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   //   Rewriter{idr}(t*sigma{ids}(Fn)*...*sigma{ids}(F1))
 106   //
 107   // In other words, from the point of view of Skolem forms, this rule
 108   // transforms t to t' by standard substitution + rewriting.
 109   //
 110   // The argument ids and idr is optional and specify the identifier of the
 111   // substitution and rewriter respectively to be used. For details, see
 112   // theory/builtin/proof_checker.h.
 113   MACRO_SR_EQ_INTRO,
 114   // ======== Substitution + Rewriting predicate introduction
 115   //
 116   // In this rule, we provide a formula F and conclude it, under the condition
 117   // that it rewrites to true under a proven substitution.
 118   //
 119   // Children: (P1:F1, ..., Pn:Fn)
 120   // Arguments: (F, (ids (idr)?)?)
 121   // ---------------------------------------------------------------
 122   // Conclusion: F
 123   // where
 124   //   Rewriter{idr}(toWitness(F)*sigma{ids}(Fn)*...*sigma{ids}(F1)) == true
 125   // where ids and idr are method identifiers.
 126   //
 127   // Notice that we apply rewriting on the witness form of F, meaning that this
 128   // rule may conclude an F whose Skolem form is justified by the definition of
 129   // its (fresh) Skolem variables. Furthermore, notice that the rewriting and
 130   // substitution is applied only within the side condition, meaning the
 131   // rewritten form of the witness form of F does not escape this rule.
 132   MACRO_SR_PRED_INTRO,
 133   // ======== Substitution + Rewriting predicate elimination
 134   //
 135   // In this rule, if we have proven a formula F, then we may conclude its
 136   // rewritten form under a proven substitution.
 137   //
 138   // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn)
 139   // Arguments: ((ids (idr)?)?)
 140   // ----------------------------------------
 141   // Conclusion: F'
 142   // where
 143   //   F' is
 144   //   Rewriter{idr}(F*sigma{ids}(Fn)*...*sigma{ids}(F1)).
 145   // where ids and idr are method identifiers.
 146   //
 147   // We rewrite only on the Skolem form of F, similar to MACRO_SR_EQ_INTRO.
 148   MACRO_SR_PRED_ELIM,
 149   // ======== Substitution + Rewriting predicate transform
 150   //
 151   // In this rule, if we have proven a formula F, then we may provide a formula
 152   // G and conclude it if F and G are equivalent after rewriting under a proven
 153   // substitution.
 154   //
 155   // Children: (P1:F, P2:F1, ..., P_{n+1}:Fn)
 156   // Arguments: (G, (ids (idr)?)?)
 157   // ----------------------------------------
 158   // Conclusion: G
 159   // where
 160   //   Rewriter{idr}(toWitness(F)*sigma{ids}(Fn)*...*sigma{ids}(F1)) ==
 161   //   Rewriter{idr}(toWitness(G)*sigma{ids}(Fn)*...*sigma{ids}(F1))
 162   //
 163   // Notice that we apply rewriting on the witness form of F and G, similar to
 164   // MACRO_SR_PRED_INTRO.
