src/expr/proof_rule.h

   1 /*********************                                                        */
   2 /*! \file proof_rule.h
   3  ** \verbatim
   4  ** Top contributors (to current version):
   5  **   Haniel Barbosa, Andrew Reynolds
   6  ** This file is part of the CVC4 project.
   7  ** Copyright (c) 2009-2020 by the authors listed in the file AUTHORS
   8  ** in the top-level source directory) and their institutional affiliations.
   9  ** All rights reserved.  See the file COPYING in the top-level source
  10  ** directory for licensing information.\endverbatim
  11  **
  12  ** \brief Proof rule enumeration
  13  **/
  14
  15 #include "cvc4_private.h"
  16
  17 #ifndef CVC4__EXPR__PROOF_RULE_H
  18 #define CVC4__EXPR__PROOF_RULE_H
  19
  20 #include <iosfwd>
  21
  22 namespace CVC4 {
  23
  24 /**
  25  * An enumeration for proof rules. This enumeration is analogous to Kind for
  26  * Node objects. In the documentation below, P:F denotes a ProofNode that
  27  * proves formula F.
  28  *
  29  * Conceptually, the following proof rules form a calculus whose target
  30  * user is the Node-level theory solvers. This means that the rules below
  31  * are designed to reason about, among other things, common operations on Node
  32  * objects like Rewriter::rewrite or Node::substitute. It is intended to be
  33  * translated or printed in other formats.
  34  *
  35  * The following PfRule values include core rules and those categorized by
  36  * theory, including the theory of equality.
  37  *
  38  * The "core rules" include two distinguished rules which have special status:
  39  * (1) ASSUME, which represents an open leaf in a proof.
  40  * (2) SCOPE, which closes the scope of assumptions.
  41  * The core rules additionally correspond to generic operations that are done
  42  * internally on nodes, e.g. calling Rewriter::rewrite.
  43  */
  44 enum class PfRule : uint32_t
  45 {
  46   //================================================= Core rules
  47   //======================== Assume and Scope
  48   // ======== Assumption (a leaf)
  49   // Children: none
  50   // Arguments: (F)
  51   // --------------
  52   // Conclusion: F
  53   //
  54   // This rule has special status, in that an application of assume is an
  55   // open leaf in a proof that is not (yet) justified. An assume leaf is
  56   // analogous to a free variable in a term, where we say "F is a free
  57   // assumption in proof P" if it contains an application of F that is not
  58   // bound by SCOPE (see below).
  59   ASSUME,
  60   // ======== Scope (a binder for assumptions)
  61   // Children: (P:F)
  62   // Arguments: (F1, ..., Fn)
  63   // --------------
  64   // Conclusion: (=> (and F1 ... Fn) F) or (not (and F1 ... Fn)) if F is false
  65   //
  66   // This rule has a dual purpose with ASSUME. It is a way to close
  67   // assumptions in a proof. We require that F1 ... Fn are free assumptions in
  68   // P and say that F1, ..., Fn are not free in (SCOPE P). In other words, they
  69   // are bound by this application. For example, the proof node:
  70   //   (SCOPE (ASSUME F) :args F)
  71   // has the conclusion (=> F F) and has no free assumptions. More generally, a
  72   // proof with no free assumptions always concludes a valid formula.
  73   SCOPE,
  74
  75   //======================== Builtin theory (common node operations)
  76   // ======== Substitution
  77   // Children: (P1:F1, ..., Pn:Fn)
  78   // Arguments: (t, (ids)?)
  79   // ---------------------------------------------------------------
  80   // Conclusion: (= t t*sigma{ids}(Fn)*...*sigma{ids}(F1))
  81   // where sigma{ids}(Fi) are substitutions, which notice are applied in
  82   // reverse order.
  83   // Notice that ids is a MethodId identifier, which determines how to convert
  84   // the formulas F1, ..., Fn into substitutions.
  85   SUBS,
  86   // ======== Rewrite
  87   // Children: none
  88   // Arguments: (t, (idr)?)
  89   // ----------------------------------------
  90   // Conclusion: (= t Rewriter{idr}(t))
  91   // where idr is a MethodId identifier, which determines the kind of rewriter
  92   // to apply, e.g. Rewriter::rewrite.
