src/HOL/TPTP/TPTP_Proof_Reconstruction.thy
author wenzelm
Sun Jan 24 15:30:32 2016 +0100 (2016-01-24)
changeset 62243 dd22d2423047
parent 60754 02924903a6fd
child 63167 0909deb8059b
permissions -rw-r--r--
clarified exception handling;
     1 (*  Title:      HOL/TPTP/TPTP_Proof_Reconstruction.thy
     2     Author:     Nik Sultana, Cambridge University Computer Laboratory
     3 
     4 Proof reconstruction for Leo-II.
     5 
     6 USAGE:
     7 * Simple call the "reconstruct_leo2" function.
     8 * For more advanced use, you could use the component functions used in
     9   "reconstruct_leo2" -- see TPTP_Proof_Reconstruction_Test.thy for
    10   examples of this.
    11 
    12 This file contains definitions describing how to interpret LEO-II's
    13 calculus in Isabelle/HOL, as well as more general proof-handling
    14 functions. The definitions in this file serve to build an intermediate
    15 proof script which is then evaluated into a tactic -- both these steps
    16 are independent of LEO-II, and are defined in the TPTP_Reconstruct SML
    17 module.
    18 
    19 CONFIG:
    20 The following attributes are mainly useful for debugging:
    21   tptp_unexceptional_reconstruction |
    22   unexceptional_reconstruction      |-- when these are true, a low-level exception
    23                                         is allowed to float to the top (instead of
    24                                         triggering a higher-level exception, or
    25                                         simply indicating that the reconstruction failed).
    26   tptp_max_term_size                --- fail of a term exceeds this size. "0" is taken
    27                                         to mean infinity.
    28   tptp_informative_failure          |
    29   informative_failure               |-- produce more output during reconstruction.
    30   tptp_trace_reconstruction         |
    31 
    32 There are also two attributes, independent of the code here, that
    33 influence the success of reconstruction: blast_depth_limit and
    34 unify_search_bound. These are documented in their respective modules,
    35 but in summary, if unify_search_bound is increased then we can
    36 handle larger terms (at the cost of performance), since the unification
    37 engine takes longer to give up the search; blast_depth_limit is
    38 a limit on proof search performed by Blast. Blast is used for
    39 the limited proof search that needs to be done to interpret
    40 instances of LEO-II's inference rules.
    41 
    42 TODO:
    43   use RemoveRedundantQuantifications instead of the ad hoc use of
    44    remove_redundant_quantification_in_lit and remove_redundant_quantification
    45 *)
    46 
    47 theory TPTP_Proof_Reconstruction
    48 imports TPTP_Parser TPTP_Interpret
    49 (* keywords "import_leo2_proof" :: thy_decl *) (*FIXME currently unused*)
    50 begin
    51 
    52 
    53 section "Setup"
    54 
    55 ML {*
    56   val tptp_unexceptional_reconstruction = Attrib.setup_config_bool @{binding tptp_unexceptional_reconstruction} (K false)
    57   fun unexceptional_reconstruction ctxt = Config.get ctxt tptp_unexceptional_reconstruction
    58   val tptp_informative_failure = Attrib.setup_config_bool @{binding tptp_informative_failure} (K false)
    59   fun informative_failure ctxt = Config.get ctxt tptp_informative_failure
    60   val tptp_trace_reconstruction = Attrib.setup_config_bool @{binding tptp_trace_reconstruction} (K false)
    61   val tptp_max_term_size = Attrib.setup_config_int @{binding tptp_max_term_size} (K 0) (*0=infinity*)
    62 
    63   fun exceeds_tptp_max_term_size ctxt size =
    64     let
    65       val max = Config.get ctxt tptp_max_term_size
    66     in
    67       if max = 0 then false
    68       else size > max
    69     end
    70 *}
    71 
    72 (*FIXME move to TPTP_Proof_Reconstruction_Test_Units*)
    73 declare [[
    74   tptp_unexceptional_reconstruction = false, (*NOTE should be "false" while testing*)
    75   tptp_informative_failure = true
    76 ]]
    77 
    78 ML_file "TPTP_Parser/tptp_reconstruct_library.ML"
    79 ML "open TPTP_Reconstruct_Library"
    80 ML_file "TPTP_Parser/tptp_reconstruct.ML"
    81 
    82 (*FIXME fudge*)
    83 declare [[
    84   blast_depth_limit = 10,
    85   unify_search_bound = 5
    86 ]]
    87 
    88 
    89 section "Proof reconstruction"
    90 text {*There are two parts to proof reconstruction:
    91 \begin{itemize}
    92   \item interpreting the inferences
    93   \item building the skeleton, which indicates how to compose
    94     individual inferences into subproofs, and then compose the
    95     subproofs to give the proof).
    96 \end{itemize}
    97 
    98 One step detects unsound inferences, and the other step detects
    99 unsound composition of inferences.  The two parts can be weakly
   100 coupled. They rely on a "proof index" which maps nodes to the
   101 inference information. This information consists of the (usually
   102 prover-specific) name of the inference step, and the Isabelle
   103 formalisation of the inference as a term. The inference interpretation
   104 then maps these terms into meta-theorems, and the skeleton is used to
   105 compose the inference-level steps into a proof.
   106 
   107 Leo2 operates on conjunctions of clauses. Each Leo2 inference has the
   108 following form, where Cx are clauses:
   109 
   110            C1 && ... && Cn
   111           -----------------
   112           C'1 && ... && C'n
   113 
   114 Clauses consist of disjunctions of literals (shown as Px below), and might
   115 have a prefix of !-bound variables, as shown below.
   116 
   117   ! X... { P1 || ... || Pn}
   118 
   119 Literals are usually assigned a polarity, but this isn't always the
   120 case; you can come across inferences looking like this (where A is an
   121 object-level formula):
   122 
   123              F
   124           --------
   125           F = true
   126 
   127 The symbol "||" represents literal-level disjunction and "&&" is
   128 clause-level conjunction. Rules will typically lift formula-level
   129 conjunctions; for instance the following rule lifts object-level
   130 disjunction:
   131 
   132           {    (A | B) = true    || ... } && ...
   133           --------------------------------------
   134           { A = true || B = true || ... } && ...
   135 
   136 
   137 Using this setup, efficiency might be gained by only interpreting
   138 inferences once, merging identical inference steps, and merging
   139 identical subproofs into single inferences thus avoiding some effort.
   140 We can also attempt to minimising proof search when interpreting
   141 inferences.
   142 
   143 It is hoped that this setup can target other provers by modifying the
   144 clause representation to fit them, and adapting the inference
   145 interpretation to handle the rules used by the prover. It should also
   146 facilitate composing together proofs found by different provers.
   147 *}
   148 
   149 
   150 subsection "Instantiation"
   151 
   152 lemma polar_allE [rule_format]:
   153   "\<lbrakk>(\<forall>x. P x) = True; (P x) = True \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   154   "\<lbrakk>(\<exists>x. P x) = False; (P x) = False \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   155 by auto
   156 
   157 lemma polar_exE [rule_format]:
   158   "\<lbrakk>(\<exists>x. P x) = True; \<And>x. (P x) = True \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   159   "\<lbrakk>(\<forall>x. P x) = False; \<And>x. (P x) = False \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   160 by auto
   161 
   162 ML {*
   163 (*This carries out an allE-like rule but on (polarised) literals.
   164  Instead of yielding a free variable (which is a hell for the
   165  matcher) it seeks to use one of the subgoals' parameters.
   166  This ought to be sufficient for emulating extcnf_combined,
   167  but note that the complexity of the problem can be enormous.*)
   168 fun inst_parametermatch_tac ctxt thms i = fn st =>
   169   let
   170     val gls =
   171       Thm.prop_of st
   172       |> Logic.strip_horn
   173       |> fst
   174 
   175     val parameters =
   176       if null gls then []
   177       else
   178         rpair (i - 1) gls
   179         |> uncurry nth
   180         |> strip_top_all_vars []
   181         |> fst
   182         |> map fst (*just get the parameter names*)
   183   in
   184     if null parameters then no_tac st
   185     else
   186       let
   187         fun instantiate param =
   188           (map (Rule_Insts.eres_inst_tac ctxt [((("x", 0), Position.none), param)] []) thms
   189                    |> FIRST')
   190         val attempts = map instantiate parameters
   191       in
   192         (fold (curry (op APPEND')) attempts (K no_tac)) i st
   193       end
   194   end
   195 
   196 (*Attempts to use the polar_allE theorems on a specific subgoal.*)
   197 fun forall_pos_tac ctxt = inst_parametermatch_tac ctxt @{thms polar_allE}
   198 *}
   199 
   200 ML {*
   201 (*This is similar to inst_parametermatch_tac, but prefers to
   202   match variables having identical names. Logically, this is
   203   a hack. But it reduces the complexity of the problem.*)
   204 fun nominal_inst_parametermatch_tac ctxt thm i = fn st =>
   205   let
   206     val gls =
   207       Thm.prop_of st
   208       |> Logic.strip_horn
   209       |> fst
   210 
   211     val parameters =
   212       if null gls then []
   213       else
   214         rpair (i - 1) gls
   215         |> uncurry nth
   216         |> strip_top_all_vars []
   217         |> fst
   218         |> map fst (*just get the parameter names*)
   219   in
   220     if null parameters then no_tac st
   221     else
   222       let
   223         fun instantiates param =
   224           Rule_Insts.eres_inst_tac ctxt [((("x", 0), Position.none), param)] [] thm
   225 
   226         val quantified_var = head_quantified_variable ctxt i st
   227       in
   228         if is_none quantified_var then no_tac st
   229         else
   230           if member (op =) parameters (the quantified_var |> fst) then
   231             instantiates (the quantified_var |> fst) i st
   232           else
   233             K no_tac i st
   234       end
   235   end
   236 *}
   237 
   238 
   239 subsection "Prefix massaging"
   240 
   241 ML {*
   242 exception NO_GOALS
   243 
   244 (*Get quantifier prefix of the hypothesis and conclusion, reorder
   245   the hypothesis' quantifiers to have the ones appearing in the
   246   conclusion first.*)
   247 fun canonicalise_qtfr_order ctxt i = fn st =>
   248   let
   249     val gls =
   250       Thm.prop_of st
   251       |> Logic.strip_horn
   252       |> fst
   253   in
   254     if null gls then raise NO_GOALS
   255     else
   256       let
   257         val (params, (hyp_clause, conc_clause)) =
   258           rpair (i - 1) gls
   259           |> uncurry nth
   260           |> strip_top_all_vars []
   261           |> apsnd Logic.dest_implies
   262 
   263         val (hyp_quants, hyp_body) =
   264           HOLogic.dest_Trueprop hyp_clause
   265           |> strip_top_All_vars
   266           |> apfst rev
   267 
   268         val conc_quants =
   269           HOLogic.dest_Trueprop conc_clause
   270           |> strip_top_All_vars
   271           |> fst
   272 
   273         val new_hyp =
   274           (* fold absfree new_hyp_prefix hyp_body *)
   275           (*HOLogic.list_all*)
   276           fold_rev (fn (v, ty) => fn t => HOLogic.mk_all (v, ty, t))
   277            (prefix_intersection_list
   278              hyp_quants conc_quants)
   279            hyp_body
   280           |> HOLogic.mk_Trueprop
   281 
   282          val thm = Goal.prove ctxt [] []
   283            (Logic.mk_implies (hyp_clause, new_hyp))
   284            (fn _ =>
   285               (REPEAT_DETERM (HEADGOAL (resolve_tac ctxt @{thms allI})))
   286               THEN (REPEAT_DETERM
   287                     (HEADGOAL
   288                      (nominal_inst_parametermatch_tac ctxt @{thm allE})))
   289               THEN HEADGOAL (assume_tac ctxt))
   290       in
   291         dresolve_tac ctxt [thm] i st
   292       end
   293     end
   294 *}
   295 
   296 
   297 subsection "Some general rules and congruences"
   298 
   299 (*this isn't an actual rule used in Leo2, but it seems to be
   300   applied implicitly during some Leo2 inferences.*)
   301 lemma polarise: "P ==> P = True" by auto
   302 
   303 ML {*
   304 fun is_polarised t =
   305   (TPTP_Reconstruct.remove_polarity true t; true)
   306   handle TPTP_Reconstruct.UNPOLARISED _ => false
   307 
   308 fun polarise_subgoal_hyps ctxt =
   309   COND' (SOME #> TERMPRED is_polarised (fn _ => true)) (K no_tac) (dresolve_tac ctxt @{thms polarise})
   310 *}
   311 
