src/HOL/TPTP/TPTP_Proof_Reconstruction.thy
author wenzelm
Thu Mar 19 22:30:57 2015 +0100 (2015-03-19)
changeset 59755 f8d164ab0dc1
parent 59639 f596ed647018
child 59760 a996f037c09d
permissions -rw-r--r--
more position information;
     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 (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            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 (rtac @{thm allI})))
   286               THEN (REPEAT_DETERM
   287                     (HEADGOAL
   288                      (nominal_inst_parametermatch_tac ctxt @{thm allE})))
   289               THEN HEADGOAL atac)
   290       in
   291         dtac 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 val polarise_subgoal_hyps =
   309   COND' (SOME #> TERMPRED is_polarised (fn _ => true)) (K no_tac) (dtac @{thm 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 val solved_all_splits_tac =
   340   TRY (etac @{thm conjE} 1)
   341   THEN rtac @{thm solved_all_splits} 1
   342   THEN atac 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 atac)
   437   in
   438     HEADGOAL (canonicalise_qtfr_order ctxt)
   439     THEN (REPEAT_DETERM (HEADGOAL (rtac @{thm 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' rtac @{thm notI}
   474       THEN' (REPEAT_DETERM o etac @{thm 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 thy = Proof_Context.theory_of ctxt
   650 
   651     val gls =
   652       Thm.prop_of st
   653       |> Logic.strip_horn
   654       |> fst
   655 
   656     val parameters =
   657       if null gls then ""
   658       else
   659         rpair (i - 1) gls
   660         |> uncurry nth
   661         |> strip_top_all_vars []
   662         |> fst
   663         |> map fst (*just get the parameter names*)
   664         |> (fn l =>
   665               if null l then ""
   666               else
   667                 space_implode " " l
   668                 |> pair " "
   669                 |> op ^)
   670 
   671   in
   672     if null gls orelse null candidate_consts then no_tac st
   673     else
   674       let
   675         fun instantiate const_name =
   676           dres_inst_tac ctxt [((("sk", 0), Position.none), const_name ^ parameters)]
   677             @{thm leo2_skolemise}
   678         val attempts = map instantiate candidate_consts
   679       in
   680         (fold (curry (op APPEND')) attempts (K no_tac)) i st
   681       end
   682   end
   683 *}
   684 
   685 ML {*
   686 exception SKOLEM_DEF of term (*The tactic wasn't pointed at a skolem definition*)
   687 exception NO_SKOLEM_DEF of (*skolem const name*)string * Binding.binding * term (*The tactic could not find a skolem definition in the theory*)
   688 fun absorb_skolem_def ctxt prob_name_opt i = fn st =>
   689   let
   690     val thy = Proof_Context.theory_of ctxt
   691 
   692     val gls =
   693       Thm.prop_of st
   694       |> Logic.strip_horn
   695       |> fst
   696 
   697     val conclusion =
   698       if null gls then
   699         (*this should never be thrown*)
   700         raise NO_GOALS
   701       else
   702         rpair (i - 1) gls
   703         |> uncurry nth
   704         |> strip_top_all_vars []
   705         |> snd
   706         |> Logic.strip_horn
   707         |> snd
   708 
   709     fun skolem_const_info_of t =
   710       case t of
   711           Const (@{const_name HOL.Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _)) =>
   712           head_of t'
   713           |> 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)*)
   714           |> head_of
   715           |> dest_Const
   716         | _ => raise SKOLEM_DEF t
   717 
   718     val const_name =
   719       skolem_const_info_of conclusion
   720       |> fst
   721 
   722     val def_name = const_name ^ "_def"
   723 
   724     val bnd_def = (*FIXME consts*)
   725       const_name
   726       |> Long_Name.implode o tl o Long_Name.explode (*FIXME hack to drop theory-name prefix*)
   727       |> Binding.qualified_name
   728       |> Binding.suffix_name "_def"
   729 
   730     val bnd_name =
   731       case prob_name_opt of
   732           NONE => bnd_def
   733         | SOME prob_name =>
   734 (*            Binding.qualify false
   735              (TPTP_Problem_Name.mangle_problem_name prob_name)
   736 *)
   737              bnd_def
   738 
   739     val thm =
   740       if Name_Space.defined_entry (Theory.axiom_space thy) def_name then
   741         Thm.axiom thy def_name
   742       else
   743         if is_none prob_name_opt then
   744           (*This mode is for testing, so we can be a bit
   745             looser with theories*)
   746           Thm.add_axiom_global (bnd_name, conclusion) thy
   747           |> fst |> snd
   748         else
   749           raise (NO_SKOLEM_DEF (def_name, bnd_name, conclusion))
   750   in
   751     rtac (Drule.export_without_context thm) i st
   752   end
   753   handle SKOLEM_DEF _ => no_tac st
   754 *}
   755 
   756 ML {*
   757 (*
   758 In current system, there should only be 2 subgoals: the one where
   759 the skolem definition is being built (with a Var in the LHS), and the other subgoal using Var.
   760 *)
   761 (*arity must be greater than 0. if arity=0 then
   762   there's no need to use this expensive matching.*)
   763 fun find_skolem_term ctxt consts_candidate arity = fn st =>
   764   let
   765     val _ = @{assert} (arity > 0)
   766 
   767     val gls =
   768       Thm.prop_of st
   769       |> Logic.strip_horn
   770       |> fst
   771 
   772     (*extract the conclusion of each subgoal*)
   773     val conclusions =
   774       if null gls then
   775         raise NO_GOALS
   776       else
   777         map (strip_top_all_vars [] #> snd #> Logic.strip_horn #> snd) gls
   778         (*Remove skolem-definition conclusion, to avoid wasting time analysing it*)
   779         |> filter (try_dest_Trueprop #> conc_is_skolem_def #> not)
   780         (*There should only be a single goal*) (*FIXME this might not always be the case, in practice*)
   781         (* |> tap (fn x => @{assert} (is_some (try the_single x))) *)
   782 
   783     (*look for subterms headed by a skolem constant, and whose
   784       arguments are all parameter Vars*)
   785     fun get_skolem_terms args (acc : term list) t =
   786       case t of
   787           (c as Const _) $ (v as Free _) =>
   788             if c = consts_candidate andalso
   789              arity = length args + 1 then
   790               (list_comb (c, v :: args)) :: acc
   791             else acc
   792         | t1 $ (v as Free _) =>
   793             get_skolem_terms (v :: args) acc t1 @
   794              get_skolem_terms [] acc t1
   795         | t1 $ t2 =>
   796             get_skolem_terms [] acc t1 @
   797              get_skolem_terms [] acc t2
   798         | Abs (_, _, t') => get_skolem_terms [] acc t'
   799         | _ => acc
   800   in
   801     map (strip_top_All_vars #> snd) conclusions
   802     |> maps (get_skolem_terms [] [])
   803     |> distinct (op =)
   804   end
   805 *}
   806 
   807 ML {*
   808 fun instantiate_skols ctxt consts_candidates i = fn st =>
   809   let
   810     val gls =
   811       Thm.prop_of st
   812       |> Logic.strip_horn
   813       |> fst
   814 
   815     val (params, conclusion) =
   816       if null gls then
   817         raise NO_GOALS
   818       else
   819         rpair (i - 1) gls
   820         |> uncurry nth
   821         |> strip_top_all_vars []
   822         |> apsnd (Logic.strip_horn #> snd)
   823 
   824     fun skolem_const_info_of t =
   825       case t of
   826           Const (@{const_name HOL.Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ lhs $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ rhs)) =>
   827           let
   828             (*the parameters we will concern ourselves with*)
   829             val params' =
   830               Term.add_frees lhs []
   831               |> distinct (op =)
   832             (*check to make sure that params' <= params*)
   833             val _ = @{assert} (forall (member (op =) params) params')
   834             val skolem_const_ty =
   835               let
   836                 val (skolem_const_prety, no_params) =
   837                   Term.strip_comb lhs
   838                   |> apfst (dest_Var #> snd) (*head of lhs consists of a logical variable. we just want its type.*)
   839                   |> apsnd length
   840 
   841                 val _ = @{assert} (length params = no_params)
   842 
   843                 (*get value type of a function type after n arguments have been supplied*)
   844                 fun get_val_ty n ty =
   845                   if n = 0 then ty
   846                   else get_val_ty (n - 1) (dest_funT ty |> snd)
   847               in
   848                 get_val_ty no_params skolem_const_prety
   849               end
   850 
   851           in
   852             (skolem_const_ty, params')
   853           end
   854         | _ => raise (SKOLEM_DEF t)
   855 
   856 (*
   857 find skolem const candidates which, after applying distinct members of params' we end up with, give us something of type skolem_const_ty.