  93   REWRITE,
  94   // ======== Substitution + Rewriting equality introduction
  95   //
  96   // In this rule, we provide a term t and conclude that it is equal to its
  97   // rewritten form under a (proven) substitution.
  98   //
  99   // Children: (P1:F1, ..., Pn:Fn)
 100   // Arguments: (t, (ids (idr)?)?)
 101   // ---------------------------------------------------------------
 102   // Conclusion: (= t t')
 103   // where
 104   //   t' is
 105   //   toWitness(Rewriter{idr}(toSkolem(t)*sigma{ids}(Fn)*...*sigma{ids}(F1)))
 106   //   toSkolem(...) converts terms from witness form to Skolem form,
 107   //   toWitness(...) converts terms from Skolem form to witness form.
 108   //
 109   // Notice that:
 110   //   toSkolem(t')=Rewriter{idr}(toSkolem(t)*sigma{ids}(Fn)*...*sigma{ids}(F1))
 111   // In other words, from the point of view of Skolem forms, this rule
 112   // transforms t to t' by standard substitution + rewriting.
 113   //
 114   // The argument ids and idr is optional and specify the identifier of the
 115   // substitution and rewriter respectively to be used. For details, see
 116   // theory/builtin/proof_checker.h.
 117   MACRO_SR_EQ_INTRO,
 118   // ======== Substitution + Rewriting predicate introduction
 119   //
 120   // In this rule, we provide a formula F and conclude it, under the condition
 121   // that it rewrites to true under a proven substitution.
 122   //
 123   // Children: (P1:F1, ..., Pn:Fn)
 124   // Arguments: (F, (ids (idr)?)?)
 125   // ---------------------------------------------------------------
 126   // Conclusion: F
 127   // where
 128   //   Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)) == true
 129   // where ids and idr are method identifiers.
 130   //
 131   // Notice that we apply rewriting on the witness form of F, meaning that this
 132   // rule may conclude an F whose Skolem form is justified by the definition of
 133   // its (fresh) Skolem variables. Furthermore, notice that the rewriting and
 134   // substitution is applied only within the side condition, meaning the
 135   // rewritten form of the witness form of F does not escape this rule.
 136   MACRO_SR_PRED_INTRO,
 137   // ======== Substitution + Rewriting predicate elimination
 138   //
 139   // In this rule, if we have proven a formula F, then we may conclude its
 140   // rewritten form under a proven substitution.
 141   //
 142   // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn)
 143   // Arguments: ((ids (idr)?)?)
 144   // ----------------------------------------
 145   // Conclusion: F'
 146   // where
 147   //   F' is
 148   //   toWitness(Rewriter{idr}(toSkolem(F)*sigma{ids}(Fn)*...*sigma{ids}(F1)).
 149   // where ids and idr are method identifiers.
 150   //
 151   // We rewrite only on the Skolem form of F, similar to MACRO_SR_EQ_INTRO.
 152   MACRO_SR_PRED_ELIM,
 153   // ======== Substitution + Rewriting predicate transform
 154   //
 155   // In this rule, if we have proven a formula F, then we may provide a formula
 156   // G and conclude it if F and G are equivalent after rewriting under a proven
 157   // substitution.
 158   //
 159   // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn)
 160   // Arguments: (G, (ids (idr)?)?)
 161   // ----------------------------------------
 162   // Conclusion: G
 163   // where
 164   //   Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)) ==
 165   //   Rewriter{idr}(G*sigma{ids}(Fn)*...*sigma{ids}(F1))
 166   //
 167   // Notice that we apply rewriting on the witness form of F and G, similar to
 168   // MACRO_SR_PRED_INTRO.