   312 lemma simp_meta [rule_format]:
   313   "(A --> B) == (~A | B)"
   314   "(A | B) | C == A | B | C"
   315   "(A & B) & C == A & B & C"
   316   "(~ (~ A)) == A"
   317   (* "(A & B) == (~ (~A | ~B))" *)
   318   "~ (A & B) == (~A | ~B)"
   319   "~(A | B) == (~A) & (~B)"
   320 by auto
   321 
   322 
   323 subsection "Emulation of Leo2's inference rules"
   324 
   325 (*this is not included in simp_meta since it would make a mess of the polarities*)
   326 lemma expand_iff [rule_format]:
   327  "((A :: bool) = B) \<equiv> (~ A | B) & (~ B | A)"
   328 by (rule eq_reflection, auto)
   329 
   330 lemma polarity_switch [rule_format]:
   331   "(\<not> P) = True \<Longrightarrow> P = False"
   332   "(\<not> P) = False \<Longrightarrow> P = True"
   333   "P = False \<Longrightarrow> (\<not> P) = True"
   334   "P = True \<Longrightarrow> (\<not> P) = False"
   335 by auto
   336 
   337 lemma solved_all_splits: "False = True \<Longrightarrow> False" by simp
   338 ML {*
   339 fun solved_all_splits_tac ctxt =
   340   TRY (eresolve_tac ctxt @{thms conjE} 1)
   341   THEN resolve_tac ctxt @{thms solved_all_splits} 1
   342   THEN assume_tac ctxt 1
   343 *}
   344 
   345 lemma lots_of_logic_expansions_meta [rule_format]:
   346   "(((A :: bool) = B) = True) == (((A \<longrightarrow> B) = True) & ((B \<longrightarrow> A) = True))"
   347   "((A :: bool) = B) = False == (((~A) | B) = False) | (((~B) | A) = False)"
   348 
   349   "((F = G) = True) == (! x. (F x = G x)) = True"
   350   "((F = G) = False) == (! x. (F x = G x)) = False"
   351 
   352   "(A | B) = True == (A = True) | (B = True)"
   353   "(A & B) = False == (A = False) | (B = False)"
   354   "(A | B) = False == (A = False) & (B = False)"
   355   "(A & B) = True == (A = True) & (B = True)"
   356   "(~ A) = True == A = False"
   357   "(~ A) = False == A = True"
   358   "~ (A = True) == A = False"
   359   "~ (A = False) == A = True"
   360 by (rule eq_reflection, auto)+
   361 
   362 (*this is used in extcnf_combined handler*)
   363 lemma eq_neg_bool: "((A :: bool) = B) = False ==> ((~ (A | B)) | ~ ((~ A) | (~ B))) = False"
   364 by auto
   365 
   366 lemma eq_pos_bool:
   367   "((A :: bool) = B) = True ==> ((~ (A | B)) | ~ (~ A | ~ B)) = True"
   368   "(A = B) = True \<Longrightarrow> A = True \<or> B = False"
   369   "(A = B) = True \<Longrightarrow> A = False \<or> B = True"
   370 by auto
   371 
   372 (*next formula is more versatile than
   373     "(F = G) = True \<Longrightarrow> \<forall>x. ((F x = G x) = True)"
   374   since it doesn't assume that clause is singleton. After splitqtfr,
   375   and after applying allI exhaustively to the conclusion, we can
   376   use the existing functions to find the "(F x = G x) = True"
   377   disjunct in the conclusion*)
   378 lemma eq_pos_func: "\<And> x. (F = G) = True \<Longrightarrow> (F x = G x) = True"
   379 by auto
   380 
   381 (*make sure the conclusion consists of just "False"*)
   382 lemma flip:
   383   "((A = True) ==> False) ==> A = False"
   384   "((A = False) ==> False) ==> A = True"
   385 by auto
   386 
   387 (*FIXME try to use Drule.equal_elim_rule1 directly for this*)
   388 lemma equal_elim_rule1: "(A \<equiv> B) \<Longrightarrow> A \<Longrightarrow> B" by auto
   389 lemmas leo2_rules =
   390  lots_of_logic_expansions_meta[THEN equal_elim_rule1]
   391 
   392 (*FIXME is there any overlap with lots_of_logic_expansions_meta or leo2_rules?*)
   393 lemma extuni_bool2 [rule_format]: "(A = B) = False \<Longrightarrow> (A = True) | (B = True)" by auto
   394 lemma extuni_bool1 [rule_format]: "(A = B) = False \<Longrightarrow> (A = False) | (B = False)" by auto
   395 lemma extuni_triv [rule_format]: "(A = A) = False \<Longrightarrow> R" by auto
   396 
   397 (*Order (of A, B, C, D) matters*)
   398 lemma dec_commut_eq [rule_format]:
   399   "((A = B) = (C = D)) = False \<Longrightarrow> (B = C) = False | (A = D) = False"
   400   "((A = B) = (C = D)) = False \<Longrightarrow> (B = D) = False | (A = C) = False"
   401 by auto
   402 lemma dec_commut_disj [rule_format]:
   403   "((A \<or> B) = (C \<or> D)) = False \<Longrightarrow> (B = C) = False \<or> (A = D) = False"
   404 by auto
   405 
   406 lemma extuni_func [rule_format]: "(F = G) = False \<Longrightarrow> (! X. (F X = G X)) = False" by auto
   407 
   408 
   409 subsection "Emulation: tactics"
   410 
   411 ML {*
   412 (*Instantiate a variable according to the info given in the
   413   proof annotation. Through this we avoid having to come up
   414   with instantiations during reconstruction.*)
   415 fun bind_tac ctxt prob_name ordered_binds =
   416   let
   417     val thy = Proof_Context.theory_of ctxt
   418     fun term_to_string t =
   419         Print_Mode.with_modes [""]
   420           (fn () => Output.output (Syntax.string_of_term ctxt t)) ()
   421     val ordered_instances =
   422       TPTP_Reconstruct.interpret_bindings prob_name thy ordered_binds []
   423       |> map (snd #> term_to_string)
   424       |> permute
   425 
   426     (*instantiate a list of variables, order matters*)
   427     fun instantiate_vars ctxt vars : tactic =
   428       map (fn var =>
   429             Rule_Insts.eres_inst_tac ctxt
   430              [((("x", 0), Position.none), var)] [] @{thm allE} 1)
   431           vars
   432       |> EVERY
   433 
   434     fun instantiate_tac vars =
   435       instantiate_vars ctxt vars
   436       THEN (HEADGOAL (assume_tac ctxt))
   437   in
   438     HEADGOAL (canonicalise_qtfr_order ctxt)
   439     THEN (REPEAT_DETERM (HEADGOAL (resolve_tac ctxt @{thms allI})))
   440     THEN REPEAT_DETERM (HEADGOAL (nominal_inst_parametermatch_tac ctxt @{thm allE}))
   441     (*now only the variable to instantiate should be left*)
   442     THEN FIRST (map instantiate_tac ordered_instances)
   443   end
   444 *}
   445 
   446 ML {*
   447 (*Simplification tactics*)
   448 local
   449   fun rew_goal_tac thms ctxt i =
   450     rewrite_goal_tac ctxt thms i
   451     |> CHANGED
   452 in
   453   val expander_animal =
   454     rew_goal_tac (@{thms simp_meta} @ @{thms lots_of_logic_expansions_meta})
   455 
   456   val simper_animal =
   457     rew_goal_tac @{thms simp_meta}
   458 end
   459 *}
   460 
   461 lemma prop_normalise [rule_format]:
   462   "(A | B) | C == A | B | C"
   463   "(A & B) & C == A & B & C"
   464   "A | B == ~(~A & ~B)"
   465   "~~ A == A"
   466 by auto
   467 ML {*
   468 (*i.e., break_conclusion*)
   469 fun flip_conclusion_tac ctxt =
   470   let
   471     val default_tac =
   472       (TRY o CHANGED o (rewrite_goal_tac ctxt @{thms prop_normalise}))
   473       THEN' resolve_tac ctxt @{thms notI}
   474       THEN' (REPEAT_DETERM o eresolve_tac ctxt @{thms conjE})
   475       THEN' (TRY o (expander_animal ctxt))
   476   in
   477     default_tac ORELSE' resolve_tac ctxt @{thms flip}
   478   end
   479 *}
   480 
   481 
   482 subsection "Skolemisation"
   483 
   484 lemma skolemise [rule_format]:
   485   "\<forall> P. (~ (! x. P x)) \<longrightarrow> ~ (P (SOME x. ~ P x))"
   486 proof -
   487   have "\<And> P. (~ (! x. P x)) \<Longrightarrow> ~ (P (SOME x. ~ P x))"
   488   proof -
   489     fix P
   490     assume ption: "~ (! x. P x)"
   491     hence a: "? x. ~ P x" by force
   492 
   493     have hilbert : "\<And> P. (? x. P x) \<Longrightarrow> (P (SOME x. P x))"
   494     proof -
   495       fix P
   496       assume "(? x. P x)"
   497       thus "(P (SOME x. P x))"
   498         apply auto
   499         apply (rule someI)
   500         apply auto
   501         done
   502     qed
   503 
   504     from a show "~ P (SOME x. ~ P x)"
   505     proof -
   506       assume "? x. ~ P x"
   507       hence "\<not> P (SOME x. \<not> P x)" by (rule hilbert)
   508       thus ?thesis .
   509     qed
   510   qed
   511   thus ?thesis by blast
   512 qed
   513 
   514 lemma polar_skolemise [rule_format]:
   515   "\<forall> P. (! x. P x) = False \<longrightarrow> (P (SOME x. ~ P x)) = False"
   516 proof -
   517   have "\<And> P. (! x. P x) = False \<Longrightarrow> (P (SOME x. ~ P x)) = False"
   518   proof -
   519     fix P
   520     assume ption: "(! x. P x) = False"
   521     hence "\<not> (\<forall> x. P x)" by force
   522     hence "\<not> All P" by force
   523     hence "\<not> (P (SOME x. \<not> P x))" by (rule skolemise)
   524     thus "(P (SOME x. \<not> P x)) = False" by force
   525   qed
   526   thus ?thesis by blast
   527 qed
   528 
   529 lemma leo2_skolemise [rule_format]:
   530   "\<forall> P sk. (! x. P x) = False \<longrightarrow> (sk = (SOME x. ~ P x)) \<longrightarrow> (P sk) = False"
   531 by (clarify, rule polar_skolemise)
   532 
   533 lemma lift_forall [rule_format]:
   534   "!! x. (! x. A x) = True ==> (A x) = True"
   535   "!! x. (? x. A x) = False ==> (A x) = False"
   536 by auto
   537 lemma lift_exists [rule_format]:
   538   "\<lbrakk>(All P) = False; sk = (SOME x. \<not> P x)\<rbrakk> \<Longrightarrow> P sk = False"
   539   "\<lbrakk>(Ex P) = True; sk = (SOME x. P x)\<rbrakk> \<Longrightarrow> P sk = True"
   540 apply (drule polar_skolemise, simp)
   541 apply (simp, drule someI_ex, simp)
   542 done
   543 
   544 ML {*
   545 (*FIXME LHS should be constant. Currently allow variables for testing. Probably should still allow Vars (but not Frees) since they'll act as intermediate values*)
   546 fun conc_is_skolem_def t =
   547   case t of
   548       Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _) =>
   549       let
   550         val (h, args) =
   551           strip_comb t'
   552           |> apfst (strip_abs #> snd #> strip_comb #> fst)
   553         val get_const_name = dest_Const #> fst
   554         val h_property =
   555           is_Free h orelse
   556           is_Var h orelse
   557           (is_Const h
   558            andalso (get_const_name h <> get_const_name @{term HOL.Ex})
   559            andalso (get_const_name h <> get_const_name @{term HOL.All})
   560            andalso (h <> @{term Hilbert_Choice.Eps})
   561            andalso (h <> @{term HOL.conj})
   562            andalso (h <> @{term HOL.disj})
   563            andalso (h <> @{term HOL.eq})
   564            andalso (h <> @{term HOL.implies})
   565            andalso (h <> @{term HOL.The})
   566            andalso (h <> @{term HOL.Ex1})
   567            andalso (h <> @{term HOL.Not})
   568            andalso (h <> @{term HOL.iff})
   569            andalso (h <> @{term HOL.not_equal}))
   570         val args_property =
   571           fold (fn t => fn b =>
   572            b andalso is_Free t) args true
   573       in
   574         h_property andalso args_property
   575       end
   576     | _ => false
   577 *}
   578 
   579 ML {*
   580 (*Hack used to detect if a Skolem definition, with an LHS Var, has had the LHS instantiated into an unacceptable term.*)
   581 fun conc_is_bad_skolem_def t =
   582   case t of
   583       Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _) =>
   584       let
   585         val (h, args) = strip_comb t'
   586         val get_const_name = dest_Const #> fst
   587         val const_h_test =
   588           if is_Const h then
   589             (get_const_name h = get_const_name @{term HOL.Ex})
   590              orelse (get_const_name h = get_const_name @{term HOL.All})
   591              orelse (h = @{term Hilbert_Choice.Eps})
   592              orelse (h = @{term HOL.conj})
   593              orelse (h = @{term HOL.disj})
   594              orelse (h = @{term HOL.eq})
   595              orelse (h = @{term HOL.implies})
   596              orelse (h = @{term HOL.The})
   597              orelse (h = @{term HOL.Ex1})
   598              orelse (h = @{term HOL.Not})
   599              orelse (h = @{term HOL.iff})
   600              orelse (h = @{term HOL.not_equal})
   601           else true
   602         val h_property =
   603           not (is_Free h) andalso
   604           not (is_Var h) andalso
   605           const_h_test
   606         val args_property =
   607           fold (fn t => fn b =>
   608            b andalso is_Free t) args true
   609       in
   610         h_property andalso args_property
   611       end
   612     | _ => false
   613 *}
   614 
   615 ML {*
   616 fun get_skolem_conc t =
   617   let
   618     val t' =
   619       strip_top_all_vars [] t
   620       |> snd
   621       |> try_dest_Trueprop
   622   in
   623     case t' of
   624         Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _) => SOME t'
   625       | _ => NONE
   626   end
   627 
   628 fun get_skolem_conc_const t =
   629   lift_option
   630    (fn t' =>
   631      head_of t'
   632      |> strip_abs_body
   633      |> head_of
   634      |> dest_Const)
   635    (get_skolem_conc t)
   636 *}
   637 
   638 (*
   639 Technique for handling quantifiers:
   640   Principles:
   641   * allE should always match with a !!