   858 
   859 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.
   860 *)
   861 (*
   862 only returns a single matching -- since terms are linear, and variable arguments are Vars, order shouldn't matter, so we can ignore permutations.
   863 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.
   864 *)
   865     fun use_candidate target_ty params acc cur_ty =
   866       if null params then
   867         if cur_ty = target_ty then
   868           SOME (rev acc)
   869         else NONE
   870       else
   871         let
   872           val (arg_ty, val_ty) = Term.dest_funT cur_ty
   873           (*now find a param of type arg_ty*)
   874           val (candidate_param, params') =
   875             find_and_remove (snd #> pair arg_ty #> op =) params
   876         in
   877           use_candidate target_ty params' (candidate_param :: acc) val_ty
   878         end
   879         handle TYPE ("dest_funT", _, _) => NONE
   880              | DEST_LIST => NONE
   881 
   882     val (skolem_const_ty, params') = skolem_const_info_of conclusion
   883 
   884 (*
   885 For each candidate, build a term and pass it to Thm.instantiate, whic in turn is chained with PRIMITIVE to give us this_tactic.
   886 
   887 Big picture:
   888   we run the following:
   889     drule leo2_skolemise THEN' this_tactic
   890 
   891 NOTE: remember to APPEND' instead of ORELSE' the two tactics relating to skolemisation
   892 *)
   893 
   894     val filtered_candidates =
   895       map (dest_Const
   896            #> snd
   897            #> use_candidate skolem_const_ty params' [])
   898        consts_candidates (* prefiltered_candidates *)
   899       |> pair consts_candidates (* prefiltered_candidates *)
   900       |> ListPair.zip
   901       |> filter (snd #> is_none #> not)
   902       |> map (apsnd the)
   903 
   904     val skolem_terms =
   905       let
   906         fun make_result_t (t, args) =
   907           (* list_comb (t, map Free args) *)
   908           if length args > 0 then
   909             hd (find_skolem_term ctxt t (length args) st)
   910           else t
   911       in
   912         map make_result_t filtered_candidates
   913       end
   914 
   915     (*prefix a skolem term with bindings for the parameters*)
   916     (* val contextualise = fold absdummy (map snd params) *)
   917     val contextualise = fold absfree params
   918 
   919     val skolem_cts = map (contextualise #> Thm.cterm_of ctxt) skolem_terms
   920 
   921 
   922 (*now the instantiation code*)
   923 
   924     (*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.*)
   925     val var_opt =
   926       let
   927         val pre_var =
   928           gls
   929           |> map
   930               (strip_top_all_vars [] #> snd #>
   931                Logic.strip_horn #> snd #>
   932                get_skolem_conc)
   933           |> switch (fold (fn x => fn l => if is_some x then the x :: l else l)) []
   934           |> maps (switch Term.add_vars [])
   935 
   936         fun make_var pre_var =
   937           the_single pre_var
   938           |> Var
   939           |> Thm.cterm_of ctxt
   940           |> SOME
   941       in
   942         if null pre_var then NONE
   943         else make_var pre_var
   944      end
   945 
   946     fun instantiate_tac from to =
   947       Thm.instantiate ([], [(from, to)])
   948       |> PRIMITIVE
   949 
   950     val tectic =
   951       if is_none var_opt then no_tac
   952       else
   953         fold (curry (op APPEND)) (map (instantiate_tac (the var_opt)) skolem_cts) no_tac
   954 
   955   in
   956     tectic st
   957   end
   958 *}
   959 
   960 ML {*
   961 fun new_skolem_tac ctxt consts_candidates =
   962   let
   963     fun tec thm =
   964       dtac thm
   965       THEN' instantiate_skols ctxt consts_candidates
   966   in
   967     if null consts_candidates then K no_tac
   968     else FIRST' (map tec @{thms lift_exists})
   969   end
   970 *}
   971 
   972 (*
   973 need a tactic to expand "? x . P" to "~ ! x. ~ P"
   974 *)
   975 ML {*
   976 fun ex_expander_tac ctxt i =
   977    let
   978      val simpset =
   979        empty_simpset ctxt (*NOTE for some reason, Bind exception gets raised if ctxt's simpset isn't emptied*)
   980        |> Simplifier.add_simp @{lemma "Ex P == (~ (! x. ~ P x))" by auto}
   981    in
   982      CHANGED (asm_full_simp_tac simpset i)
   983    end
   984 *}
   985 
   986 
   987 subsubsection "extuni_dec"
   988 
   989 ML {*
   990 (*n-ary decomposition. Code is based on the n-ary arg_cong generator*)
   991 fun extuni_dec_n ctxt arity =
   992   let
   993     val _ = @{assert} (arity > 0)
   994     val is =
   995       upto (1, arity)
   996       |> map Int.toString
   997     val arg_tys = map (fn i => TFree ("arg" ^ i ^ "_ty", @{sort type})) is
   998     val res_ty = TFree ("res" ^ "_ty", @{sort type})
   999     val f_ty = arg_tys ---> res_ty
  1000     val f = Free ("f", f_ty)
  1001     val xs = map (fn i =>
  1002       Free ("x" ^ i, TFree ("arg" ^ i ^ "_ty", @{sort type}))) is
  1003     (*FIXME DRY principle*)
  1004     val ys = map (fn i =>
  1005       Free ("y" ^ i, TFree ("arg" ^ i ^ "_ty", @{sort type}))) is
  1006 
  1007     val hyp_lhs = list_comb (f, xs)
  1008     val hyp_rhs = list_comb (f, ys)
  1009     val hyp_eq =
  1010       HOLogic.eq_const res_ty $ hyp_lhs $ hyp_rhs
  1011     val hyp =
  1012       HOLogic.eq_const HOLogic.boolT $ hyp_eq $ @{term False}
  1013       |> HOLogic.mk_Trueprop
  1014     fun conc_eq i =
  1015       let
  1016         val ty = TFree ("arg" ^ i ^ "_ty", @{sort type})
  1017         val x = Free ("x" ^ i, ty)
  1018         val y = Free ("y" ^ i, ty)
  1019         val eq = HOLogic.eq_const ty $ x $ y
  1020       in
  1021         HOLogic.eq_const HOLogic.boolT $ eq $ @{term False}
  1022       end
  1023 
  1024     val conc_disjs = map conc_eq is
  1025 
  1026     val conc =
  1027       if length conc_disjs = 1 then
  1028         the_single conc_disjs
  1029       else
  1030         fold
  1031          (fn t => fn t_conc => HOLogic.mk_disj (t_conc, t))
  1032          (tl conc_disjs) (hd conc_disjs)
  1033 
  1034     val t =
  1035       Logic.mk_implies (hyp, HOLogic.mk_Trueprop conc)
  1036   in
  1037     Goal.prove ctxt [] [] t (fn _ => auto_tac ctxt)
  1038     |> Drule.export_without_context
  1039   end
  1040 *}
  1041 
  1042 ML {*
  1043 (*Determine the arity of a function which the "dec"
  1044   unification rule is about to be applied.