 165   MACRO_SR_PRED_TRANSFORM,
 166
 167   //================================================= Boolean rules
 168   // ======== Split
 169   // Children: none
 170   // Arguments: (F)
 171   // ---------------------
 172   // Conclusion: (or F (not F))
 173   SPLIT,
 174   // ======== And elimination
 175   // Children: (P:(and F1 ... Fn))
 176   // Arguments: (i)
 177   // ---------------------
 178   // Conclusion: (Fi)
 179   AND_ELIM,
 180   // ======== And introduction
 181   // Children: (P1:F1 ... Pn:Fn))
 182   // Arguments: ()
 183   // ---------------------
 184   // Conclusion: (and P1 ... Pn)
 185   AND_INTRO,
 186   // ======== Not Or elimination
 187   // Children: (P:(not (or F1 ... Fn)))
 188   // Arguments: (i)
 189   // ---------------------
 190   // Conclusion: (not Fi)
 191   NOT_OR_ELIM,
 192   // ======== Implication elimination
 193   // Children: (P:(=> F1 F2))
 194   // Arguments: ()
 195   // ---------------------
 196   // Conclusion: (or (not F1) F2)
 197   IMPLIES_ELIM,
 198   // ======== Not Implication elimination version 1
 199   // Children: (P:(not (=> F1 F2)))
 200   // Arguments: ()
 201   // ---------------------
 202   // Conclusion: (F1)
 203   NOT_IMPLIES_ELIM1,
 204   // ======== Not Implication elimination version 2
 205   // Children: (P:(not (=> F1 F2)))
 206   // Arguments: ()
 207   // ---------------------
 208   // Conclusion: (not F2)
 209   NOT_IMPLIES_ELIM2,
 210   // ======== Equivalence elimination version 1
 211   // Children: (P:(= F1 F2))
 212   // Arguments: ()
 213   // ---------------------
 214   // Conclusion: (or (not F1) F2)
 215   EQUIV_ELIM1,
 216   // ======== Equivalence elimination version 2
 217   // Children: (P:(= F1 F2))
 218   // Arguments: ()
 219   // ---------------------
 220   // Conclusion: (or F1 (not F2))
 221   EQUIV_ELIM2,
 222   // ======== Not Equivalence elimination version 1
 223   // Children: (P:(not (= F1 F2)))
 224   // Arguments: ()
 225   // ---------------------
 226   // Conclusion: (or F1 F2)
 227   NOT_EQUIV_ELIM1,
 228   // ======== Not Equivalence elimination version 2
 229   // Children: (P:(not (= F1 F2)))
 230   // Arguments: ()
 231   // ---------------------
 232   // Conclusion: (or (not F1) (not F2))
 233   NOT_EQUIV_ELIM2,
 234   // ======== XOR elimination version 1
 235   // Children: (P:(xor F1 F2)))
 236   // Arguments: ()
 237   // ---------------------
 238   // Conclusion: (or F1 F2)
 239   XOR_ELIM1,
 240   // ======== XOR elimination version 2
 241   // Children: (P:(xor F1 F2)))
 242   // Arguments: ()
 243   // ---------------------
 244   // Conclusion: (or (not F1) (not F2))
 245   XOR_ELIM2,
 246   // ======== Not XOR elimination version 1
 247   // Children: (P:(not (xor F1 F2)))
 248   // Arguments: ()
 249   // ---------------------
 250   // Conclusion: (or F1 (not F2))
 251   NOT_XOR_ELIM1,
 252   // ======== Not XOR elimination version 2
 253   // Children: (P:(not (xor F1 F2)))
 254   // Arguments: ()
 255   // ---------------------
 256   // Conclusion: (or (not F1) F2)
 257   NOT_XOR_ELIM2,
 258   // ======== ITE elimination version 1
 259   // Children: (P:(ite C F1 F2))
 260   // Arguments: ()
 261   // ---------------------
 262   // Conclusion: (or (not C) F1)
 263   ITE_ELIM1,
 264   // ======== ITE elimination version 2
 265   // Children: (P:(ite C F1 F2))
 266   // Arguments: ()
 267   // ---------------------
 268   // Conclusion: (or C F2)
 269   ITE_ELIM2,