   642   * exE should match with a constant,
   643      or bind a fresh !! -- currently not doing the latter since it never seems to arised in normal Leo2 proofs.
   644 *)
   645 
   646 ML {*
   647 fun forall_neg_tac candidate_consts ctxt i = fn st =>
   648   let
   649     val gls =
   650       Thm.prop_of st
   651       |> Logic.strip_horn
   652       |> fst
   653 
   654     val parameters =
   655       if null gls then ""
   656       else
   657         rpair (i - 1) gls
   658         |> uncurry nth
   659         |> strip_top_all_vars []
   660         |> fst
   661         |> map fst (*just get the parameter names*)
   662         |> (fn l =>
   663               if null l then ""
   664               else
   665                 space_implode " " l
   666                 |> pair " "
   667                 |> op ^)
   668 
   669   in
   670     if null gls orelse null candidate_consts then no_tac st
   671     else
   672       let
   673         fun instantiate const_name =
   674           Rule_Insts.dres_inst_tac ctxt [((("sk", 0), Position.none), const_name ^ parameters)] []
   675             @{thm leo2_skolemise}
   676         val attempts = map instantiate candidate_consts
   677       in
   678         (fold (curry (op APPEND')) attempts (K no_tac)) i st
   679       end
   680   end
   681 *}
   682 
   683 ML {*
   684 exception SKOLEM_DEF of term (*The tactic wasn't pointed at a skolem definition*)
   685 exception NO_SKOLEM_DEF of (*skolem const name*)string * Binding.binding * term (*The tactic could not find a skolem definition in the theory*)
   686 fun absorb_skolem_def ctxt prob_name_opt i = fn st =>
   687   let
   688     val thy = Proof_Context.theory_of ctxt
   689 
   690     val gls =
   691       Thm.prop_of st
   692       |> Logic.strip_horn
   693       |> fst
   694 
   695     val conclusion =
   696       if null gls then
   697         (*this should never be thrown*)
   698         raise NO_GOALS
   699       else
   700         rpair (i - 1) gls
   701         |> uncurry nth
   702         |> strip_top_all_vars []
   703         |> snd
   704         |> Logic.strip_horn
   705         |> snd
   706 
   707     fun skolem_const_info_of t =
   708       case t of
   709           Const (@{const_name HOL.Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _)) =>
   710           head_of t'
   711           |> strip_abs_body (*since in general might have a skolem term, so we want to rip out the prefixing lambdas to get to the constant (which should be at head position)*)
   712           |> head_of
   713           |> dest_Const
   714         | _ => raise SKOLEM_DEF t
   715 
   716     val const_name =
   717       skolem_const_info_of conclusion
   718       |> fst
   719 
   720     val def_name = const_name ^ "_def"
   721 
   722     val bnd_def = (*FIXME consts*)
   723       const_name
   724       |> Long_Name.implode o tl o Long_Name.explode (*FIXME hack to drop theory-name prefix*)
   725       |> Binding.qualified_name
   726       |> Binding.suffix_name "_def"
   727 
   728     val bnd_name =
   729       case prob_name_opt of
   730           NONE => bnd_def
   731         | SOME prob_name =>
   732 (*            Binding.qualify false
   733              (TPTP_Problem_Name.mangle_problem_name prob_name)
   734 *)
   735              bnd_def
   736 
   737     val thm =
   738       (case try (Thm.axiom thy) def_name of
   739         SOME thm => thm
   740       | NONE =>
   741           if is_none prob_name_opt then
   742             (*This mode is for testing, so we can be a bit
   743               looser with theories*)
   744             (* FIXME bad theory context!? *)
   745             Thm.add_axiom_global (bnd_name, conclusion) thy
   746             |> fst |> snd
   747           else
   748             raise (NO_SKOLEM_DEF (def_name, bnd_name, conclusion)))
   749   in
   750     resolve_tac ctxt [Drule.export_without_context thm] i st
   751   end
   752   handle SKOLEM_DEF _ => no_tac st
   753 *}
   754 
   755 ML {*
   756 (*
   757 In current system, there should only be 2 subgoals: the one where
   758 the skolem definition is being built (with a Var in the LHS), and the other subgoal using Var.
   759 *)
   760 (*arity must be greater than 0. if arity=0 then
   761   there's no need to use this expensive matching.*)
   762 fun find_skolem_term ctxt consts_candidate arity = fn st =>
   763   let
   764     val _ = @{assert} (arity > 0)
   765 
   766     val gls =
   767       Thm.prop_of st
   768       |> Logic.strip_horn
   769       |> fst
   770 
   771     (*extract the conclusion of each subgoal*)
   772     val conclusions =
   773       if null gls then
   774         raise NO_GOALS
   775       else
   776         map (strip_top_all_vars [] #> snd #> Logic.strip_horn #> snd) gls
   777         (*Remove skolem-definition conclusion, to avoid wasting time analysing it*)
   778         |> filter (try_dest_Trueprop #> conc_is_skolem_def #> not)
   779         (*There should only be a single goal*) (*FIXME this might not always be the case, in practice*)
   780         (* |> tap (fn x => @{assert} (is_some (try the_single x))) *)
   781 
   782     (*look for subterms headed by a skolem constant, and whose
   783       arguments are all parameter Vars*)
   784     fun get_skolem_terms args (acc : term list) t =
   785       case t of
   786           (c as Const _) $ (v as Free _) =>
   787             if c = consts_candidate andalso
   788              arity = length args + 1 then
   789               (list_comb (c, v :: args)) :: acc
   790             else acc
   791         | t1 $ (v as Free _) =>
   792             get_skolem_terms (v :: args) acc t1 @
   793              get_skolem_terms [] acc t1
   794         | t1 $ t2 =>
   795             get_skolem_terms [] acc t1 @
   796              get_skolem_terms [] acc t2
   797         | Abs (_, _, t') => get_skolem_terms [] acc t'
   798         | _ => acc
   799   in
   800     map (strip_top_All_vars #> snd) conclusions
   801     |> maps (get_skolem_terms [] [])
   802     |> distinct (op =)
   803   end
   804 *}
   805 
   806 ML {*
   807 fun instantiate_skols ctxt consts_candidates i = fn st =>
   808   let
   809     val gls =
   810       Thm.prop_of st
   811       |> Logic.strip_horn
   812       |> fst
   813 
   814     val (params, conclusion) =
   815       if null gls then
   816         raise NO_GOALS
   817       else
   818         rpair (i - 1) gls
   819         |> uncurry nth
   820         |> strip_top_all_vars []
   821         |> apsnd (Logic.strip_horn #> snd)
   822 
   823     fun skolem_const_info_of t =
   824       case t of
   825           Const (@{const_name HOL.Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ lhs $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ rhs)) =>
   826           let
   827             (*the parameters we will concern ourselves with*)
   828             val params' =
   829               Term.add_frees lhs []
   830               |> distinct (op =)
   831             (*check to make sure that params' <= params*)
   832             val _ = @{assert} (forall (member (op =) params) params')
   833             val skolem_const_ty =
   834               let
   835                 val (skolem_const_prety, no_params) =
   836                   Term.strip_comb lhs
   837                   |> apfst (dest_Var #> snd) (*head of lhs consists of a logical variable. we just want its type.*)
   838                   |> apsnd length
   839 
   840                 val _ = @{assert} (length params = no_params)
   841 
   842                 (*get value type of a function type after n arguments have been supplied*)
   843                 fun get_val_ty n ty =
   844                   if n = 0 then ty
   845                   else get_val_ty (n - 1) (dest_funT ty |> snd)
   846               in
   847                 get_val_ty no_params skolem_const_prety
   848               end
   849 
   850           in
   851             (skolem_const_ty, params')
   852           end
   853         | _ => raise (SKOLEM_DEF t)
   854 
   855 (*
   856 find skolem const candidates which, after applying distinct members of params' we end up with, give us something of type skolem_const_ty.
   857 
   858 given a candidate's type, skolem_const_ty, and params', we get some pemutations of params' (i.e. the order in which they can be given to the candidate in order to get skolem_const_ty). If the list of permutations is empty, then we cannot use that candidate.
   859 *)
   860 (*
   861 only returns a single matching -- since terms are linear, and variable arguments are Vars, order shouldn't matter, so we can ignore permutations.
   862 doesn't work with polymorphism (for which we'd need to use type unification) -- this is OK since no terms should be polymorphic, since Leo2 proofs aren't.
   863 *)
   864     fun use_candidate target_ty params acc cur_ty =
   865       if null params then
   866         if cur_ty = target_ty then
   867           SOME (rev acc)
   868         else NONE
   869       else
   870         let
   871           val (arg_ty, val_ty) = Term.dest_funT cur_ty
   872           (*now find a param of type arg_ty*)
   873           val (candidate_param, params') =
   874             find_and_remove (snd #> pair arg_ty #> op =) params
   875         in
   876           use_candidate target_ty params' (candidate_param :: acc) val_ty
   877         end
   878         handle TYPE ("dest_funT", _, _) => NONE  (* FIXME fragile *)
   879              | _ => NONE  (* FIXME avoid catch-all handler *)
   880 
   881     val (skolem_const_ty, params') = skolem_const_info_of conclusion
   882 
   883 (*
   884 For each candidate, build a term and pass it to Thm.instantiate, whic in turn is chained with PRIMITIVE to give us this_tactic.