  1045   NOTE:
  1046     * Assumes that there is a single hypothesis
  1047 *)
  1048 fun find_dec_arity i = fn st =>
  1049   let
  1050     val gls =
  1051       Thm.prop_of st
  1052       |> Logic.strip_horn
  1053       |> fst
  1054   in
  1055     if null gls then raise NO_GOALS
  1056     else
  1057       let
  1058         val (params, (literal, conc_clause)) =
  1059           rpair (i - 1) gls
  1060           |> uncurry nth
  1061           |> strip_top_all_vars []
  1062           |> apsnd Logic.strip_horn
  1063           |> apsnd (apfst the_single)
  1064 
  1065         val get_ty =
  1066           HOLogic.dest_Trueprop
  1067           #> strip_top_All_vars
  1068           #> snd
  1069           #> HOLogic.dest_eq (*polarity's "="*)
  1070           #> fst
  1071           #> HOLogic.dest_eq (*the unification constraint's "="*)
  1072           #> fst
  1073           #> head_of
  1074           #> dest_Const
  1075           #> snd
  1076 
  1077        fun arity_of ty =
  1078          let
  1079            val (_, res_ty) = dest_funT ty
  1080 
  1081          in
  1082            1 + arity_of res_ty
  1083          end
  1084          handle (TYPE ("dest_funT", _, _)) => 0
  1085 
  1086       in
  1087         arity_of (get_ty literal)
  1088       end
  1089   end
  1090 
  1091 (*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*)
  1092 fun breakdown_inference i = fn st =>
  1093   let
  1094     val gls =
  1095       Thm.prop_of st
  1096       |> Logic.strip_horn
  1097       |> fst
  1098   in
  1099     if null gls then raise NO_GOALS
  1100     else
  1101       rpair (i - 1) gls
  1102       |> uncurry nth
  1103       |> strip_top_all_vars []
  1104   end
  1105 
  1106 (*build a custom elimination rule for extuni_dec, and instantiate it to match a specific subgoal*)
  1107 fun extuni_dec_elim_rule ctxt arity i = fn st =>
  1108   let
  1109     val rule = extuni_dec_n ctxt arity
  1110 
  1111     val rule_hyp =
  1112       Thm.prop_of rule
  1113       |> Logic.dest_implies
  1114       |> fst (*assuming that rule has single hypothesis*)
  1115 
  1116     (*having run break_hypothesis earlier, we know that the hypothesis
  1117       now consists of a single literal. We can (and should)
  1118       disregard the conclusion, since it hasn't been "broken",
  1119       and it might include some unwanted literals -- the latter
  1120       could cause "diff" to fail (since they won't agree with the
  1121       rule we have generated.*)
  1122 
  1123     val inference_hyp =
  1124       snd (breakdown_inference i st)
  1125       |> Logic.dest_implies
  1126       |> fst (*assuming that inference has single hypothesis,
  1127                as explained above.*)
  1128   in
  1129     TPTP_Reconstruct_Library.diff_and_instantiate ctxt rule rule_hyp inference_hyp
  1130   end
  1131 
  1132 fun extuni_dec_tac ctxt i = fn st =>
  1133   let
  1134     val arity = find_dec_arity i st
  1135 
  1136     fun elim_tac i st =
  1137       let
  1138         val rule =
  1139           extuni_dec_elim_rule ctxt arity i st
  1140           (*in case we itroduced free variables during
  1141             instantiation, we generalise the rule to make
  1142             those free variables into logical variables.*)
  1143           |> Thm.forall_intr_frees
  1144           |> Drule.export_without_context
  1145       in dtac rule i st end
  1146       handle NO_GOALS => no_tac st
  1147 
  1148     fun closure tac =
  1149      (*batter fails if there's no toplevel disjunction in the
  1150        hypothesis, so we also try atac*)
  1151       SOLVE o (tac THEN' (batter_tac ctxt ORELSE' assume_tac ctxt))
  1152     val search_tac =
  1153       ASAP
  1154         (rtac @{thm disjI1} APPEND' rtac @{thm disjI2})
  1155         (FIRST' (map closure
  1156                   [dresolve_tac ctxt @{thms dec_commut_eq},
  1157                    dtac @{thm dec_commut_disj},
  1158                    elim_tac]))
  1159   in
  1160     (CHANGED o search_tac) i st
  1161   end
  1162 *}
  1163 
  1164 
  1165 subsubsection "standard_cnf"
  1166 (*Given a standard_cnf inference, normalise it
  1167      e.g. ((A & B) & C \<longrightarrow> D & E \<longrightarrow> F \<longrightarrow> G) = False
  1168      is changed to
  1169           (A & B & C & D & E & F \<longrightarrow> G) = False
  1170  then custom-build a metatheorem which validates this:
  1171           (A & B & C & D & E & F \<longrightarrow> G) = False
  1172        -------------------------------------------
  1173           (A = True) & (B = True) & (C = True) &
  1174           (D = True) & (E = True) & (F = True) & (G = False)
  1175  and apply this metatheorem.
  1176 
  1177 There aren't any "positive" standard_cnfs in Leo2's calculus:
  1178   e.g.,  "(A \<longrightarrow> B) = True \<Longrightarrow> A = False | (A = True & B = True)"
  1179 since "standard_cnf" seems to be applied at the preprocessing
  1180 stage, together with splitting.