 169   MACRO_SR_PRED_TRANSFORM,
 170
 171   //================================================= Boolean rules
 172   // ======== Split
 173   // Children: none
 174   // Arguments: (F)
 175   // ---------------------
 176   // Conclusion: (or F (not F))
 177   SPLIT,
 178   // ======== And elimination
 179   // Children: (P:(and F1 ... Fn))
 180   // Arguments: (i)
 181   // ---------------------
 182   // Conclusion: (Fi)
 183   AND_ELIM,
 184   // ======== And introduction
 185   // Children: (P1:F1 ... Pn:Fn))
 186   // Arguments: ()
 187   // ---------------------
 188   // Conclusion: (and P1 ... Pn)
 189   AND_INTRO,
 190   // ======== Not Or elimination
 191   // Children: (P:(not (or F1 ... Fn)))
 192   // Arguments: (i)
 193   // ---------------------
 194   // Conclusion: (not Fi)
 195   NOT_OR_ELIM,
 196   // ======== Implication elimination
 197   // Children: (P:(=> F1 F2))
 198   // Arguments: ()
 199   // ---------------------
 200   // Conclusion: (or (not F1) F2)
 201   IMPLIES_ELIM,
 202   // ======== Not Implication elimination version 1
 203   // Children: (P:(not (=> F1 F2)))
 204   // Arguments: ()
 205   // ---------------------
 206   // Conclusion: (F1)
 207   NOT_IMPLIES_ELIM1,
 208   // ======== Not Implication elimination version 2
 209   // Children: (P:(not (=> F1 F2)))
 210   // Arguments: ()
 211   // ---------------------
 212   // Conclusion: (not F2)
 213   NOT_IMPLIES_ELIM2,
 214   // ======== Equivalence elimination version 1
 215   // Children: (P:(= F1 F2))
 216   // Arguments: ()
 217   // ---------------------
 218   // Conclusion: (or (not F1) F2)
 219   EQUIV_ELIM1,
 220   // ======== Equivalence elimination version 2
 221   // Children: (P:(= F1 F2))
 222   // Arguments: ()
 223   // ---------------------
 224   // Conclusion: (or F1 (not F2))
 225   EQUIV_ELIM2,
 226   // ======== Not Equivalence elimination version 1
 227   // Children: (P:(not (= F1 F2)))
 228   // Arguments: ()
 229   // ---------------------
 230   // Conclusion: (or F1 F2)
 231   NOT_EQUIV_ELIM1,
 232   // ======== Not Equivalence elimination version 2
 233   // Children: (P:(not (= F1 F2)))
 234   // Arguments: ()
 235   // ---------------------
 236   // Conclusion: (or (not F1) (not F2))
 237   NOT_EQUIV_ELIM2,
 238   // ======== XOR elimination version 1
 239   // Children: (P:(xor F1 F2)))
 240   // Arguments: ()
 241   // ---------------------
 242   // Conclusion: (or F1 F2)
 243   XOR_ELIM1,
 244   // ======== XOR elimination version 2
 245   // Children: (P:(xor F1 F2)))
 246   // Arguments: ()
 247   // ---------------------
 248   // Conclusion: (or (not F1) (not F2))
 249   XOR_ELIM2,
 250   // ======== Not XOR elimination version 1
 251   // Children: (P:(not (xor F1 F2)))
 252   // Arguments: ()
 253   // ---------------------
 254   // Conclusion: (or F1 (not F2))
 255   NOT_XOR_ELIM1,
 256   // ======== Not XOR elimination version 2
 257   // Children: (P:(not (xor F1 F2)))
 258   // Arguments: ()
 259   // ---------------------
 260   // Conclusion: (or (not F1) F2)
 261   NOT_XOR_ELIM2,
 262   // ======== ITE elimination version 1
 263   // Children: (P:(ite C F1 F2))
 264   // Arguments: ()
 265   // ---------------------
 266   // Conclusion: (or (not C) F1)
 267   ITE_ELIM1,
 268   // ======== ITE elimination version 2
 269   // Children: (P:(ite C F1 F2))
 270   // Arguments: ()
 271   // ---------------------
 272   // Conclusion: (or C F2)
 273   ITE_ELIM2,
 274   // ======== Not ITE elimination version 1
 275   // Children: (P:(not (ite C F1 F2)))
 276   // Arguments: ()
 277   // ---------------------
 278   // Conclusion: (or (not C) (not F1))
 279   NOT_ITE_ELIM1,