   885 
   886 Big picture:
   887   we run the following:
   888     drule leo2_skolemise THEN' this_tactic
   889 
   890 NOTE: remember to APPEND' instead of ORELSE' the two tactics relating to skolemisation
   891 *)
   892 
   893     val filtered_candidates =
   894       map (dest_Const
   895            #> snd
   896            #> use_candidate skolem_const_ty params' [])
   897        consts_candidates (* prefiltered_candidates *)
   898       |> pair consts_candidates (* prefiltered_candidates *)
   899       |> ListPair.zip
   900       |> filter (snd #> is_none #> not)
   901       |> map (apsnd the)
   902 
   903     val skolem_terms =
   904       let
   905         fun make_result_t (t, args) =
   906           (* list_comb (t, map Free args) *)
   907           if length args > 0 then
   908             hd (find_skolem_term ctxt t (length args) st)
   909           else t
   910       in
   911         map make_result_t filtered_candidates
   912       end
   913 
   914     (*prefix a skolem term with bindings for the parameters*)
   915     (* val contextualise = fold absdummy (map snd params) *)
   916     val contextualise = fold absfree params
   917 
   918     val skolem_cts = map (contextualise #> Thm.cterm_of ctxt) skolem_terms
   919 
   920 
   921 (*now the instantiation code*)
   922 
   923     (*there should only be one Var -- that is from the previous application of drule leo2_skolemise. We look for it at the head position in some equation at a conclusion of a subgoal.*)
   924     val var_opt =
   925       let
   926         val pre_var =
   927           gls
   928           |> map
   929               (strip_top_all_vars [] #> snd #>
   930                Logic.strip_horn #> snd #>
   931                get_skolem_conc)
   932           |> switch (fold (fn x => fn l => if is_some x then the x :: l else l)) []
   933           |> maps (switch Term.add_vars [])
   934 
   935         fun make_var pre_var =
   936           the_single pre_var
   937           |> Var
   938           |> Thm.cterm_of ctxt
   939           |> SOME
   940       in
   941         if null pre_var then NONE
   942         else make_var pre_var
   943      end
   944 
   945     fun instantiate_tac from to =
   946       PRIMITIVE (Thm.instantiate ([], [(from, to)]))
   947 
   948     val tactic =
   949       if is_none var_opt then no_tac
   950       else
   951         fold (curry (op APPEND))
   952           (map (instantiate_tac (dest_Var (Thm.term_of (the var_opt)))) skolem_cts) no_tac
   953   in
   954     tactic st
   955   end
   956 *}
   957 
   958 ML {*
   959 fun new_skolem_tac ctxt consts_candidates =
   960   let
   961     fun tac thm =
   962       dresolve_tac ctxt [thm]
   963       THEN' instantiate_skols ctxt consts_candidates
   964   in
   965     if null consts_candidates then K no_tac
   966     else FIRST' (map tac @{thms lift_exists})
   967   end
   968 *}
   969 
   970 (*
   971 need a tactic to expand "? x . P" to "~ ! x. ~ P"
   972 *)
   973 ML {*
   974 fun ex_expander_tac ctxt i =
   975    let
   976      val simpset =
   977        empty_simpset ctxt (*NOTE for some reason, Bind exception gets raised if ctxt's simpset isn't emptied*)
   978        |> Simplifier.add_simp @{lemma "Ex P == (~ (! x. ~ P x))" by auto}
   979    in
   980      CHANGED (asm_full_simp_tac simpset i)
   981    end
   982 *}
   983 
   984 
   985 subsubsection "extuni_dec"
   986 
   987 ML {*
   988 (*n-ary decomposition. Code is based on the n-ary arg_cong generator*)
   989 fun extuni_dec_n ctxt arity =
   990   let
   991     val _ = @{assert} (arity > 0)
   992     val is =
   993       upto (1, arity)
   994       |> map Int.toString
   995     val arg_tys = map (fn i => TFree ("arg" ^ i ^ "_ty", @{sort type})) is
   996     val res_ty = TFree ("res" ^ "_ty", @{sort type})
   997     val f_ty = arg_tys ---> res_ty
   998     val f = Free ("f", f_ty)
   999     val xs = map (fn i =>
  1000       Free ("x" ^ i, TFree ("arg" ^ i ^ "_ty", @{sort type}))) is
  1001     (*FIXME DRY principle*)
  1002     val ys = map (fn i =>
  1003       Free ("y" ^ i, TFree ("arg" ^ i ^ "_ty", @{sort type}))) is
  1004 
  1005     val hyp_lhs = list_comb (f, xs)
  1006     val hyp_rhs = list_comb (f, ys)
  1007     val hyp_eq =
  1008       HOLogic.eq_const res_ty $ hyp_lhs $ hyp_rhs
  1009     val hyp =
  1010       HOLogic.eq_const HOLogic.boolT $ hyp_eq $ @{term False}
  1011       |> HOLogic.mk_Trueprop
  1012     fun conc_eq i =
  1013       let
  1014         val ty = TFree ("arg" ^ i ^ "_ty", @{sort type})
  1015         val x = Free ("x" ^ i, ty)
  1016         val y = Free ("y" ^ i, ty)
  1017         val eq = HOLogic.eq_const ty $ x $ y
  1018       in
  1019         HOLogic.eq_const HOLogic.boolT $ eq $ @{term False}
  1020       end
  1021 
  1022     val conc_disjs = map conc_eq is
  1023 
  1024     val conc =
  1025       if length conc_disjs = 1 then
  1026         the_single conc_disjs
  1027       else
  1028         fold
  1029          (fn t => fn t_conc => HOLogic.mk_disj (t_conc, t))
  1030          (tl conc_disjs) (hd conc_disjs)
  1031 
  1032     val t =
  1033       Logic.mk_implies (hyp, HOLogic.mk_Trueprop conc)
  1034   in
  1035     Goal.prove ctxt [] [] t (fn _ => auto_tac ctxt)
  1036     |> Drule.export_without_context
  1037   end
  1038 *}
  1039 
  1040 ML {*
  1041 (*Determine the arity of a function which the "dec"
  1042   unification rule is about to be applied.
  1043   NOTE:
  1044     * Assumes that there is a single hypothesis
  1045 *)
  1046 fun find_dec_arity i = fn st =>
  1047   let
  1048     val gls =
  1049       Thm.prop_of st
  1050       |> Logic.strip_horn
  1051       |> fst
  1052   in
  1053     if null gls then raise NO_GOALS
  1054     else
  1055       let
  1056         val (params, (literal, conc_clause)) =
  1057           rpair (i - 1) gls
  1058           |> uncurry nth
  1059           |> strip_top_all_vars []
  1060           |> apsnd Logic.strip_horn
  1061           |> apsnd (apfst the_single)
  1062 
  1063         val get_ty =
  1064           HOLogic.dest_Trueprop
  1065           #> strip_top_All_vars
  1066           #> snd
  1067           #> HOLogic.dest_eq (*polarity's "="*)
  1068           #> fst
  1069           #> HOLogic.dest_eq (*the unification constraint's "="*)
  1070           #> fst
  1071           #> head_of
  1072           #> dest_Const
  1073           #> snd
  1074 
  1075        fun arity_of ty =
  1076          let
  1077            val (_, res_ty) = dest_funT ty
  1078 
  1079          in
  1080            1 + arity_of res_ty
  1081          end
  1082          handle (TYPE ("dest_funT", _, _)) => 0
  1083 
  1084       in
  1085         arity_of (get_ty literal)
  1086       end
  1087   end
  1088 
  1089 (*given an inference, it returns the parameters (i.e., we've already matched the leading & shared quantification in the hypothesis & conclusion clauses), and the "raw" inference*)
  1090 fun breakdown_inference i = fn st =>
  1091   let
  1092     val gls =
  1093       Thm.prop_of st
  1094       |> Logic.strip_horn
  1095       |> fst
  1096   in
  1097     if null gls then raise NO_GOALS
  1098     else
  1099       rpair (i - 1) gls
  1100       |> uncurry nth
  1101       |> strip_top_all_vars []
  1102   end
  1103 
  1104 (*build a custom elimination rule for extuni_dec, and instantiate it to match a specific subgoal*)
  1105 fun extuni_dec_elim_rule ctxt arity i = fn st =>
  1106   let
  1107     val rule = extuni_dec_n ctxt arity
  1108 
  1109     val rule_hyp =
  1110       Thm.prop_of rule
  1111       |> Logic.dest_implies
  1112       |> fst (*assuming that rule has single hypothesis*)
  1113 
  1114     (*having run break_hypothesis earlier, we know that the hypothesis
  1115       now consists of a single literal. We can (and should)
  1116       disregard the conclusion, since it hasn't been "broken",
  1117       and it might include some unwanted literals -- the latter
  1118       could cause "diff" to fail (since they won't agree with the
  1119       rule we have generated.*)
  1120 
  1121     val inference_hyp =
  1122       snd (breakdown_inference i st)
  1123       |> Logic.dest_implies
  1124       |> fst (*assuming that inference has single hypothesis,
  1125                as explained above.*)
  1126   in
  1127     TPTP_Reconstruct_Library.diff_and_instantiate ctxt rule rule_hyp inference_hyp
  1128   end
  1129 
  1130 fun extuni_dec_tac ctxt i = fn st =>
  1131   let
  1132     val arity = find_dec_arity i st
  1133 
  1134     fun elim_tac i st =
  1135       let
  1136         val rule =
  1137           extuni_dec_elim_rule ctxt arity i st
  1138           (*in case we itroduced free variables during
  1139             instantiation, we generalise the rule to make
  1140             those free variables into logical variables.*)
  1141           |> Thm.forall_intr_frees
  1142           |> Drule.export_without_context
  1143       in dresolve_tac ctxt [rule] i st end
  1144       handle NO_GOALS => no_tac st
  1145 
  1146     fun closure tac =
  1147      (*batter fails if there's no toplevel disjunction in the
  1148        hypothesis, so we also try atac*)
  1149       SOLVE o (tac THEN' (batter_tac ctxt ORELSE' assume_tac ctxt))
  1150     val search_tac =
  1151       ASAP
  1152         (resolve_tac ctxt @{thms disjI1} APPEND' resolve_tac ctxt @{thms disjI2})
  1153         (FIRST' (map closure
  1154                   [dresolve_tac ctxt @{thms dec_commut_eq},
  1155                    dresolve_tac ctxt @{thms dec_commut_disj},
  1156                    elim_tac]))
  1157   in
  1158     (CHANGED o search_tac) i st
  1159   end
  1160 *}
  1161 
  1162 
  1163 subsubsection "standard_cnf"
  1164 (*Given a standard_cnf inference, normalise it
  1165      e.g. ((A & B) & C \<longrightarrow> D & E \<longrightarrow> F \<longrightarrow> G) = False
  1166      is changed to
  1167           (A & B & C & D & E & F \<longrightarrow> G) = False
  1168  then custom-build a metatheorem which validates this:
  1169           (A & B & C & D & E & F \<longrightarrow> G) = False
  1170        -------------------------------------------
  1171           (A = True) & (B = True) & (C = True) &
  1172           (D = True) & (E = True) & (F = True) & (G = False)
  1173  and apply this metatheorem.
  1174 
  1175 There aren't any "positive" standard_cnfs in Leo2's calculus:
  1176   e.g.,  "(A \<longrightarrow> B) = True \<Longrightarrow> A = False | (A = True & B = True)"
  1177 since "standard_cnf" seems to be applied at the preprocessing
  1178 stage, together with splitting.