  1181 *)
  1182 
  1183 ML {*
  1184 (*Conjunctive counterparts to Term.disjuncts_aux and Term.disjuncts*)
  1185 fun conjuncts_aux (Const (@{const_name HOL.conj}, _) $ t $ t') conjs =
  1186      conjuncts_aux t (conjuncts_aux t' conjs)
  1187   | conjuncts_aux t conjs = t :: conjs
  1188 
  1189 fun conjuncts t = conjuncts_aux t []
  1190 
  1191 (*HOL equivalent of Logic.strip_horn*)
  1192 local
  1193   fun imp_strip_horn' acc (Const (@{const_name HOL.implies}, _) $ A $ B) =
  1194         imp_strip_horn' (A :: acc) B
  1195     | imp_strip_horn' acc t = (acc, t)
  1196 in
  1197   fun imp_strip_horn t =
  1198     imp_strip_horn' [] t
  1199     |> apfst rev
  1200 end
  1201 *}
  1202 
  1203 ML {*
  1204 (*Returns whether the antecedents are separated by conjunctions
  1205   or implications; the number of antecedents; and the polarity
  1206   of the original clause -- I think this will always be "false".*)
  1207 fun standard_cnf_type ctxt i : thm -> (TPTP_Reconstruct.formula_kind * int * bool) option = fn st =>
  1208   let
  1209     val gls =
  1210       Thm.prop_of st
  1211       |> Logic.strip_horn
  1212       |> fst
  1213 
  1214     val hypos =
  1215       if null gls then raise NO_GOALS
  1216       else
  1217         rpair (i - 1) gls
  1218         |> uncurry nth
  1219         |> TPTP_Reconstruct.strip_top_all_vars []
  1220         |> snd
  1221         |> Logic.strip_horn
  1222         |> fst
  1223 
  1224     (*hypothesis clause should be singleton*)
  1225     val _ = @{assert} (length hypos = 1)
  1226 
  1227     val (t, pol) = the_single hypos
  1228       |> try_dest_Trueprop
  1229       |> TPTP_Reconstruct.strip_top_All_vars
  1230       |> snd
  1231       |> TPTP_Reconstruct.remove_polarity true
  1232 
  1233     (*literal is negative*)
  1234     val _ = @{assert} (not pol)
  1235 
  1236     val (antes, conc) = imp_strip_horn t
  1237 
  1238     val (ante_type, antes') =
  1239       if length antes = 1 then
  1240         let
  1241           val conjunctive_antes =
  1242             the_single antes
  1243             |> conjuncts
  1244         in
  1245           if length conjunctive_antes > 1 then
  1246             (TPTP_Reconstruct.Conjunctive NONE,
  1247              conjunctive_antes)
  1248           else
  1249             (TPTP_Reconstruct.Implicational NONE,
  1250              antes)
  1251         end
  1252       else
  1253         (TPTP_Reconstruct.Implicational NONE,
  1254          antes)
  1255   in
  1256     if null antes then NONE
  1257     else SOME (ante_type, length antes', pol)
  1258   end
  1259 *}
  1260 
  1261 ML {*
  1262 (*Given a certain standard_cnf type, build a metatheorem that would
  1263   validate it*)
  1264 fun mk_standard_cnf ctxt kind arity =
  1265   let
  1266     val _ = @{assert} (arity > 0)
  1267     val vars =
  1268       upto (1, arity + 1)
  1269       |> map (fn i => Free ("x" ^ Int.toString i, HOLogic.boolT))
  1270 
  1271     val consequent = hd vars
  1272     val antecedents = tl vars
  1273 
  1274     val conc =
  1275       fold
  1276        (curry HOLogic.mk_conj)
  1277        (map (fn var => HOLogic.mk_eq (var, @{term True})) antecedents)
  1278        (HOLogic.mk_eq (consequent, @{term False}))
  1279 
  1280     val pre_hyp =
  1281       case kind of
  1282           TPTP_Reconstruct.Conjunctive NONE =>
  1283             curry HOLogic.mk_imp
  1284              (if length antecedents = 1 then the_single antecedents
  1285               else
  1286                 fold (curry HOLogic.mk_conj) (tl antecedents) (hd antecedents))
  1287              (hd vars)
  1288         | TPTP_Reconstruct.Implicational NONE =>
  1289             fold (curry HOLogic.mk_imp) antecedents consequent
  1290 
  1291     val hyp = HOLogic.mk_eq (pre_hyp, @{term False})
  1292 
  1293     val t =
  1294       Logic.mk_implies (HOLogic.mk_Trueprop  hyp, HOLogic.mk_Trueprop conc)
  1295   in
  1296     Goal.prove ctxt [] [] t (fn _ => HEADGOAL (blast_tac ctxt))
  1297     |> Drule.export_without_context
  1298   end
  1299 *}
  1300 
  1301 ML {*
  1302 (*Applies a d-tactic, then breaks it up conjunctively.