 270   // ======== Not ITE elimination version 1
 271   // Children: (P:(not (ite C F1 F2)))
 272   // Arguments: ()
 273   // ---------------------
 274   // Conclusion: (or (not C) (not F1))
 275   NOT_ITE_ELIM1,
 276   // ======== Not ITE elimination version 1
 277   // Children: (P:(not (ite C F1 F2)))
 278   // Arguments: ()
 279   // ---------------------
 280   // Conclusion: (or C (not F2))
 281   NOT_ITE_ELIM2,
 282   // ======== Not ITE elimination version 1
 283   // Children: (P1:P P2:(not P))
 284   // Arguments: ()
 285   // ---------------------
 286   // Conclusion: (false)
 287   CONTRA,
 288
 289   //================================================= De Morgan rules
 290   // ======== Not And
 291   // Children: (P:(not (and F1 ... Fn))
 292   // Arguments: ()
 293   // ---------------------
 294   // Conclusion: (or (not F1) ... (not Fn))
 295   NOT_AND,
 296   //================================================= CNF rules
 297   // ======== CNF And Pos
 298   // Children: ()
 299   // Arguments: ((and F1 ... Fn), i)
 300   // ---------------------
 301   // Conclusion: (or (not (and F1 ... Fn)) Fi)
 302   CNF_AND_POS,
 303   // ======== CNF And Neg
 304   // Children: ()
 305   // Arguments: ((and F1 ... Fn))
 306   // ---------------------
 307   // Conclusion: (or (and F1 ... Fn) (not F1) ... (not Fn))
 308   CNF_AND_NEG,
 309   // ======== CNF Or Pos
 310   // Children: ()
 311   // Arguments: ((or F1 ... Fn))
 312   // ---------------------
 313   // Conclusion: (or (not (or F1 ... Fn)) F1 ... Fn)
 314   CNF_OR_POS,
 315   // ======== CNF Or Neg
 316   // Children: ()
 317   // Arguments: ((or F1 ... Fn), i)
 318   // ---------------------
 319   // Conclusion: (or (or F1 ... Fn) (not Fi))
 320   CNF_OR_NEG,
 321   // ======== CNF Implies Pos
 322   // Children: ()
 323   // Arguments: ((implies F1 F2))
 324   // ---------------------
 325   // Conclusion: (or (not (implies F1 F2)) (not F1) F2)
 326   CNF_IMPLIES_POS,
 327   // ======== CNF Implies Neg version 1
 328   // Children: ()
 329   // Arguments: ((implies F1 F2))
 330   // ---------------------
 331   // Conclusion: (or (implies F1 F2) F1)
 332   CNF_IMPLIES_NEG1,
 333   // ======== CNF Implies Neg version 2
 334   // Children: ()
 335   // Arguments: ((implies F1 F2))
 336   // ---------------------
 337   // Conclusion: (or (implies F1 F2) (not F2))
 338   CNF_IMPLIES_NEG2,
 339   // ======== CNF Equiv Pos version 1
 340   // Children: ()
 341   // Arguments: ((= F1 F2))
 342   // ---------------------
 343   // Conclusion: (or (not (= F1 F2)) (not F1) F2)
 344   CNF_EQUIV_POS1,
 345   // ======== CNF Equiv Pos version 2
 346   // Children: ()
 347   // Arguments: ((= F1 F2))
 348   // ---------------------
 349   // Conclusion: (or (not (= F1 F2)) F1 (not F2))
 350   CNF_EQUIV_POS2,
 351   // ======== CNF Equiv Neg version 1
 352   // Children: ()
 353   // Arguments: ((= F1 F2))
 354   // ---------------------
 355   // Conclusion: (or (= F1 F2) F1 F2)
 356   CNF_EQUIV_NEG1,
 357   // ======== CNF Equiv Neg version 2
 358   // Children: ()
 359   // Arguments: ((= F1 F2))
 360   // ---------------------
 361   // Conclusion: (or (= F1 F2) (not F1) (not F2))
 362   CNF_EQUIV_NEG2,
 363   // ======== CNF Xor Pos version 1
 364   // Children: ()
 365   // Arguments: ((xor F1 F2))
 366   // ---------------------
 367   // Conclusion: (or (not (xor F1 F2)) F1 F2)
 368   CNF_XOR_POS1,
 369   // ======== CNF Xor Pos version 2