  1179 *)
  1180 
  1181 ML {*
  1182 (*Conjunctive counterparts to Term.disjuncts_aux and Term.disjuncts*)
  1183 fun conjuncts_aux (Const (@{const_name HOL.conj}, _) $ t $ t') conjs =
  1184      conjuncts_aux t (conjuncts_aux t' conjs)
  1185   | conjuncts_aux t conjs = t :: conjs
  1186 
  1187 fun conjuncts t = conjuncts_aux t []
  1188 
  1189 (*HOL equivalent of Logic.strip_horn*)
  1190 local
  1191   fun imp_strip_horn' acc (Const (@{const_name HOL.implies}, _) $ A $ B) =
  1192         imp_strip_horn' (A :: acc) B
  1193     | imp_strip_horn' acc t = (acc, t)
  1194 in
  1195   fun imp_strip_horn t =
  1196     imp_strip_horn' [] t
  1197     |> apfst rev
  1198 end
  1199 *}
  1200 
  1201 ML {*
  1202 (*Returns whether the antecedents are separated by conjunctions
  1203   or implications; the number of antecedents; and the polarity
  1204   of the original clause -- I think this will always be "false".*)
  1205 fun standard_cnf_type ctxt i : thm -> (TPTP_Reconstruct.formula_kind * int * bool) option = fn st =>
  1206   let
  1207     val gls =
  1208       Thm.prop_of st
  1209       |> Logic.strip_horn
  1210       |> fst
  1211 
  1212     val hypos =
  1213       if null gls then raise NO_GOALS
  1214       else
  1215         rpair (i - 1) gls
  1216         |> uncurry nth
  1217         |> TPTP_Reconstruct.strip_top_all_vars []
  1218         |> snd
  1219         |> Logic.strip_horn
  1220         |> fst
  1221 
  1222     (*hypothesis clause should be singleton*)
  1223     val _ = @{assert} (length hypos = 1)
  1224 
  1225     val (t, pol) = the_single hypos
  1226       |> try_dest_Trueprop
  1227       |> TPTP_Reconstruct.strip_top_All_vars
  1228       |> snd
  1229       |> TPTP_Reconstruct.remove_polarity true
  1230 
  1231     (*literal is negative*)
  1232     val _ = @{assert} (not pol)
  1233 
  1234     val (antes, conc) = imp_strip_horn t
  1235 
  1236     val (ante_type, antes') =
  1237       if length antes = 1 then
  1238         let
  1239           val conjunctive_antes =
  1240             the_single antes
  1241             |> conjuncts
  1242         in
  1243           if length conjunctive_antes > 1 then
  1244             (TPTP_Reconstruct.Conjunctive NONE,
  1245              conjunctive_antes)
  1246           else
  1247             (TPTP_Reconstruct.Implicational NONE,
  1248              antes)
  1249         end
  1250       else
  1251         (TPTP_Reconstruct.Implicational NONE,
  1252          antes)
  1253   in
  1254     if null antes then NONE
  1255     else SOME (ante_type, length antes', pol)
  1256   end
  1257 *}
  1258 
  1259 ML {*
  1260 (*Given a certain standard_cnf type, build a metatheorem that would
  1261   validate it*)
  1262 fun mk_standard_cnf ctxt kind arity =
  1263   let
  1264     val _ = @{assert} (arity > 0)
  1265     val vars =
  1266       upto (1, arity + 1)
  1267       |> map (fn i => Free ("x" ^ Int.toString i, HOLogic.boolT))
  1268 
  1269     val consequent = hd vars
  1270     val antecedents = tl vars
  1271 
  1272     val conc =
  1273       fold
  1274        (curry HOLogic.mk_conj)
  1275        (map (fn var => HOLogic.mk_eq (var, @{term True})) antecedents)
  1276        (HOLogic.mk_eq (consequent, @{term False}))
  1277 
  1278     val pre_hyp =
  1279       case kind of
  1280           TPTP_Reconstruct.Conjunctive NONE =>
  1281             curry HOLogic.mk_imp
  1282              (if length antecedents = 1 then the_single antecedents
  1283               else
  1284                 fold (curry HOLogic.mk_conj) (tl antecedents) (hd antecedents))
  1285              (hd vars)
  1286         | TPTP_Reconstruct.Implicational NONE =>
  1287             fold (curry HOLogic.mk_imp) antecedents consequent
  1288 
  1289     val hyp = HOLogic.mk_eq (pre_hyp, @{term False})
  1290 
  1291     val t =
  1292       Logic.mk_implies (HOLogic.mk_Trueprop  hyp, HOLogic.mk_Trueprop conc)
  1293   in
  1294     Goal.prove ctxt [] [] t (fn _ => HEADGOAL (blast_tac ctxt))
  1295     |> Drule.export_without_context
  1296   end
  1297 *}
  1298 
  1299 ML {*
  1300 (*Applies a d-tactic, then breaks it up conjunctively.
  1301   This can be used to transform subgoals as follows:
  1302      (A \<longrightarrow> B) = False  \<Longrightarrow> R
  1303               |
  1304               v
  1305   \<lbrakk>A = True; B = False\<rbrakk> \<Longrightarrow> R
  1306 *)
  1307 fun weak_conj_tac ctxt drule =
  1308   dresolve_tac ctxt [drule] THEN'
  1309   (REPEAT_DETERM o eresolve_tac ctxt @{thms conjE})
  1310 *}
  1311 
  1312 ML {*
  1313 fun uncurry_lit_neg_tac ctxt =
  1314   REPEAT_DETERM o
  1315     dresolve_tac ctxt [@{lemma "(A \<longrightarrow> B \<longrightarrow> C) = False \<Longrightarrow> (A & B \<longrightarrow> C) = False" by auto}]
  1316 *}
  1317 
  1318 ML {*
  1319 fun standard_cnf_tac ctxt i = fn st =>
  1320   let
  1321     fun core_tactic i = fn st =>
  1322       case standard_cnf_type ctxt i st of
  1323           NONE => no_tac st
  1324         | SOME (kind, arity, _) =>
  1325             let
  1326               val rule = mk_standard_cnf ctxt kind arity;
  1327             in
  1328               (weak_conj_tac ctxt rule THEN' assume_tac ctxt) i st
  1329             end
  1330   in
  1331     (uncurry_lit_neg_tac ctxt
  1332      THEN' TPTP_Reconstruct_Library.reassociate_conjs_tac ctxt
  1333      THEN' core_tactic) i st
  1334   end
  1335 *}
  1336 
  1337 
  1338 subsubsection "Emulator prep"
  1339 
  1340 ML {*
  1341 datatype cleanup_feature =
  1342     RemoveHypothesesFromSkolemDefs
  1343   | RemoveDuplicates
  1344 
  1345 datatype loop_feature =
  1346     Close_Branch
  1347   | ConjI
  1348   | King_Cong
  1349   | Break_Hypotheses
  1350   | Donkey_Cong (*simper_animal + ex_expander_tac*)
  1351   | RemoveRedundantQuantifications
  1352   | Assumption
  1353 
  1354   (*Closely based on Leo2 calculus*)
  1355   | Existential_Free
  1356   | Existential_Var
  1357   | Universal
  1358   | Not_pos
  1359   | Not_neg
  1360   | Or_pos
  1361   | Or_neg
  1362   | Equal_pos
  1363   | Equal_neg
  1364   | Extuni_Bool2
  1365   | Extuni_Bool1
  1366   | Extuni_Dec
  1367   | Extuni_Bind
  1368   | Extuni_Triv
  1369   | Extuni_FlexRigid
  1370   | Extuni_Func
  1371   | Polarity_switch
  1372   | Forall_special_pos
  1373 
  1374 datatype feature =
  1375     ConstsDiff
  1376   | StripQuantifiers
  1377   | Flip_Conclusion
  1378   | Loop of loop_feature list
  1379   | LoopOnce of loop_feature list
  1380   | InnerLoopOnce of loop_feature list
  1381   | CleanUp of cleanup_feature list
  1382   | AbsorbSkolemDefs
  1383 *}
  1384 
  1385 ML {*
  1386 fun can_feature x l =
  1387   let
  1388     fun sublist_of_clean_up el =
  1389       case el of
  1390           CleanUp l'' => SOME l''
  1391         | _ => NONE
  1392     fun sublist_of_loop el =
  1393       case el of
  1394           Loop l'' => SOME l''
  1395         | _ => NONE
  1396     fun sublist_of_loop_once el =
  1397       case el of
  1398           LoopOnce l'' => SOME l''
  1399         | _ => NONE
  1400     fun sublist_of_inner_loop_once el =
  1401       case el of
  1402           InnerLoopOnce l'' => SOME l''
  1403         | _ => NONE
  1404 
  1405     fun check_sublist sought_sublist opt_list =
  1406       if forall is_none opt_list then false
  1407       else
  1408         fold_options opt_list
  1409         |> flat
  1410         |> pair sought_sublist
  1411         |> subset (op =)
  1412   in
  1413     case x of
  1414         CleanUp l' =>
  1415           map sublist_of_clean_up l
  1416           |> check_sublist l'
  1417       | Loop l' =>
  1418           map sublist_of_loop l
  1419           |> check_sublist l'
  1420       | LoopOnce l' =>
  1421           map sublist_of_loop_once l
  1422           |> check_sublist l'
  1423       | InnerLoopOnce l' =>
  1424           map sublist_of_inner_loop_once l
  1425           |> check_sublist l'
  1426       | _ => exists (curry (op =) x) l
  1427   end;
  1428 
  1429 fun loop_can_feature loop_feats l =
  1430   can_feature (Loop loop_feats) l orelse
  1431   can_feature (LoopOnce loop_feats) l orelse
  1432   can_feature (InnerLoopOnce loop_feats) l;
  1433 
  1434 @{assert} (can_feature ConstsDiff [StripQuantifiers, ConstsDiff]);
  1435 
  1436 @{assert}
  1437   (can_feature (CleanUp [RemoveHypothesesFromSkolemDefs])
  1438     [CleanUp [RemoveHypothesesFromSkolemDefs, RemoveDuplicates]]);
  1439 
  1440 @{assert}
  1441   (can_feature (Loop []) [Loop [Existential_Var]]);
  1442 
  1443 @{assert}
  1444   (not (can_feature (Loop []) [InnerLoopOnce [Existential_Var]]));
  1445 *}
  1446 
  1447 ML {*
  1448 exception NO_LOOP_FEATS
  1449 fun get_loop_feats (feats : feature list) =
  1450   let
  1451     val loop_find =
  1452       fold (fn x => fn loop_feats_acc =>
  1453         if is_some loop_feats_acc then loop_feats_acc
  1454         else
  1455           case x of
  1456               Loop loop_feats => SOME loop_feats
  1457             | LoopOnce loop_feats => SOME loop_feats
  1458             | InnerLoopOnce loop_feats => SOME loop_feats
  1459             | _ => NONE)
  1460        feats
  1461        NONE
  1462   in
  1463     if is_some loop_find then the loop_find
  1464     else raise NO_LOOP_FEATS
  1465   end;
  1466 
  1467 @{assert}
  1468   (get_loop_feats [Loop [King_Cong, Break_Hypotheses, Existential_Free, Existential_Var, Universal]] =
  1469    [King_Cong, Break_Hypotheses, Existential_Free, Existential_Var, Universal])
  1470 *}
  1471 
  1472 (*use as elim rule to remove premises*)
  1473 lemma insa_prems: "\<lbrakk>Q; P\<rbrakk> \<Longrightarrow> P" by auto
  1474 ML {*
  1475 fun cleanup_skolem_defs ctxt feats =
  1476   let
  1477     (*remove hypotheses from skolem defs,
  1478      after testing that they look like skolem defs*)
  1479     val dehypothesise_skolem_defs =
  1480       COND' (SOME #> TERMPRED (fn _ => true) conc_is_skolem_def)
  1481         (REPEAT_DETERM o eresolve_tac ctxt @{thms insa_prems})
  1482         (K no_tac)
  1483   in
  1484     if can_feature (CleanUp [RemoveHypothesesFromSkolemDefs]) feats then
  1485       ALLGOALS (TRY o dehypothesise_skolem_defs)
  1486     else all_tac
  1487   end
  1488 *}
  1489 
  1490 ML {*
  1491 fun remove_duplicates_tac feats =
  1492   (if can_feature (CleanUp [RemoveDuplicates]) feats then
  1493      ALLGOALS distinct_subgoal_tac
  1494    else all_tac)
  1495 *}
  1496 
  1497 ML {*
  1498 (*given a goal state, indicates the skolem constants committed-to in it (i.e. appearing in LHS of a skolem definition)*)
  1499 fun which_skolem_concs_used ctxt = fn st =>
  1500   let
  1501     val feats = [CleanUp [RemoveHypothesesFromSkolemDefs, RemoveDuplicates]]
  1502     val scrubup_tac =
  1503       cleanup_skolem_defs ctxt feats
  1504       THEN remove_duplicates_tac feats
  1505   in
  1506     scrubup_tac st
  1507     |> break_seq
  1508     |> tap (fn (_, rest) => @{assert} (null (Seq.list_of rest)))
  1509     |> fst
  1510     |> TERMFUN (snd (*discard hypotheses*)
  1511                  #> get_skolem_conc_const) NONE
  1512     |> switch (fold (fn x => fn l => if is_some x then the x :: l else l)) []
  1513     |> map Const
  1514   end
  1515 *}
  1516 
  1517 ML {*
  1518 fun exists_tac ctxt feats consts_diff =
  1519   let
  1520     val ex_var =
  1521       if loop_can_feature [Existential_Var] feats andalso consts_diff <> [] then
  1522         new_skolem_tac ctxt consts_diff
  1523         (*We're making sure that each skolem constant is used once in instantiations.*)
  1524       else K no_tac
  1525 
  1526     val ex_free =
  1527       if loop_can_feature [Existential_Free] feats andalso consts_diff = [] then
  1528         eresolve_tac ctxt @{thms polar_exE}
  1529       else K no_tac
  1530   in
  1531     ex_var APPEND' ex_free
  1532   end
  1533 
  1534 fun forall_tac ctxt feats =
  1535   if loop_can_feature [Universal] feats then
  1536     forall_pos_tac ctxt
  1537   else K no_tac
  1538 *}
  1539 
  1540 
  1541 subsubsection "Finite types"
  1542 (*lift quantification from a singleton literal to a singleton clause*)
  1543 lemma forall_pos_lift:
  1544 "\<lbrakk>(! X. P X) = True; ! X. (P X = True) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" by auto