  1303   This can be used to transform subgoals as follows:
  1304      (A \<longrightarrow> B) = False  \<Longrightarrow> R
  1305               |
  1306               v
  1307   \<lbrakk>A = True; B = False\<rbrakk> \<Longrightarrow> R
  1308 *)
  1309 fun weak_conj_tac drule =
  1310   dtac drule THEN' (REPEAT_DETERM o etac @{thm conjE})
  1311 *}
  1312 
  1313 ML {*
  1314 val uncurry_lit_neg_tac =
  1315   dtac @{lemma "(A \<longrightarrow> B \<longrightarrow> C) = False \<Longrightarrow> (A & B \<longrightarrow> C) = False" by auto}
  1316   #> REPEAT_DETERM
  1317 *}
  1318 
  1319 ML {*
  1320 fun standard_cnf_tac ctxt i = fn st =>
  1321   let
  1322     fun core_tactic i = fn st =>
  1323       case standard_cnf_type ctxt i st of
  1324           NONE => no_tac st
  1325         | SOME (kind, arity, _) =>
  1326             let
  1327               val rule = mk_standard_cnf ctxt kind arity;
  1328             in
  1329               (weak_conj_tac rule THEN' atac) i st
  1330             end
  1331   in
  1332     (uncurry_lit_neg_tac
  1333      THEN' TPTP_Reconstruct_Library.reassociate_conjs_tac ctxt
  1334      THEN' core_tactic) i st
  1335   end
  1336 *}
  1337 
  1338 
  1339 subsubsection "Emulator prep"
  1340 
  1341 ML {*
  1342 datatype cleanup_feature =
  1343     RemoveHypothesesFromSkolemDefs
  1344   | RemoveDuplicates
  1345 
  1346 datatype loop_feature =
  1347     Close_Branch
  1348   | ConjI
  1349   | King_Cong
  1350   | Break_Hypotheses
  1351   | Donkey_Cong (*simper_animal + ex_expander_tac*)
  1352   | RemoveRedundantQuantifications
  1353   | Assumption
  1354 
  1355   (*Closely based on Leo2 calculus*)
  1356   | Existential_Free
  1357   | Existential_Var
  1358   | Universal
  1359   | Not_pos
  1360   | Not_neg
  1361   | Or_pos
  1362   | Or_neg
  1363   | Equal_pos
  1364   | Equal_neg
  1365   | Extuni_Bool2
  1366   | Extuni_Bool1
  1367   | Extuni_Dec
  1368   | Extuni_Bind
  1369   | Extuni_Triv
  1370   | Extuni_FlexRigid
  1371   | Extuni_Func
  1372   | Polarity_switch
  1373   | Forall_special_pos
  1374 
  1375 datatype feature =
  1376     ConstsDiff
  1377   | StripQuantifiers
  1378   | Flip_Conclusion
  1379   | Loop of loop_feature list
  1380   | LoopOnce of loop_feature list
  1381   | InnerLoopOnce of loop_feature list
  1382   | CleanUp of cleanup_feature list
  1383   | AbsorbSkolemDefs
  1384 *}
  1385 
  1386 ML {*
  1387 fun can_feature x l =
  1388   let
  1389     fun sublist_of_clean_up el =
  1390       case el of
  1391           CleanUp l'' => SOME l''
  1392         | _ => NONE
  1393     fun sublist_of_loop el =
  1394       case el of
  1395           Loop l'' => SOME l''
  1396         | _ => NONE
  1397     fun sublist_of_loop_once el =
  1398       case el of
  1399           LoopOnce l'' => SOME l''
  1400         | _ => NONE
  1401     fun sublist_of_inner_loop_once el =
  1402       case el of
  1403           InnerLoopOnce l'' => SOME l''
  1404         | _ => NONE
  1405 
  1406     fun check_sublist sought_sublist opt_list =
  1407       if forall is_none opt_list then false
  1408       else
  1409         fold_options opt_list
  1410         |> flat
  1411         |> pair sought_sublist
  1412         |> subset (op =)
  1413   in
  1414     case x of
  1415         CleanUp l' =>
  1416           map sublist_of_clean_up l
  1417           |> check_sublist l'
  1418       | Loop l' =>
  1419           map sublist_of_loop l
  1420           |> check_sublist l'
  1421       | LoopOnce l' =>
  1422           map sublist_of_loop_once l
  1423           |> check_sublist l'
  1424       | InnerLoopOnce l' =>
  1425           map sublist_of_inner_loop_once l
  1426           |> check_sublist l'
  1427       | _ => exists (curry (op =) x) l
  1428   end;
  1429 
  1430 fun loop_can_feature loop_feats l =
  1431   can_feature (Loop loop_feats) l orelse
  1432   can_feature (LoopOnce loop_feats) l orelse
  1433   can_feature (InnerLoopOnce loop_feats) l;
  1434 
  1435 @{assert} (can_feature ConstsDiff [StripQuantifiers, ConstsDiff]);
  1436 
  1437 @{assert}
  1438   (can_feature (CleanUp [RemoveHypothesesFromSkolemDefs])
  1439     [CleanUp [RemoveHypothesesFromSkolemDefs, RemoveDuplicates]]);
  1440 
  1441 @{assert}
  1442   (can_feature (Loop []) [Loop [Existential_Var]]);
  1443 
  1444 @{assert}
  1445   (not (can_feature (Loop []) [InnerLoopOnce [Existential_Var]]));
  1446 *}
  1447 
  1448 ML {*
  1449 exception NO_LOOP_FEATS
  1450 fun get_loop_feats (feats : feature list) =
  1451   let
  1452     val loop_find =
  1453       fold (fn x => fn loop_feats_acc =>
  1454         if is_some loop_feats_acc then loop_feats_acc
  1455         else
  1456           case x of
  1457               Loop loop_feats => SOME loop_feats
  1458             | LoopOnce loop_feats => SOME loop_feats
  1459             | InnerLoopOnce loop_feats => SOME loop_feats
  1460             | _ => NONE)
  1461        feats
  1462        NONE
  1463   in
  1464     if is_some loop_find then the loop_find
  1465     else raise NO_LOOP_FEATS
  1466   end;
  1467 
  1468 @{assert}
  1469   (get_loop_feats [Loop [King_Cong, Break_Hypotheses, Existential_Free, Existential_Var, Universal]] =
  1470    [King_Cong, Break_Hypotheses, Existential_Free, Existential_Var, Universal])
  1471 *}
  1472 
  1473 (*use as elim rule to remove premises*)
  1474 lemma insa_prems: "\<lbrakk>Q; P\<rbrakk> \<Longrightarrow> P" by auto
  1475 ML {*
  1476 fun cleanup_skolem_defs feats =
  1477   let
  1478     (*remove hypotheses from skolem defs,
  1479      after testing that they look like skolem defs*)
  1480     val dehypothesise_skolem_defs =
  1481       COND' (SOME #> TERMPRED (fn _ => true) conc_is_skolem_def)
  1482         (REPEAT_DETERM o etac @{thm insa_prems})
  1483         (K no_tac)
  1484   in
  1485     if can_feature (CleanUp [RemoveHypothesesFromSkolemDefs]) feats then
  1486       ALLGOALS (TRY o dehypothesise_skolem_defs)
  1487     else all_tac
  1488   end
  1489 *}
  1490 
  1491 ML {*
  1492 fun remove_duplicates_tac feats =
  1493   (if can_feature (CleanUp [RemoveDuplicates]) feats then
  1494      ALLGOALS distinct_subgoal_tac
  1495    else all_tac)
  1496 *}
  1497 
  1498 ML {*
  1499 (*given a goal state, indicates the skolem constants committed-to in it (i.e. appearing in LHS of a skolem definition)*)
  1500 val which_skolem_concs_used = fn st =>
  1501   let
  1502     val feats = [CleanUp [RemoveHypothesesFromSkolemDefs, RemoveDuplicates]]
  1503     val scrubup_tac =
  1504       cleanup_skolem_defs feats
  1505       THEN remove_duplicates_tac feats
  1506   in
  1507     scrubup_tac st
  1508     |> break_seq