 280   // ======== Not ITE elimination version 1
 281   // Children: (P:(not (ite C F1 F2)))
 282   // Arguments: ()
 283   // ---------------------
 284   // Conclusion: (or C (not F2))
 285   NOT_ITE_ELIM2,
 286   // ======== Not ITE elimination version 1
 287   // Children: (P1:P P2:(not P))
 288   // Arguments: ()
 289   // ---------------------
 290   // Conclusion: (false)
 291   CONTRA,
 292
 293   //================================================= De Morgan rules
 294   // ======== Not And
 295   // Children: (P:(not (and F1 ... Fn))
 296   // Arguments: ()
 297   // ---------------------
 298   // Conclusion: (or (not F1) ... (not Fn))
 299   NOT_AND,
 300   //================================================= CNF rules
 301   // ======== CNF And Pos
 302   // Children: ()
 303   // Arguments: ((and F1 ... Fn), i)
 304   // ---------------------
 305   // Conclusion: (or (not (and F1 ... Fn)) Fi)
 306   CNF_AND_POS,
 307   // ======== CNF And Neg
 308   // Children: ()
 309   // Arguments: ((and F1 ... Fn))
 310   // ---------------------
 311   // Conclusion: (or (and F1 ... Fn) (not F1) ... (not Fn))
 312   CNF_AND_NEG,
 313   // ======== CNF Or Pos
 314   // Children: ()
 315   // Arguments: ((or F1 ... Fn))
 316   // ---------------------
 317   // Conclusion: (or (not (or F1 ... Fn)) F1 ... Fn)
 318   CNF_OR_POS,
 319   // ======== CNF Or Neg
 320   // Children: ()
 321   // Arguments: ((or F1 ... Fn), i)
 322   // ---------------------
 323   // Conclusion: (or (or F1 ... Fn) (not Fi))
 324   CNF_OR_NEG,
 325   // ======== CNF Implies Pos
 326   // Children: ()
 327   // Arguments: ((implies F1 F2))
 328   // ---------------------
 329   // Conclusion: (or (not (implies F1 F2)) (not F1) F2)
 330   CNF_IMPLIES_POS,
 331   // ======== CNF Implies Neg version 1
 332   // Children: ()
 333   // Arguments: ((implies F1 F2))
 334   // ---------------------
 335   // Conclusion: (or (implies F1 F2) F1)
 336   CNF_IMPLIES_NEG1,
 337   // ======== CNF Implies Neg version 2
 338   // Children: ()
 339   // Arguments: ((implies F1 F2))
 340   // ---------------------
 341   // Conclusion: (or (implies F1 F2) (not F2))
 342   CNF_IMPLIES_NEG2,
 343   // ======== CNF Equiv Pos version 1
 344   // Children: ()
 345   // Arguments: ((= F1 F2))
 346   // ---------------------
 347   // Conclusion: (or (not (= F1 F2)) (not F1) F2)
 348   CNF_EQUIV_POS1,
 349   // ======== CNF Equiv Pos version 2
 350   // Children: ()
 351   // Arguments: ((= F1 F2))
 352   // ---------------------
 353   // Conclusion: (or (not (= F1 F2)) F1 (not F2))
 354   CNF_EQUIV_POS2,
 355   // ======== CNF Equiv Neg version 1
 356   // Children: ()
 357   // Arguments: ((= F1 F2))
 358   // ---------------------
 359   // Conclusion: (or (= F1 F2) F1 F2)
 360   CNF_EQUIV_NEG1,
 361   // ======== CNF Equiv Neg version 2
 362   // Children: ()
 363   // Arguments: ((= F1 F2))
 364   // ---------------------
 365   // Conclusion: (or (= F1 F2) (not F1) (not F2))
 366   CNF_EQUIV_NEG2,
 367   // ======== CNF Xor Pos version 1
 368   // Children: ()
 369   // Arguments: ((xor F1 F2))
 370   // ---------------------
 371   // Conclusion: (or (not (xor F1 F2)) F1 F2)
 372   CNF_XOR_POS1,
 373   // ======== CNF Xor Pos version 2
 374   // Children: ()
 375   // Arguments: ((xor F1 F2))
 376   // ---------------------
 377   // Conclusion: (or (not (xor F1 F2)) (not F1) (not F2))
 378   CNF_XOR_POS2,
 379   // ======== CNF Xor Neg version 1
 380   // Children: ()
 381   // Arguments: ((xor F1 F2))
 382   // ---------------------
 383   // Conclusion: (or (xor F1 F2) (not F1) F2)
 384   CNF_XOR_NEG1,
 385   // ======== CNF Xor Neg version 2