  1545 
  1546 (*predicate over the type of the leading quantified variable*)
  1547 
  1548 ML {*
  1549 fun extcnf_forall_special_pos_tac ctxt =
  1550   let
  1551     val bool =
  1552       ["True", "False"]
  1553 
  1554     val bool_to_bool =
  1555       ["% _ . True", "% _ . False", "% x . x", "Not"]
  1556 
  1557     val tacs =
  1558       map (fn t_s =>  (* FIXME proper context!? *)
  1559        Rule_Insts.eres_inst_tac @{context} [((("x", 0), Position.none), t_s)] [] @{thm allE}
  1560        THEN' assume_tac ctxt)
  1561   in
  1562     (TRY o eresolve_tac ctxt @{thms forall_pos_lift})
  1563     THEN' (assume_tac ctxt
  1564            ORELSE' FIRST'
  1565             (*FIXME could check the type of the leading quantified variable, instead of trying everything*)
  1566             (tacs (bool @ bool_to_bool)))
  1567   end
  1568 *}
  1569 
  1570 
  1571 subsubsection "Emulator"
  1572 
  1573 lemma efq: "[|A = True; A = False|] ==> R" by auto
  1574 ML {*
  1575 fun efq_tac ctxt =
  1576   (eresolve_tac ctxt @{thms efq} THEN' assume_tac ctxt)
  1577   ORELSE' assume_tac ctxt
  1578 *}
  1579 
  1580 ML {*
  1581 (*This is applied to all subgoals, repeatedly*)
  1582 fun extcnf_combined_main ctxt feats consts_diff =
  1583   let
  1584     (*This is applied to subgoals which don't have a conclusion
  1585       consisting of a Skolem definition*)
  1586     fun extcnf_combined_tac' ctxt i = fn st =>
  1587       let
  1588         val skolem_consts_used_so_far = which_skolem_concs_used ctxt st
  1589         val consts_diff' = subtract (op =) skolem_consts_used_so_far consts_diff
  1590 
  1591         fun feat_to_tac feat =
  1592           case feat of
  1593               Close_Branch => trace_tac' ctxt "mark: closer" (efq_tac ctxt)
  1594             | ConjI => trace_tac' ctxt "mark: conjI" (resolve_tac ctxt @{thms conjI})
  1595             | King_Cong => trace_tac' ctxt "mark: expander_animal" (expander_animal ctxt)
  1596             | Break_Hypotheses => trace_tac' ctxt "mark: break_hypotheses" (break_hypotheses_tac ctxt)
  1597             | RemoveRedundantQuantifications => K all_tac
  1598 (*
  1599 FIXME Building this into the loop instead.. maybe not the ideal choice
  1600             | RemoveRedundantQuantifications =>
  1601                 trace_tac' ctxt "mark: strip_unused_variable_hyp"
  1602                  (REPEAT_DETERM o remove_redundant_quantification_in_lit)
  1603 *)
  1604 
  1605             | Assumption => assume_tac ctxt
  1606 (*FIXME both Existential_Free and Existential_Var run same code*)
  1607             | Existential_Free => trace_tac' ctxt "mark: forall_neg" (exists_tac ctxt feats consts_diff')
  1608             | Existential_Var => trace_tac' ctxt "mark: forall_neg" (exists_tac ctxt feats consts_diff')
  1609             | Universal => trace_tac' ctxt "mark: forall_pos" (forall_tac ctxt feats)
  1610             | Not_pos => trace_tac' ctxt "mark: not_pos" (dresolve_tac ctxt @{thms leo2_rules(9)})
  1611             | Not_neg => trace_tac' ctxt "mark: not_neg" (dresolve_tac ctxt @{thms leo2_rules(10)})
  1612             | Or_pos => trace_tac' ctxt "mark: or_pos" (dresolve_tac ctxt @{thms leo2_rules(5)}) (*could add (6) for negated conjunction*)
  1613             | Or_neg => trace_tac' ctxt "mark: or_neg" (dresolve_tac ctxt @{thms leo2_rules(7)})
  1614             | Equal_pos => trace_tac' ctxt "mark: equal_pos" (dresolve_tac ctxt (@{thms eq_pos_bool} @ [@{thm leo2_rules(3)}, @{thm eq_pos_func}]))
  1615             | Equal_neg => trace_tac' ctxt "mark: equal_neg" (dresolve_tac ctxt [@{thm eq_neg_bool}, @{thm leo2_rules(4)}])
  1616             | Donkey_Cong => trace_tac' ctxt "mark: donkey_cong" (simper_animal ctxt THEN' ex_expander_tac ctxt)
  1617 
  1618             | Extuni_Bool2 => trace_tac' ctxt "mark: extuni_bool2" (dresolve_tac ctxt @{thms extuni_bool2})
  1619             | Extuni_Bool1 => trace_tac' ctxt "mark: extuni_bool1" (dresolve_tac ctxt @{thms extuni_bool1})
  1620             | Extuni_Bind => trace_tac' ctxt "mark: extuni_triv" (eresolve_tac ctxt @{thms extuni_triv})
  1621             | Extuni_Triv => trace_tac' ctxt "mark: extuni_triv" (eresolve_tac ctxt @{thms extuni_triv})
  1622             | Extuni_Dec => trace_tac' ctxt "mark: extuni_dec_tac" (extuni_dec_tac ctxt)
  1623             | Extuni_FlexRigid => trace_tac' ctxt "mark: extuni_flex_rigid" (assume_tac ctxt ORELSE' asm_full_simp_tac ctxt)
  1624             | Extuni_Func => trace_tac' ctxt "mark: extuni_func" (dresolve_tac ctxt @{thms extuni_func})
  1625             | Polarity_switch => trace_tac' ctxt "mark: polarity_switch" (eresolve_tac ctxt @{thms polarity_switch})
  1626             | Forall_special_pos => trace_tac' ctxt "mark: dorall_special_pos" (extcnf_forall_special_pos_tac ctxt)
  1627 
  1628         val core_tac =
  1629           get_loop_feats feats
  1630           |> map feat_to_tac
  1631           |> FIRST'
  1632       in
  1633         core_tac i st
  1634       end
  1635 
  1636     (*This is applied to all subgoals, repeatedly*)
  1637     fun extcnf_combined_tac ctxt i =
  1638       COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1639         no_tac
  1640         (extcnf_combined_tac' ctxt i)
  1641 
  1642     val core_tac = CHANGED (ALLGOALS (IF_UNSOLVED o TRY o extcnf_combined_tac ctxt))
  1643 
  1644     val full_tac = REPEAT core_tac
  1645 
  1646   in
  1647     CHANGED
  1648       (if can_feature (InnerLoopOnce []) feats then
  1649          core_tac
  1650        else full_tac)
  1651   end
  1652 
  1653 val interpreted_consts =
  1654   [@{const_name HOL.All}, @{const_name HOL.Ex},
  1655    @{const_name Hilbert_Choice.Eps},
  1656    @{const_name HOL.conj},
  1657    @{const_name HOL.disj},
  1658    @{const_name HOL.eq},
  1659    @{const_name HOL.implies},
  1660    @{const_name HOL.The},
  1661    @{const_name HOL.Ex1},
  1662    @{const_name HOL.Not},
  1663    (* @{const_name HOL.iff}, *) (*FIXME do these exist?*)
  1664    (* @{const_name HOL.not_equal}, *)
  1665    @{const_name HOL.False},
  1666    @{const_name HOL.True},
  1667    @{const_name Pure.imp}]
  1668 
  1669 fun strip_qtfrs_tac ctxt =
  1670   REPEAT_DETERM (HEADGOAL (resolve_tac ctxt @{thms allI}))
  1671   THEN REPEAT_DETERM (HEADGOAL (eresolve_tac ctxt @{thms exE}))
  1672   THEN HEADGOAL (canonicalise_qtfr_order ctxt)
  1673   THEN
  1674     ((REPEAT (HEADGOAL (nominal_inst_parametermatch_tac ctxt @{thm allE})))
  1675      APPEND (REPEAT (HEADGOAL (inst_parametermatch_tac ctxt [@{thm allE}]))))
  1676   (*FIXME need to handle "@{thm exI}"?*)
  1677 
  1678 (*difference in constants between the hypothesis clause and the conclusion clause*)
  1679 fun clause_consts_diff thm =
  1680   let
  1681     val t =
  1682       Thm.prop_of thm
  1683       |> Logic.dest_implies
  1684       |> fst
  1685 
  1686       (*This bit should not be needed, since Leo2 inferences don't have parameters*)
  1687       |> TPTP_Reconstruct.strip_top_all_vars []
  1688       |> snd
  1689 
  1690     val do_diff =
  1691       Logic.dest_implies
  1692       #> uncurry TPTP_Reconstruct.new_consts_between
  1693       #> filter
  1694            (fn Const (n, _) =>
  1695              not (member (op =) interpreted_consts n))
  1696   in
  1697     if head_of t = Logic.implies then do_diff t
  1698     else []
  1699   end
  1700 *}
  1701 
  1702 ML {*
  1703 (*remove quantification in hypothesis clause (! X. t), if
  1704   X not free in t*)
  1705 fun remove_redundant_quantification ctxt i = fn st =>
  1706   let
  1707     val gls =
  1708       Thm.prop_of st
  1709       |> Logic.strip_horn
  1710       |> fst
  1711   in
  1712     if null gls then raise NO_GOALS
  1713     else
  1714       let
  1715         val (params, (hyp_clauses, conc_clause)) =
  1716           rpair (i - 1) gls
  1717           |> uncurry nth
  1718           |> TPTP_Reconstruct.strip_top_all_vars []
  1719           |> apsnd Logic.strip_horn
  1720       in
  1721         (*this is to fail gracefully in case this tactic is applied to a goal which doesn't have a single hypothesis*)
  1722         if length hyp_clauses > 1 then no_tac st
  1723         else
  1724           let
  1725             val hyp_clause = the_single hyp_clauses
  1726             val sep_prefix =
  1727               HOLogic.dest_Trueprop
  1728               #> TPTP_Reconstruct.strip_top_All_vars
  1729               #> apfst rev
  1730             val (hyp_prefix, hyp_body) = sep_prefix hyp_clause
  1731             val (conc_prefix, conc_body) = sep_prefix conc_clause
  1732           in
  1733             if null hyp_prefix orelse
  1734               member (op =) conc_prefix (hd hyp_prefix) orelse
  1735               member (op =)  (Term.add_frees hyp_body []) (hd hyp_prefix) then
  1736               no_tac st
  1737             else
  1738               Rule_Insts.eres_inst_tac ctxt [((("x", 0), Position.none), "(@X. False)")] []
  1739                 @{thm allE} i st
  1740           end
  1741      end
  1742   end
  1743 *}
  1744 
  1745 ML {*
  1746 fun remove_redundant_quantification_ignore_skolems ctxt i =
  1747   COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1748     no_tac
  1749     (remove_redundant_quantification ctxt i)
  1750 *}
  1751 
  1752 lemma drop_redundant_literal_qtfr:
  1753   "(! X. P) = True \<Longrightarrow> P = True"
  1754   "(? X. P) = True \<Longrightarrow> P = True"
  1755   "(! X. P) = False \<Longrightarrow> P = False"
  1756   "(? X. P) = False \<Longrightarrow> P = False"
  1757 by auto
  1758 
  1759 ML {*
  1760 (*remove quantification in the literal "(! X. t) = True/False"
  1761   in the singleton hypothesis clause, if X not free in t*)
  1762 fun remove_redundant_quantification_in_lit ctxt i = fn st =>
  1763   let
  1764     val gls =
  1765       Thm.prop_of st
  1766       |> Logic.strip_horn
  1767       |> fst
  1768   in
  1769     if null gls then raise NO_GOALS
  1770     else
  1771       let
  1772         val (params, (hyp_clauses, conc_clause)) =
  1773           rpair (i - 1) gls
  1774           |> uncurry nth
  1775           |> TPTP_Reconstruct.strip_top_all_vars []
  1776           |> apsnd Logic.strip_horn
  1777       in
  1778         (*this is to fail gracefully in case this tactic is applied to a goal which doesn't have a single hypothesis*)
  1779         if length hyp_clauses > 1 then no_tac st
  1780         else
  1781           let
  1782             fun literal_content (Const (@{const_name HOL.eq}, _) $ lhs $ (rhs as @{term True})) = SOME (lhs, rhs)
  1783               | literal_content (Const (@{const_name HOL.eq}, _) $ lhs $ (rhs as @{term False})) = SOME (lhs, rhs)
  1784               | literal_content t = NONE
  1785 
  1786             val hyp_clause =
  1787               the_single hyp_clauses
  1788               |> HOLogic.dest_Trueprop
  1789               |> literal_content
  1790 
  1791           in
  1792             if is_none hyp_clause then
  1793               no_tac st
  1794             else
  1795               let
  1796                 val (hyp_lit_prefix, hyp_lit_body) =
  1797                   the hyp_clause
  1798                   |> (fn (t, polarity) =>
  1799                        TPTP_Reconstruct.strip_top_All_vars t
  1800                        |> apfst rev)
  1801               in
  1802                 if null hyp_lit_prefix orelse
  1803                   member (op =)  (Term.add_frees hyp_lit_body []) (hd hyp_lit_prefix) then
  1804                   no_tac st
  1805                 else
  1806                   dresolve_tac ctxt @{thms drop_redundant_literal_qtfr} i st
  1807               end
  1808           end
  1809      end
  1810   end
  1811 *}
  1812 
  1813 ML {*
  1814 fun remove_redundant_quantification_in_lit_ignore_skolems ctxt i =
  1815   COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1816     no_tac
  1817     (remove_redundant_quantification_in_lit ctxt i)
  1818 *}
  1819 
  1820 ML {*
  1821 fun extcnf_combined_tac ctxt prob_name_opt feats skolem_consts = fn st =>
  1822   let
  1823     val thy = Proof_Context.theory_of ctxt
  1824 
  1825     (*Initially, st consists of a single goal, showing the
  1826       hypothesis clause implying the conclusion clause.