  1509     |> tap (fn (_, rest) => @{assert} (null (Seq.list_of rest)))
  1510     |> fst
  1511     |> TERMFUN (snd (*discard hypotheses*)
  1512                  #> get_skolem_conc_const) NONE
  1513     |> switch (fold (fn x => fn l => if is_some x then the x :: l else l)) []
  1514     |> map Const
  1515   end
  1516 *}
  1517 
  1518 ML {*
  1519 fun exists_tac ctxt feats consts_diff =
  1520   let
  1521     val ex_var =
  1522       if loop_can_feature [Existential_Var] feats andalso consts_diff <> [] then
  1523         new_skolem_tac ctxt consts_diff
  1524         (*We're making sure that each skolem constant is used once in instantiations.*)
  1525       else K no_tac
  1526 
  1527     val ex_free =
  1528       if loop_can_feature [Existential_Free] feats andalso consts_diff = [] then
  1529         eresolve_tac ctxt @{thms polar_exE}
  1530       else K no_tac
  1531   in
  1532     ex_var APPEND' ex_free
  1533   end
  1534 
  1535 fun forall_tac ctxt feats =
  1536   if loop_can_feature [Universal] feats then
  1537     forall_pos_tac ctxt
  1538   else K no_tac
  1539 *}
  1540 
  1541 
  1542 subsubsection "Finite types"
  1543 (*lift quantification from a singleton literal to a singleton clause*)
  1544 lemma forall_pos_lift:
  1545 "\<lbrakk>(! X. P X) = True; ! X. (P X = True) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" by auto
  1546 
  1547 (*predicate over the type of the leading quantified variable*)
  1548 
  1549 ML {*
  1550 val extcnf_forall_special_pos_tac =
  1551   let
  1552     val bool =
  1553       ["True", "False"]
  1554 
  1555     val bool_to_bool =
  1556       ["% _ . True", "% _ . False", "% x . x", "Not"]
  1557 
  1558     val tecs =
  1559       map (fn t_s =>
  1560        eres_inst_tac @{context} [((("x", 0), Position.none), t_s)] @{thm allE}
  1561        THEN' atac)
  1562   in
  1563     (TRY o etac @{thm forall_pos_lift})
  1564     THEN' (atac
  1565            ORELSE' FIRST'
  1566             (*FIXME could check the type of the leading quantified variable, instead of trying everything*)
  1567             (tecs (bool @ bool_to_bool)))
  1568   end
  1569 *}
  1570 
  1571 
  1572 subsubsection "Emulator"
  1573 
  1574 lemma efq: "[|A = True; A = False|] ==> R" by auto
  1575 ML {*
  1576 val efq_tac =
  1577   (etac @{thm efq} THEN' atac)
  1578   ORELSE' atac
  1579 *}
  1580 
  1581 ML {*
  1582 (*This is applied to all subgoals, repeatedly*)
  1583 fun extcnf_combined_main ctxt feats consts_diff =
  1584   let
  1585     (*This is applied to subgoals which don't have a conclusion
  1586       consisting of a Skolem definition*)
  1587     fun extcnf_combined_tac' ctxt i = fn st =>
  1588       let
  1589         val skolem_consts_used_so_far = which_skolem_concs_used st
  1590         val consts_diff' = subtract (op =) skolem_consts_used_so_far consts_diff
  1591 
  1592         fun feat_to_tac feat =
  1593           case feat of
  1594               Close_Branch => trace_tac' ctxt "mark: closer" efq_tac
  1595             | ConjI => trace_tac' ctxt "mark: conjI" (rtac @{thm conjI})
  1596             | King_Cong => trace_tac' ctxt "mark: expander_animal" (expander_animal ctxt)
  1597             | Break_Hypotheses => trace_tac' ctxt "mark: break_hypotheses" (break_hypotheses_tac ctxt)
  1598             | RemoveRedundantQuantifications => K all_tac
  1599 (*
  1600 FIXME Building this into the loop instead.. maybe not the ideal choice
  1601             | RemoveRedundantQuantifications =>
  1602                 trace_tac' ctxt "mark: strip_unused_variable_hyp"
  1603                  (REPEAT_DETERM o remove_redundant_quantification_in_lit)
  1604 *)
  1605 
  1606             | Assumption => atac
  1607 (*FIXME both Existential_Free and Existential_Var run same code*)
  1608             | Existential_Free => trace_tac' ctxt "mark: forall_neg" (exists_tac ctxt feats consts_diff')
  1609             | Existential_Var => trace_tac' ctxt "mark: forall_neg" (exists_tac ctxt feats consts_diff')
  1610             | Universal => trace_tac' ctxt "mark: forall_pos" (forall_tac ctxt feats)
  1611             | Not_pos => trace_tac' ctxt "mark: not_pos" (dtac @{thm leo2_rules(9)})
  1612             | Not_neg => trace_tac' ctxt "mark: not_neg" (dtac @{thm leo2_rules(10)})
  1613             | Or_pos => trace_tac' ctxt "mark: or_pos" (dtac @{thm leo2_rules(5)}) (*could add (6) for negated conjunction*)
  1614             | Or_neg => trace_tac' ctxt "mark: or_neg" (dtac @{thm leo2_rules(7)})
  1615             | Equal_pos => trace_tac' ctxt "mark: equal_pos" (dresolve_tac ctxt (@{thms eq_pos_bool} @ [@{thm leo2_rules(3)}, @{thm eq_pos_func}]))
  1616             | Equal_neg => trace_tac' ctxt "mark: equal_neg" (dresolve_tac ctxt [@{thm eq_neg_bool}, @{thm leo2_rules(4)}])
  1617             | Donkey_Cong => trace_tac' ctxt "mark: donkey_cong" (simper_animal ctxt THEN' ex_expander_tac ctxt)
  1618 
  1619             | Extuni_Bool2 => trace_tac' ctxt "mark: extuni_bool2" (dtac @{thm extuni_bool2})
  1620             | Extuni_Bool1 => trace_tac' ctxt "mark: extuni_bool1" (dtac @{thm extuni_bool1})
  1621             | Extuni_Bind => trace_tac' ctxt "mark: extuni_triv" (etac @{thm extuni_triv})
  1622             | Extuni_Triv => trace_tac' ctxt "mark: extuni_triv" (etac @{thm extuni_triv})
  1623             | Extuni_Dec => trace_tac' ctxt "mark: extuni_dec_tac" (extuni_dec_tac ctxt)
  1624             | Extuni_FlexRigid => trace_tac' ctxt "mark: extuni_flex_rigid" (atac ORELSE' asm_full_simp_tac ctxt)
  1625             | Extuni_Func => trace_tac' ctxt "mark: extuni_func" (dtac @{thm extuni_func})
  1626             | Polarity_switch => trace_tac' ctxt "mark: polarity_switch" (eresolve_tac ctxt @{thms polarity_switch})
  1627             | Forall_special_pos => trace_tac' ctxt "mark: dorall_special_pos" extcnf_forall_special_pos_tac
  1628 
  1629         val core_tac =
  1630           get_loop_feats feats
  1631           |> map feat_to_tac
  1632           |> FIRST'
  1633       in
  1634         core_tac i st
  1635       end
  1636 
  1637     (*This is applied to all subgoals, repeatedly*)
  1638     fun extcnf_combined_tac ctxt i =
  1639       COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1640         no_tac
  1641         (extcnf_combined_tac' ctxt i)
  1642 
  1643     val core_tac = CHANGED (ALLGOALS (IF_UNSOLVED o TRY o extcnf_combined_tac ctxt))
  1644 
  1645     val full_tac = REPEAT core_tac
  1646 
  1647   in
  1648     CHANGED
  1649       (if can_feature (InnerLoopOnce []) feats then
  1650          core_tac
  1651        else full_tac)
  1652   end
  1653 
  1654 val interpreted_consts =