 386   // Children: ()
 387   // Arguments: ((xor F1 F2))
 388   // ---------------------
 389   // Conclusion: (or (xor F1 F2) F1 (not F2))
 390   CNF_XOR_NEG2,
 391   // ======== CNF ITE Pos version 1
 392   // Children: ()
 393   // Arguments: ((ite C F1 F2))
 394   // ---------------------
 395   // Conclusion: (or (not (ite C F1 F2)) (not C) F1)
 396   CNF_ITE_POS1,
 397   // ======== CNF ITE Pos version 2
 398   // Children: ()
 399   // Arguments: ((ite C F1 F2))
 400   // ---------------------
 401   // Conclusion: (or (not (ite C F1 F2)) C F2)
 402   CNF_ITE_POS2,
 403   // ======== CNF ITE Pos version 3
 404   // Children: ()
 405   // Arguments: ((ite C F1 F2))
 406   // ---------------------
 407   // Conclusion: (or (not (ite C F1 F2)) F1 F2)
 408   CNF_ITE_POS3,
 409   // ======== CNF ITE Neg version 1
 410   // Children: ()
 411   // Arguments: ((ite C F1 F2))
 412   // ---------------------
 413   // Conclusion: (or (ite C F1 F2) (not C) (not F1))
 414   CNF_ITE_NEG1,
 415   // ======== CNF ITE Neg version 2
 416   // Children: ()
 417   // Arguments: ((ite C F1 F2))
 418   // ---------------------
 419   // Conclusion: (or (ite C F1 F2) C (not F2))
 420   CNF_ITE_NEG2,
 421   // ======== CNF ITE Neg version 3
 422   // Children: ()
 423   // Arguments: ((ite C F1 F2))
 424   // ---------------------
 425   // Conclusion: (or (ite C F1 F2) (not F1) (not F2))
 426   CNF_ITE_NEG3,
 427
 428   //================================================= Equality rules
 429   // ======== Reflexive
 430   // Children: none
 431   // Arguments: (t)
 432   // ---------------------
 433   // Conclusion: (= t t)
 434   REFL,
 435   // ======== Symmetric
 436   // Children: (P:(= t1 t2)) or (P:(not (= t1 t2)))
 437   // Arguments: none
 438   // -----------------------
 439   // Conclusion: (= t2 t1) or (not (= t2 t1))
 440   SYMM,
 441   // ======== Transitivity
 442   // Children: (P1:(= t1 t2), ..., Pn:(= t{n-1} tn))
 443   // Arguments: none
 444   // -----------------------
 445   // Conclusion: (= t1 tn)
 446   TRANS,
 447   // ======== Congruence  (subsumed by Substitute?)
 448   // Children: (P1:(= t1 s1), ..., Pn:(= tn sn))
 449   // Arguments: (f)
 450   // ---------------------------------------------
 451   // Conclusion: (= (f t1 ... tn) (f s1 ... sn))
 452   CONG,
 453   // ======== True intro
 454   // Children: (P:F)
 455   // Arguments: none
 456   // ----------------------------------------
 457   // Conclusion: (= F true)
 458   TRUE_INTRO,
 459   // ======== True elim
 460   // Children: (P:(= F true)
 461   // Arguments: none
 462   // ----------------------------------------
 463   // Conclusion: F
 464   TRUE_ELIM,
 465   // ======== False intro
 466   // Children: (P:(not F))
 467   // Arguments: none
 468   // ----------------------------------------
 469   // Conclusion: (= F false)
 470   FALSE_INTRO,
 471   // ======== False elim
 472   // Children: (P:(= F false)
 473   // Arguments: none
 474   // ----------------------------------------
 475   // Conclusion: (not F)
 476   FALSE_ELIM,
 477
 478   //================================================= Quantifiers rules
 479   // ======== Witness intro
 480   // Children: (P:F[t])
 481   // Arguments: (t)
 482   // ----------------------------------------
 483   // Conclusion: (= t (witness ((x T)) F[x]))
 484   // where x is a BOUND_VARIABLE unique to the pair F,t.
 485   WITNESS_INTRO,
 486   // ======== Exists intro
 487   // Children: (P:F[t])
 488   // Arguments: (t)
 489   // ----------------------------------------
 490   // Conclusion: (exists ((x T)) F[x])
 491   // where x is a BOUND_VARIABLE unique to the pair F,t.