  1827       There are no parameters.*)
  1828     val consts_diff =
  1829       union (op =) skolem_consts
  1830        (if can_feature ConstsDiff feats then
  1831           clause_consts_diff st
  1832         else [])
  1833 
  1834     val main_tac =
  1835       if can_feature (LoopOnce []) feats orelse can_feature (InnerLoopOnce []) feats then
  1836         extcnf_combined_main ctxt feats consts_diff
  1837       else if can_feature (Loop []) feats then
  1838         BEST_FIRST (TERMPRED (fn _ => true) conc_is_skolem_def NONE, size_of_thm)
  1839 (*FIXME maybe need to weaken predicate to include "solved form"?*)
  1840          (extcnf_combined_main ctxt feats consts_diff)
  1841       else all_tac (*to allow us to use the cleaning features*)
  1842 
  1843     (*Remove hypotheses from Skolem definitions,
  1844       then remove duplicate subgoals,
  1845       then we should be left with skolem definitions:
  1846         absorb them as axioms into the theory.*)
  1847     val cleanup =
  1848       cleanup_skolem_defs ctxt feats
  1849       THEN remove_duplicates_tac feats
  1850       THEN (if can_feature AbsorbSkolemDefs feats then
  1851               ALLGOALS (absorb_skolem_def ctxt prob_name_opt)
  1852             else all_tac)
  1853 
  1854     val have_loop_feats =
  1855       (get_loop_feats feats; true)
  1856       handle NO_LOOP_FEATS => false
  1857 
  1858     val tec =
  1859       (if can_feature StripQuantifiers feats then
  1860          (REPEAT (CHANGED (strip_qtfrs_tac ctxt)))
  1861        else all_tac)
  1862       THEN (if can_feature Flip_Conclusion feats then
  1863              HEADGOAL (flip_conclusion_tac ctxt)
  1864            else all_tac)
  1865 
  1866       (*after stripping the quantifiers any remaining quantifiers
  1867         can be simply eliminated -- they're redundant*)
  1868       (*FIXME instead of just using allE, instantiate to a silly
  1869          term, to remove opportunities for unification.*)
  1870       THEN (REPEAT_DETERM (eresolve_tac ctxt @{thms allE} 1))
  1871 
  1872       THEN (REPEAT_DETERM (resolve_tac ctxt @{thms allI} 1))
  1873 
  1874       THEN (if have_loop_feats then
  1875               REPEAT (CHANGED
  1876               ((ALLGOALS (TRY o clause_breaker_tac ctxt)) (*brush away literals which don't change*)
  1877                THEN
  1878                 (*FIXME move this to a different level?*)
  1879                 (if loop_can_feature [Polarity_switch] feats then
  1880                    all_tac
  1881                  else
  1882                    (TRY (IF_UNSOLVED (HEADGOAL (remove_redundant_quantification_ignore_skolems ctxt))))
  1883                    THEN (TRY (IF_UNSOLVED (HEADGOAL (remove_redundant_quantification_in_lit_ignore_skolems ctxt)))))
  1884                THEN (TRY main_tac)))
  1885             else
  1886               all_tac)
  1887       THEN IF_UNSOLVED cleanup
  1888 
  1889   in
  1890     DEPTH_SOLVE (CHANGED tec) st
  1891   end
  1892 *}
  1893 
  1894 
  1895 subsubsection "unfold_def"
  1896 
  1897 (*this is used when handling unfold_tac, because the skeleton includes the definitions conjoined with the goal. it turns out that, for my tactic, the definitions are harmful. instead of modifying the skeleton (which may be nontrivial) i'm just dropping the information using this lemma. obviously, and from the name, order matters here.*)
  1898 lemma drop_first_hypothesis [rule_format]: "\<lbrakk>A; B\<rbrakk> \<Longrightarrow> B" by auto
  1899 
  1900 (*Unfold_def works by reducing the goal to a meta equation,
  1901   then working on it until it can be discharged by atac,
  1902   or reflexive, or else turned back into an object equation
  1903   and broken down further.*)
  1904 lemma un_meta_polarise: "(X \<equiv> True) \<Longrightarrow> X" by auto
  1905 lemma meta_polarise: "X \<Longrightarrow> X \<equiv> True" by auto
  1906 
  1907 ML {*
  1908 fun unfold_def_tac ctxt depends_on_defs = fn st =>
  1909   let
  1910     (*This is used when we end up with something like
  1911         (A & B) \<equiv> True \<Longrightarrow> (B & A) \<equiv> True.
  1912       It breaks down this subgoal until it can be trivially
  1913       discharged.
  1914      *)
  1915     val kill_meta_eqs_tac =
  1916       dresolve_tac ctxt @{thms un_meta_polarise}
  1917       THEN' resolve_tac ctxt @{thms meta_polarise}
  1918       THEN' (REPEAT_DETERM o (eresolve_tac ctxt @{thms conjE}))
  1919       THEN' (REPEAT_DETERM o (resolve_tac ctxt @{thms conjI} ORELSE' assume_tac ctxt))
  1920 
  1921     val continue_reducing_tac =
  1922       resolve_tac ctxt @{thms meta_eq_to_obj_eq} 1
  1923       THEN (REPEAT_DETERM (ex_expander_tac ctxt 1))
  1924       THEN TRY (polarise_subgoal_hyps ctxt 1) (*no need to REPEAT_DETERM here, since there should only be one hypothesis*)
  1925       THEN TRY (dresolve_tac ctxt @{thms eq_reflection} 1)
  1926       THEN (TRY ((CHANGED o rewrite_goal_tac ctxt
  1927               (@{thm expand_iff} :: @{thms simp_meta})) 1))
  1928       THEN HEADGOAL (resolve_tac ctxt @{thms reflexive}
  1929                      ORELSE' assume_tac ctxt
  1930                      ORELSE' kill_meta_eqs_tac)
  1931 
  1932     val tactic =
  1933       (resolve_tac ctxt @{thms polarise} 1 THEN assume_tac ctxt 1)
  1934       ORELSE
  1935         (REPEAT_DETERM (eresolve_tac ctxt @{thms conjE} 1 THEN
  1936           eresolve_tac ctxt @{thms drop_first_hypothesis} 1)
  1937          THEN PRIMITIVE (Conv.fconv_rule Thm.eta_long_conversion)
  1938          THEN (REPEAT_DETERM (ex_expander_tac ctxt 1))
  1939          THEN (TRY ((CHANGED o rewrite_goal_tac ctxt @{thms simp_meta}) 1))
  1940          THEN PRIMITIVE (Conv.fconv_rule Thm.eta_long_conversion)
  1941          THEN
  1942            (HEADGOAL (assume_tac ctxt)
  1943            ORELSE
  1944             (unfold_tac ctxt depends_on_defs
  1945              THEN IF_UNSOLVED continue_reducing_tac)))
  1946   in
  1947     tactic st
  1948   end
  1949 *}
  1950 
  1951 
  1952 subsection "Handling split 'preprocessing'"
  1953 
  1954 lemma split_tranfs:
  1955   "! x. P x & Q x \<equiv> (! x. P x) & (! x. Q x)"
  1956   "~ (~ A) \<equiv> A"
  1957   "? x. A \<equiv> A"
  1958   "(A & B) & C \<equiv> A & B & C"
  1959   "A = B \<equiv> (A --> B) & (B --> A)"
  1960 by (rule eq_reflection, auto)+
  1961 
  1962 (*Same idiom as ex_expander_tac*)
  1963 ML {*
  1964 fun split_simp_tac (ctxt : Proof.context) i =
  1965    let
  1966      val simpset =
  1967        fold Simplifier.add_simp @{thms split_tranfs} (empty_simpset ctxt)
  1968    in
  1969      CHANGED (asm_full_simp_tac simpset i)
  1970    end
  1971 *}
  1972 
  1973 
  1974 subsection "Alternative reconstruction tactics"
  1975 ML {*
  1976 (*An "auto"-based proof reconstruction, where we attempt to reconstruct each inference
  1977   using auto_tac. A realistic tactic would inspect the inference name and act
  1978   accordingly.*)
  1979 fun auto_based_reconstruction_tac ctxt prob_name n =
  1980   let
  1981     val thy = Proof_Context.theory_of ctxt
  1982     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  1983   in
  1984     TPTP_Reconstruct.inference_at_node
  1985      thy
  1986      prob_name (#meta pannot) n
  1987       |> the
  1988       |> (fn {inference_fmla, ...} =>
  1989           Goal.prove ctxt [] [] inference_fmla
  1990            (fn pdata => auto_tac (#context pdata)))
  1991   end
  1992 *}
  1993 
  1994 (*An oracle-based reconstruction, which is only used to test the shunting part of the system*)
  1995 oracle oracle_iinterp = "fn t => t"
  1996 ML {*
  1997 fun oracle_based_reconstruction_tac ctxt prob_name n =
  1998   let
  1999     val thy = Proof_Context.theory_of ctxt
  2000     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  2001   in
  2002     TPTP_Reconstruct.inference_at_node
  2003      thy
  2004      prob_name (#meta pannot) n
  2005       |> the
  2006       |> (fn {inference_fmla, ...} => Thm.cterm_of ctxt inference_fmla)
  2007       |> oracle_iinterp
  2008   end
  2009 *}
  2010 
  2011 
  2012 subsection "Leo2 reconstruction tactic"
  2013 
  2014 ML {*
  2015 exception UNSUPPORTED_ROLE
  2016 exception INTERPRET_INFERENCE
  2017 
  2018 (*Failure reports can be adjusted to avoid interrupting
  2019   an overall reconstruction process*)
  2020 fun fail ctxt x =
  2021   if unexceptional_reconstruction ctxt then
  2022     (warning x; raise INTERPRET_INFERENCE)
  2023   else error x
  2024 
  2025 fun interpret_leo2_inference_tac ctxt prob_name node =
  2026   let
  2027     val thy = Proof_Context.theory_of ctxt
  2028 
  2029     val _ =
  2030       if Config.get ctxt tptp_trace_reconstruction then
  2031         tracing ("interpret_inference reconstructing node" ^ node ^ " of " ^ TPTP_Problem_Name.mangle_problem_name prob_name)
  2032       else ()
  2033 
  2034     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  2035 
  2036     fun nonfull_extcnf_combined_tac feats =
  2037       extcnf_combined_tac ctxt (SOME prob_name)
  2038        [ConstsDiff,
  2039         StripQuantifiers,
  2040         InnerLoopOnce (Break_Hypotheses :: (*FIXME RemoveRedundantQuantifications :: *) feats),
  2041         AbsorbSkolemDefs]
  2042        []
  2043 
  2044     val source_inf_opt =
  2045       AList.lookup (op =) (#meta pannot)
  2046       #> the
  2047       #> #source_inf_opt
  2048 
  2049     (*FIXME integrate this with other lookup code, or in the early analysis*)
  2050     local
  2051       fun node_is_of_role role node =
  2052         AList.lookup (op =) (#meta pannot) node |> the
  2053         |> #role
  2054         |> (fn role' => role = role')
  2055 
  2056       fun roled_dependencies_names role =
  2057         let
  2058           fun values () =
  2059             case role of
  2060                 TPTP_Syntax.Role_Definition =>
  2061                   map (apsnd Binding.name_of) (#defs pannot)
  2062               | TPTP_Syntax.Role_Axiom =>
  2063                   map (apsnd Binding.name_of) (#axs pannot)