  1655   [@{const_name HOL.All}, @{const_name HOL.Ex},
  1656    @{const_name Hilbert_Choice.Eps},
  1657    @{const_name HOL.conj},
  1658    @{const_name HOL.disj},
  1659    @{const_name HOL.eq},
  1660    @{const_name HOL.implies},
  1661    @{const_name HOL.The},
  1662    @{const_name HOL.Ex1},
  1663    @{const_name HOL.Not},
  1664    (* @{const_name HOL.iff}, *) (*FIXME do these exist?*)
  1665    (* @{const_name HOL.not_equal}, *)
  1666    @{const_name HOL.False},
  1667    @{const_name HOL.True},
  1668    @{const_name Pure.imp}]
  1669 
  1670 fun strip_qtfrs_tac ctxt =
  1671   REPEAT_DETERM (HEADGOAL (rtac @{thm allI}))
  1672   THEN REPEAT_DETERM (HEADGOAL (etac @{thm exE}))
  1673   THEN HEADGOAL (canonicalise_qtfr_order ctxt)
  1674   THEN
  1675     ((REPEAT (HEADGOAL (nominal_inst_parametermatch_tac ctxt @{thm allE})))
  1676      APPEND (REPEAT (HEADGOAL (inst_parametermatch_tac ctxt [@{thm allE}]))))
  1677   (*FIXME need to handle "@{thm exI}"?*)
  1678 
  1679 (*difference in constants between the hypothesis clause and the conclusion clause*)
  1680 fun clause_consts_diff thm =
  1681   let
  1682     val t =
  1683       Thm.prop_of thm
  1684       |> Logic.dest_implies
  1685       |> fst
  1686 
  1687       (*This bit should not be needed, since Leo2 inferences don't have parameters*)
  1688       |> TPTP_Reconstruct.strip_top_all_vars []
  1689       |> snd
  1690 
  1691     val do_diff =
  1692       Logic.dest_implies
  1693       #> uncurry TPTP_Reconstruct.new_consts_between
  1694       #> filter
  1695            (fn Const (n, _) =>
  1696              not (member (op =) interpreted_consts n))
  1697   in
  1698     if head_of t = Logic.implies then do_diff t
  1699     else []
  1700   end
  1701 *}
  1702 
  1703 ML {*
  1704 (*remove quantification in hypothesis clause (! X. t), if
  1705   X not free in t*)
  1706 fun remove_redundant_quantification ctxt i = fn st =>
  1707   let
  1708     val gls =
  1709       Thm.prop_of st
  1710       |> Logic.strip_horn
  1711       |> fst
  1712   in
  1713     if null gls then raise NO_GOALS
  1714     else
  1715       let
  1716         val (params, (hyp_clauses, conc_clause)) =
  1717           rpair (i - 1) gls
  1718           |> uncurry nth
  1719           |> TPTP_Reconstruct.strip_top_all_vars []
  1720           |> apsnd Logic.strip_horn
  1721       in
  1722         (*this is to fail gracefully in case this tactic is applied to a goal which doesn't have a single hypothesis*)
  1723         if length hyp_clauses > 1 then no_tac st
  1724         else
  1725           let
  1726             val hyp_clause = the_single hyp_clauses
  1727             val sep_prefix =
  1728               HOLogic.dest_Trueprop
  1729               #> TPTP_Reconstruct.strip_top_All_vars
  1730               #> apfst rev
  1731             val (hyp_prefix, hyp_body) = sep_prefix hyp_clause
  1732             val (conc_prefix, conc_body) = sep_prefix conc_clause
  1733           in
  1734             if null hyp_prefix orelse
  1735               member (op =) conc_prefix (hd hyp_prefix) orelse
  1736               member (op =)  (Term.add_frees hyp_body []) (hd hyp_prefix) then
  1737               no_tac st
  1738             else
  1739               eres_inst_tac ctxt [((("x", 0), Position.none), "(@X. False)")]
  1740                 @{thm allE} i st
  1741           end
  1742      end
  1743   end
  1744 *}
  1745 
  1746 ML {*
  1747 fun remove_redundant_quantification_ignore_skolems ctxt i =
  1748   COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1749     no_tac
  1750     (remove_redundant_quantification ctxt i)
  1751 *}
  1752 
  1753 lemma drop_redundant_literal_qtfr:
  1754   "(! X. P) = True \<Longrightarrow> P = True"
  1755   "(? X. P) = True \<Longrightarrow> P = True"
  1756   "(! X. P) = False \<Longrightarrow> P = False"
  1757   "(? X. P) = False \<Longrightarrow> P = False"
  1758 by auto
  1759 
  1760 ML {*
  1761 (*remove quantification in the literal "(! X. t) = True/False"
  1762   in the singleton hypothesis clause, if X not free in t*)
  1763 fun remove_redundant_quantification_in_lit ctxt i = fn st =>
  1764   let
  1765     val gls =
  1766       Thm.prop_of st
  1767       |> Logic.strip_horn
  1768       |> fst
  1769   in
  1770     if null gls then raise NO_GOALS
  1771     else
  1772       let
  1773         val (params, (hyp_clauses, conc_clause)) =
  1774           rpair (i - 1) gls
  1775           |> uncurry nth
  1776           |> TPTP_Reconstruct.strip_top_all_vars []
  1777           |> apsnd Logic.strip_horn
  1778       in
  1779         (*this is to fail gracefully in case this tactic is applied to a goal which doesn't have a single hypothesis*)
  1780         if length hyp_clauses > 1 then no_tac st
  1781         else
  1782           let
  1783             fun literal_content (Const (@{const_name HOL.eq}, _) $ lhs $ (rhs as @{term True})) = SOME (lhs, rhs)
  1784               | literal_content (Const (@{const_name HOL.eq}, _) $ lhs $ (rhs as @{term False})) = SOME (lhs, rhs)
  1785               | literal_content t = NONE
  1786 
  1787             val hyp_clause =
  1788               the_single hyp_clauses
  1789               |> HOLogic.dest_Trueprop
  1790               |> literal_content
  1791 
  1792           in
  1793             if is_none hyp_clause then
  1794               no_tac st
  1795             else
  1796               let
  1797                 val (hyp_lit_prefix, hyp_lit_body) =
  1798                   the hyp_clause
  1799                   |> (fn (t, polarity) =>
  1800                        TPTP_Reconstruct.strip_top_All_vars t
  1801                        |> apfst rev)
  1802               in
  1803                 if null hyp_lit_prefix orelse
  1804                   member (op =)  (Term.add_frees hyp_lit_body []) (hd hyp_lit_prefix) then
  1805                   no_tac st
  1806                 else
  1807                   dresolve_tac ctxt @{thms drop_redundant_literal_qtfr} i st
  1808               end
  1809           end
  1810      end
  1811   end
  1812 *}
  1813 
  1814 ML {*
  1815 fun remove_redundant_quantification_in_lit_ignore_skolems ctxt i =
  1816   COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1817     no_tac
  1818     (remove_redundant_quantification_in_lit ctxt i)
  1819 *}
  1820 
  1821 ML {*
  1822 fun extcnf_combined_tac ctxt prob_name_opt feats skolem_consts = fn st =>
  1823   let
  1824     val thy = Proof_Context.theory_of ctxt
  1825 
  1826     (*Initially, st consists of a single goal, showing the
  1827       hypothesis clause implying the conclusion clause.