 492   EXISTS_INTRO,
 493   // ======== Skolemize
 494   // Children: (P:(exists ((x1 T1) ... (xn Tn)) F))
 495   // Arguments: none
 496   // ----------------------------------------
 497   // Conclusion: F*sigma
 498   // sigma maps x1 ... xn to their representative skolems obtained by
 499   // SkolemManager::mkSkolemize, returned in the skolems argument of that
 500   // method.
 501   SKOLEMIZE,
 502   // ======== Instantiate
 503   // Children: (P:(forall ((x1 T1) ... (xn Tn)) F))
 504   // Arguments: (t1 ... tn)
 505   // ----------------------------------------
 506   // Conclusion: F*sigma
 507   // sigma maps x1 ... xn to t1 ... tn.
 508   INSTANTIATE,
 509
 510   // ======== Adding Inequalities
 511   // Note: an ArithLiteral is a term of the form (>< poly const)
 512   // where
 513   //   >< is >=, >, ==, <, <=, or not(== ...).
 514   //   poly is a polynomial
 515   //   const is a rational constant
 516
 517   // Children: (P1:l1, ..., Pn:ln)
 518   //           where each li is an ArithLiteral
 519   //           not(= ...) is dis-allowed!
 520   //
 521   // Arguments: (k1, ..., kn), non-zero reals
 522   // ---------------------
 523   // Conclusion: (>< (* k t1) (* k t2))
 524   //    where >< is the fusion of the combination of the ><i, (flipping each it
 525   //    its ki is negative). >< is always one of <, <=
 526   //    NB: this implies that lower bounds must have negative ki,
 527   //                      and upper bounds must have positive ki.
 528   //    t1 is the sum of the polynomials.
 529   //    t2 is the sum of the constants.
 530   ARITH_SCALE_SUM_UPPER_BOUNDS,
 531
 532   // ======== Tightening Strict Integer Upper Bounds
 533   // Children: (P:(< i c))
 534   //         where i has integer type.
 535   // Arguments: none
 536   // ---------------------
 537   // Conclusion: (<= i greatestIntLessThan(c)})
 538   INT_TIGHT_UB,
 539
 540   // ======== Tightening Strict Integer Lower Bounds
 541   // Children: (P:(> i c))
 542   //         where i has integer type.
 543   // Arguments: none
 544   // ---------------------
 545   // Conclusion: (>= i leastIntGreaterThan(c)})
 546   INT_TIGHT_LB,
 547
 548   // ======== Trichotomy of the reals
 549   // Children: (A B)
 550   // Arguments: (C)
 551   // ---------------------
 552   // Conclusion: (C),
 553   //                 where (not A) (not B) and C
 554   //                   are (> x c) (< x c) and (= x c)
 555   //                   in some order
 556   //                 note that "not" here denotes arithmetic negation, flipping
 557   //                 >= to <, etc.
 558   ARITH_TRICHOTOMY,
 559
 560   // ======== Arithmetic operator elimination
 561   // Children: none
 562   // Arguments: (t)
 563   // ---------------------
 564   // Conclusion: arith::OperatorElim::getAxiomFor(t)
 565   ARITH_OP_ELIM_AXIOM,
 566
 567   // ======== Int Trust
 568   // Children: (P1 ... Pn)
 569   // Arguments: (Q)
 570   // ---------------------
 571   // Conclusion: (Q)
 572   INT_TRUST,
 573
 574   //================================================= Unknown rule
 575   UNKNOWN,
 576 };
 577
 578 /**
 579  * Converts a proof rule to a string. Note: This function is also used in
 580  * `safe_print()`. Changing this function name or signature will result in
 581  * `safe_print()` printing "<unsupported>" instead of the proper strings for
 582  * the enum values.
 583  *
 584  * @param id The proof rule
 585  * @return The name of the proof rule
 586  */
 587 const char* toString(PfRule id);
 588
 589 /**
 590  * Writes a proof rule name to a stream.
 591  *
 592  * @param out The stream to write to
 593  * @param id The proof rule to write to the stream
 594  * @return The stream
 595  */
 596 std::ostream& operator<<(std::ostream& out, PfRule id);
 597
 598 }  // namespace CVC4
 599
 600 #endif /* CVC4__EXPR__PROOF_RULE_H */