  2064               | _ => raise UNSUPPORTED_ROLE
  2065           in
  2066             if is_none (source_inf_opt node) then []
  2067             else
  2068               case the (source_inf_opt node) of
  2069                   TPTP_Proof.Inference (_, _, parent_inf) =>
  2070                     map TPTP_Proof.parent_name parent_inf
  2071                     |> filter (node_is_of_role role)
  2072                     |> (*FIXME currently definitions are not
  2073                          included in the proof annotations, so
  2074                          i'm using all the definitions available
  2075                          in the proof. ideally i should only
  2076                          use the ones in the proof annotation.*)
  2077                        (fn x =>
  2078                          if role = TPTP_Syntax.Role_Definition then
  2079                            let fun values () = map (apsnd Binding.name_of) (#defs pannot)
  2080                            in
  2081                              map snd (values ())
  2082                            end
  2083                          else
  2084                          map (fn node => AList.lookup (op =) (values ()) node |> the) x)
  2085                 | _ => []
  2086          end
  2087 
  2088       val roled_dependencies =
  2089         roled_dependencies_names
  2090         #> map (Global_Theory.get_thm thy)
  2091     in
  2092       val depends_on_defs = roled_dependencies TPTP_Syntax.Role_Definition
  2093       val depends_on_axs = roled_dependencies TPTP_Syntax.Role_Axiom
  2094       val depends_on_defs_names = roled_dependencies_names TPTP_Syntax.Role_Definition
  2095     end
  2096 
  2097     fun get_binds source_inf_opt =
  2098       case the source_inf_opt of
  2099           TPTP_Proof.Inference (_, _, parent_inf) =>
  2100             maps
  2101               (fn TPTP_Proof.Parent _ => []
  2102                 | TPTP_Proof.ParentWithDetails (_, parent_details) => parent_details)
  2103               parent_inf
  2104         | _ => []
  2105 
  2106     val inference_name =
  2107       case TPTP_Reconstruct.inference_at_node thy prob_name (#meta pannot) node of
  2108           NONE => fail ctxt "Cannot reconstruct rule: no information"
  2109         | SOME {inference_name, ...} => inference_name
  2110     val default_tac = HEADGOAL (blast_tac ctxt)
  2111   in
  2112     case inference_name of
  2113       "fo_atp_e" =>
  2114         HEADGOAL (Metis_Tactic.metis_tac [] ATP_Problem_Generate.combs_or_liftingN ctxt [])
  2115         (*NOTE To treat E as an oracle use the following line:
  2116         HEADGOAL (etac (oracle_based_reconstruction_tac ctxt prob_name node))
  2117         *)
  2118     | "copy" =>
  2119          HEADGOAL
  2120           (assume_tac ctxt
  2121            ORELSE'
  2122               (resolve_tac ctxt @{thms polarise}
  2123                THEN' assume_tac ctxt))
  2124     | "polarity_switch" => nonfull_extcnf_combined_tac [Polarity_switch]
  2125     | "solved_all_splits" => solved_all_splits_tac ctxt
  2126     | "extcnf_not_pos" => nonfull_extcnf_combined_tac [Not_pos]
  2127     | "extcnf_forall_pos" => nonfull_extcnf_combined_tac [Universal]
  2128     | "negate_conjecture" => fail ctxt "Should not handle negate_conjecture here"
  2129     | "unfold_def" => unfold_def_tac ctxt depends_on_defs
  2130     | "extcnf_not_neg" => nonfull_extcnf_combined_tac [Not_neg]
  2131     | "extcnf_or_neg" => nonfull_extcnf_combined_tac [Or_neg]
  2132     | "extcnf_equal_pos" => nonfull_extcnf_combined_tac [Equal_pos]
  2133     | "extcnf_equal_neg" => nonfull_extcnf_combined_tac [Equal_neg]
  2134     | "extcnf_forall_special_pos" =>
  2135          nonfull_extcnf_combined_tac [Forall_special_pos]
  2136          ORELSE HEADGOAL (blast_tac ctxt)
  2137     | "extcnf_or_pos" => nonfull_extcnf_combined_tac [Or_pos]
  2138     | "extuni_bool2" => nonfull_extcnf_combined_tac [Extuni_Bool2]
  2139     | "extuni_bool1" => nonfull_extcnf_combined_tac [Extuni_Bool1]
  2140     | "extuni_dec" =>
  2141         HEADGOAL (assume_tac ctxt)
  2142         ORELSE nonfull_extcnf_combined_tac [Extuni_Dec]
  2143     | "extuni_bind" => nonfull_extcnf_combined_tac [Extuni_Bind]
  2144     | "extuni_triv" => nonfull_extcnf_combined_tac [Extuni_Triv]
  2145     | "extuni_flex_rigid" => nonfull_extcnf_combined_tac [Extuni_FlexRigid]
  2146     | "prim_subst" => nonfull_extcnf_combined_tac [Assumption]
  2147     | "bind" =>
  2148         let
  2149           val ordered_binds = get_binds (source_inf_opt node)
  2150         in
  2151           bind_tac ctxt prob_name ordered_binds
  2152         end
  2153     | "standard_cnf" => HEADGOAL (standard_cnf_tac ctxt)
  2154     | "extcnf_forall_neg" =>
  2155         nonfull_extcnf_combined_tac
  2156          [Existential_Var(* , RemoveRedundantQuantifications *)] (*FIXME RemoveRedundantQuantifications*)
  2157     | "extuni_func" =>
  2158         nonfull_extcnf_combined_tac [Extuni_Func, Existential_Var]
  2159     | "replace_leibnizEQ" => nonfull_extcnf_combined_tac [Assumption]
  2160     | "replace_andrewsEQ" => nonfull_extcnf_combined_tac [Assumption]
  2161     | "split_preprocessing" =>
  2162          (REPEAT (HEADGOAL (split_simp_tac ctxt)))
  2163          THEN TRY (PRIMITIVE (Conv.fconv_rule Thm.eta_long_conversion))
  2164          THEN HEADGOAL (assume_tac ctxt)
  2165 
  2166     (*FIXME some of these could eventually be handled specially*)
  2167     | "fac_restr" => default_tac
  2168     | "sim" => default_tac
  2169     | "res" => default_tac
  2170     | "rename" => default_tac
  2171     | "flexflex" => default_tac
  2172     | other => fail ctxt ("Unknown inference rule: " ^ other)
  2173   end
  2174 *}
  2175 
  2176 ML {*
  2177 fun interpret_leo2_inference ctxt prob_name node =
  2178   let
  2179     val thy = Proof_Context.theory_of ctxt
  2180     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  2181 
  2182     val (inference_name, inference_fmla) =
  2183       case TPTP_Reconstruct.inference_at_node thy prob_name (#meta pannot) node of
  2184           NONE => fail ctxt "Cannot reconstruct rule: no information"
  2185         | SOME {inference_name, inference_fmla, ...} =>
  2186             (inference_name, inference_fmla)
  2187 
  2188     val proof_outcome =
  2189       let
  2190         fun prove () =
  2191           Goal.prove ctxt [] [] inference_fmla
  2192            (fn pdata => interpret_leo2_inference_tac
  2193             (#context pdata) prob_name node)
  2194       in
  2195         if informative_failure ctxt then SOME (prove ())
  2196         else try prove ()
  2197       end
  2198 
  2199   in case proof_outcome of
  2200       NONE => fail ctxt (Pretty.string_of
  2201         (Pretty.block
  2202           [Pretty.str ("Failed inference reconstruction for '" ^
  2203             inference_name ^ "' at node " ^ node ^ ":\n"),
  2204            Syntax.pretty_term ctxt inference_fmla]))
  2205     | SOME thm => thm
  2206   end
  2207 *}
  2208 
  2209 ML {*
  2210 (*filter a set of nodes based on which inference rule was used to
  2211   derive a node*)
  2212 fun nodes_by_inference (fms : TPTP_Reconstruct.formula_meaning list) inference_rule =
  2213   let
  2214     fun fold_fun n l =
  2215       case TPTP_Reconstruct.node_info fms #source_inf_opt n of
  2216           NONE => l
  2217         | SOME (TPTP_Proof.File _) => l
  2218         | SOME (TPTP_Proof.Inference (rule_name, _, _)) =>
  2219             if rule_name = inference_rule then n :: l
  2220             else l
  2221   in
  2222     fold fold_fun (map fst fms) []
  2223   end
  2224 *}
  2225 
  2226 
  2227 section "Importing proofs and reconstructing theorems"
  2228 
  2229 ML {*
  2230 (*Preprocessing carried out on a LEO-II proof.*)
  2231 fun leo2_on_load (pannot : TPTP_Reconstruct.proof_annotation) thy =
  2232   let
  2233     val ctxt = Proof_Context.init_global thy
  2234     val dud = ("", Binding.empty, @{term False})
  2235     val pre_skolem_defs =
  2236       nodes_by_inference (#meta pannot) "extcnf_forall_neg" @
  2237        nodes_by_inference (#meta pannot) "extuni_func"
  2238       |> map (fn x =>
  2239               (interpret_leo2_inference ctxt (#problem_name pannot) x; dud)
  2240                handle NO_SKOLEM_DEF (s, bnd, t) => (s, bnd, t))
  2241       |> filter (fn (x, _, _) => x <> "") (*In case no skolem constants were introduced in that inference*)
  2242     val skolem_defs = map (fn (x, y, _) => (x, y)) pre_skolem_defs
  2243     val thy' =
  2244       fold (fn skolem_def => fn thy =>
  2245              let
  2246                val ((s, thm), thy') = Thm.add_axiom_global skolem_def thy
  2247                (* val _ = warning ("Added skolem definition " ^ s ^ ": " ^  @{make_string thm}) *) (*FIXME use of make_string*)
  2248              in thy' end)
  2249        (map (fn (_, y, z) => (y, z)) pre_skolem_defs)
  2250        thy
  2251   in
  2252     ({problem_name = #problem_name pannot,
  2253       skolem_defs = skolem_defs,
  2254       defs = #defs pannot,
  2255       axs = #axs pannot,
  2256       meta = #meta pannot},
  2257      thy')
  2258   end
  2259 *}
  2260 
  2261 ML {*
  2262 (*Imports and reconstructs a LEO-II proof.*)
  2263 fun reconstruct_leo2 path thy =
  2264   let
  2265     val prob_file = Path.base path
  2266     val dir = Path.dir path
  2267     val thy' = TPTP_Reconstruct.import_thm true [dir, prob_file] path leo2_on_load thy
  2268     val ctxt =
  2269       Context.Theory thy'
  2270       |> Context.proof_of
  2271     val prob_name =
  2272       Path.implode prob_file
  2273       |> TPTP_Problem_Name.parse_problem_name
  2274     val theorem =
  2275       TPTP_Reconstruct.reconstruct ctxt
  2276        (TPTP_Reconstruct.naive_reconstruct_tac ctxt interpret_leo2_inference)
  2277        prob_name
  2278   in
  2279     (*NOTE we could return the theorem value alone, since
  2280        users could get the thy value from the thm value.*)
  2281     (thy', theorem)
  2282   end
  2283 *}
  2284 
  2285 end