  1828       There are no parameters.*)
  1829     val consts_diff =
  1830       union (op =) skolem_consts
  1831        (if can_feature ConstsDiff feats then
  1832           clause_consts_diff st
  1833         else [])
  1834 
  1835     val main_tac =
  1836       if can_feature (LoopOnce []) feats orelse can_feature (InnerLoopOnce []) feats then
  1837         extcnf_combined_main ctxt feats consts_diff
  1838       else if can_feature (Loop []) feats then
  1839         BEST_FIRST (TERMPRED (fn _ => true) conc_is_skolem_def NONE, size_of_thm)
  1840 (*FIXME maybe need to weaken predicate to include "solved form"?*)
  1841          (extcnf_combined_main ctxt feats consts_diff)
  1842       else all_tac (*to allow us to use the cleaning features*)
  1843 
  1844     (*Remove hypotheses from Skolem definitions,
  1845       then remove duplicate subgoals,
  1846       then we should be left with skolem definitions:
  1847         absorb them as axioms into the theory.*)
  1848     val cleanup =
  1849       cleanup_skolem_defs feats
  1850       THEN remove_duplicates_tac feats
  1851       THEN (if can_feature AbsorbSkolemDefs feats then
  1852               ALLGOALS (absorb_skolem_def ctxt prob_name_opt)
  1853             else all_tac)
  1854 
  1855     val have_loop_feats =
  1856       (get_loop_feats feats; true)
  1857       handle NO_LOOP_FEATS => false
  1858 
  1859     val tec =
  1860       (if can_feature StripQuantifiers feats then
  1861          (REPEAT (CHANGED (strip_qtfrs_tac ctxt)))
  1862        else all_tac)
  1863       THEN (if can_feature Flip_Conclusion feats then
  1864              HEADGOAL (flip_conclusion_tac ctxt)
  1865            else all_tac)
  1866 
  1867       (*after stripping the quantifiers any remaining quantifiers
  1868         can be simply eliminated -- they're redundant*)
  1869       (*FIXME instead of just using allE, instantiate to a silly
  1870          term, to remove opportunities for unification.*)
  1871       THEN (REPEAT_DETERM (etac @{thm allE} 1))
  1872 
  1873       THEN (REPEAT_DETERM (rtac @{thm allI} 1))
  1874 
  1875       THEN (if have_loop_feats then
  1876               REPEAT (CHANGED
  1877               ((ALLGOALS (TRY o clause_breaker_tac ctxt)) (*brush away literals which don't change*)
  1878                THEN
  1879                 (*FIXME move this to a different level?*)
  1880                 (if loop_can_feature [Polarity_switch] feats then
  1881                    all_tac
  1882                  else
  1883                    (TRY (IF_UNSOLVED (HEADGOAL (remove_redundant_quantification_ignore_skolems ctxt))))
  1884                    THEN (TRY (IF_UNSOLVED (HEADGOAL (remove_redundant_quantification_in_lit_ignore_skolems ctxt)))))
  1885                THEN (TRY main_tac)))
  1886             else
  1887               all_tac)
  1888       THEN IF_UNSOLVED cleanup
  1889 
  1890   in
  1891     DEPTH_SOLVE (CHANGED tec) st
  1892   end
  1893 *}
  1894 
  1895 
  1896 subsubsection "unfold_def"
  1897 
  1898 (*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.*)
  1899 lemma drop_first_hypothesis [rule_format]: "\<lbrakk>A; B\<rbrakk> \<Longrightarrow> B" by auto
  1900 
  1901 (*Unfold_def works by reducing the goal to a meta equation,
  1902   then working on it until it can be discharged by atac,
  1903   or reflexive, or else turned back into an object equation
  1904   and broken down further.*)
  1905 lemma un_meta_polarise: "(X \<equiv> True) \<Longrightarrow> X" by auto
  1906 lemma meta_polarise: "X \<Longrightarrow> X \<equiv> True" by auto
  1907 
  1908 ML {*
  1909 fun unfold_def_tac ctxt depends_on_defs = fn st =>
  1910   let
  1911     (*This is used when we end up with something like
  1912         (A & B) \<equiv> True \<Longrightarrow> (B & A) \<equiv> True.
  1913       It breaks down this subgoal until it can be trivially
  1914       discharged.
  1915      *)
  1916     val kill_meta_eqs_tac =
  1917       dtac @{thm un_meta_polarise}
  1918       THEN' rtac @{thm meta_polarise}
  1919       THEN' (REPEAT_DETERM o (etac @{thm conjE}))
  1920       THEN' (REPEAT_DETERM o (rtac @{thm conjI} ORELSE' atac))
  1921 
  1922     val continue_reducing_tac =
  1923       rtac @{thm meta_eq_to_obj_eq} 1
  1924       THEN (REPEAT_DETERM (ex_expander_tac ctxt 1))
  1925       THEN TRY (polarise_subgoal_hyps 1) (*no need to REPEAT_DETERM here, since there should only be one hypothesis*)
  1926       THEN TRY (dtac @{thm eq_reflection} 1)
  1927       THEN (TRY ((CHANGED o rewrite_goal_tac ctxt
  1928               (@{thm expand_iff} :: @{thms simp_meta})) 1))
  1929       THEN HEADGOAL (rtac @{thm reflexive}
  1930                      ORELSE' atac
  1931                      ORELSE' kill_meta_eqs_tac)
  1932 
  1933     val tectic =
  1934       (rtac @{thm polarise} 1 THEN atac 1)
  1935       ORELSE
  1936         (REPEAT_DETERM (etac @{thm conjE} 1 THEN etac @{thm drop_first_hypothesis} 1)
  1937          THEN PRIMITIVE (Conv.fconv_rule Drule.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 Drule.eta_long_conversion)
  1941          THEN
  1942            (HEADGOAL atac
  1943            ORELSE
  1944             (unfold_tac ctxt depends_on_defs
  1945              THEN IF_UNSOLVED continue_reducing_tac)))
  1946   in
  1947     tectic 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.dest) (#defs pannot)
  2062               | TPTP_Syntax.Role_Axiom =>
  2063                   map (apsnd Binding.dest) (#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.dest) (#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 (#3 #> 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           (atac
  2121            ORELSE'
  2122               (rtac @{thm polarise}
  2123                THEN' atac))
  2124     | "polarity_switch" => nonfull_extcnf_combined_tac [Polarity_switch]
  2125     | "solved_all_splits" => solved_all_splits_tac
  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 atac
  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 Drule.eta_long_conversion))
  2164          THEN HEADGOAL atac
  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