R): "\<lambda>p. default"
 
   352     "\<Lambda> P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
 
   349     "\<Lambda> (c: _) (d: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
 
   353 
   350 
   354   notE (P): "Null"
 
   351   notE (P): "Null"
 
   355     "\<Lambda> P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
 
   352     "\<Lambda> (c: _) P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> c \<bullet> h)"
 
   356 
   353 
   357   notE (R): "default"
 
   354   notE (R): "default"
 
   358     "\<Lambda> P R. notE \<cdot> _ \<cdot> _"
 
   355     "\<Lambda> (c: _) P R. notE \<cdot> _ \<cdot> _"
 
   359 
   356 
   360   notE: "Null" "notE"
 
   357   notE: "Null" "notE"
 
   361 
   358 
   362   subst (P): "\<lambda>s t ps. ps"
 
   359   subst (P): "\<lambda>s t ps. ps"
 
   363     "\<Lambda> s t P (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> h"
 
   360     "\<Lambda> (c: _) s t P (d: _) (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> d \<bullet> h"
 
   364 
   361 
   365   subst: "Null" "subst"
 
   362   subst: "Null" "subst"
 
   366 
   363 
   367   iffD1 (P, Q): "fst"
 
   364   iffD1 (P, Q): "fst"
 
   368     "\<Lambda> Q P pq (h: _) p.
 
   365     "\<Lambda> (d: _) (c: _) Q P pq (h: _) p.
 
   369        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   366        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> d \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   370 
   367 
   371   iffD1 (P): "\<lambda>p. p"
 
   368   iffD1 (P): "\<lambda>p. p"
 
   372     "\<Lambda> Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
 
   369     "\<Lambda> (c: _) Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
 
   373 
   370 
   374   iffD1 (Q): "Null"
 
   371   iffD1 (Q): "Null"
 
   375     "\<Lambda> Q P q1 (h: _) q2.
 
   372     "\<Lambda> (c: _) Q P q1 (h: _) q2.
 
   376        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   373        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   377 
   374 
   378   iffD1: "Null" "iffD1"
 
   375   iffD1: "Null" "iffD1"
 
   379 
   376 
   380   iffD2 (P, Q): "snd"
 
   377   iffD2 (P, Q): "snd"
 
   381     "\<Lambda> P Q pq (h: _) q.
 
   378     "\<Lambda> (c: _) (d: _) P Q pq (h: _) q.
 
   382        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   379        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> d \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   383 
   380 
   384   iffD2 (P): "\<lambda>p. p"
 
   381   iffD2 (P): "\<lambda>p. p"
 
   385     "\<Lambda> P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
 
   382     "\<Lambda> (c: _) P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
 
   386 
   383 
   387   iffD2 (Q): "Null"
 
   384   iffD2 (Q): "Null"
 
   388     "\<Lambda> P Q q1 (h: _) q2.
 
   385     "\<Lambda> (c: _) P Q q1 (h: _) q2.
 
   389        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   386        mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> c \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
 
   390 
   387 
   391   iffD2: "Null" "iffD2"
 
   388   iffD2: "Null" "iffD2"
 
   392 
   389 
   393   iffI (P, Q): "Pair"
 
   390   iffI (P, Q): "Pair"
 
   394     "\<Lambda> P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
 
   391     "\<Lambda> (c: _) (d: _) P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
 
   395        (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
 
   392        (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
 
   396        (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
 
   393        (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
 
   397        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
 
   394        (arity_type_fun \<bullet> c \<bullet> d) \<bullet>
 
   398        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
 
   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)))"
 
   399 
   398 
   400   iffI (P): "\<lambda>p. p"
 
   399   iffI (P): "\<lambda>p. p"
 
   401     "\<Lambda> P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
 
   400     "\<Lambda> (c: _) P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
 
   402        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
 
   401        (allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
 
   403        (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
 
   402        (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
 
   404 
   403 
   405   iffI (Q): "\<lambda>q. q"
 
   404   iffI (Q): "\<lambda>q. q"
 
   406     "\<Lambda> P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
 
   405     "\<Lambda> (c: _) P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
 
   407        (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
 
   406        (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
 
   408        (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
 
   407        (allI \<cdot> _ \<bullet> c \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
 
   409 
   408 
   410   iffI: "Null" "iffI"
 
   409   iffI: "Null" "iffI"
 
   411 
   410 
   412 (*
 
       
   413   classical: "Null"
 
       
   414     "\<Lambda> P. classical \<cdot> _"
 
       
   415 *)
 
       
   416 
       
   417 setup {*
 
       
   418   Sign.add_const_constraint (@{const_name "default"}, SOME @{typ "'a::default"})
 
       
   419 *}
 
       
   420 
       
   421 end
 
   411 end
isabelle