 370   // Children: ()
 371   // Arguments: ((xor F1 F2))
 372   // ---------------------
 373   // Conclusion: (or (not (xor F1 F2)) (not F1) (not F2))
 374   CNF_XOR_POS2,
 375   // ======== CNF Xor Neg version 1
 376   // Children: ()
 377   // Arguments: ((xor F1 F2))
 378   // ---------------------
 379   // Conclusion: (or (xor F1 F2) (not F1) F2)
 380   CNF_XOR_NEG1,
 381   // ======== CNF Xor Neg version 2
 382   // Children: ()
 383   // Arguments: ((xor F1 F2))
 384   // ---------------------
 385   // Conclusion: (or (xor F1 F2) F1 (not F2))
 386   CNF_XOR_NEG2,
 387   // ======== CNF ITE Pos version 1
 388   // Children: ()
 389   // Arguments: ((ite C F1 F2))
 390   // ---------------------
 391   // Conclusion: (or (not (ite C F1 F2)) (not C) F1)
 392   CNF_ITE_POS1,
 393   // ======== CNF ITE Pos version 2
 394   // Children: ()
 395   // Arguments: ((ite C F1 F2))
 396   // ---------------------
 397   // Conclusion: (or (not (ite C F1 F2)) C F2)
 398   CNF_ITE_POS2,
 399   // ======== CNF ITE Pos version 3
 400   // Children: ()
 401   // Arguments: ((ite C F1 F2))
 402   // ---------------------
 403   // Conclusion: (or (not (ite C F1 F2)) F1 F2)
 404   CNF_ITE_POS3,
 405   // ======== CNF ITE Neg version 1
 406   // Children: ()
 407   // Arguments: ((ite C F1 F2))
 408   // ---------------------
 409   // Conclusion: (or (ite C F1 F2) (not C) (not F1))
 410   CNF_ITE_NEG1,
 411   // ======== CNF ITE Neg version 2
 412   // Children: ()
 413   // Arguments: ((ite C F1 F2))
 414   // ---------------------
 415   // Conclusion: (or (ite C F1 F2) C (not F2))
 416   CNF_ITE_NEG2,
 417   // ======== CNF ITE Neg version 3
 418   // Children: ()
 419   // Arguments: ((ite C F1 F2))
 420   // ---------------------
 421   // Conclusion: (or (ite C F1 F2) (not F1) (not F2))
 422   CNF_ITE_NEG3,
 423
 424   //================================================= Equality rules
 425   // ======== Reflexive
 426   // Children: none
 427   // Arguments: (t)
 428   // ---------------------
 429   // Conclusion: (= t t)
 430   REFL,
 431   // ======== Symmetric
 432   // Children: (P:(= t1 t2)) or (P:(not (= t1 t2)))
 433   // Arguments: none
 434   // -----------------------
 435   // Conclusion: (= t2 t1) or (not (= t2 t1))
 436   SYMM,
 437   // ======== Transitivity
 438   // Children: (P1:(= t1 t2), ..., Pn:(= t{n-1} tn))
 439   // Arguments: none
 440   // -----------------------
 441   // Conclusion: (= t1 tn)
 442   TRANS,
 443   // ======== Congruence  (subsumed by Substitute?)
 444   // Children: (P1:(= t1 s1), ..., Pn:(= tn sn))
 445   // Arguments: (f)
 446   // ---------------------------------------------
 447   // Conclusion: (= (f t1 ... tn) (f s1 ... sn))
 448   CONG,
 449   // ======== True intro
 450   // Children: (P:F)
 451   // Arguments: none
 452   // ----------------------------------------
 453   // Conclusion: (= F true)
 454   TRUE_INTRO,
 455   // ======== True elim
 456   // Children: (P:(= F true)
 457   // Arguments: none
 458   // ----------------------------------------
 459   // Conclusion: F
 460   TRUE_ELIM,
 461   // ======== False intro
 462   // Children: (P:(not F))
 463   // Arguments: none
 464   // ----------------------------------------
 465   // Conclusion: (= F false)
 466   FALSE_INTRO,
 467   // ======== False elim
 468   // Children: (P:(= F false)
 469   // Arguments: none
 470   // ----------------------------------------
 471   // Conclusion: (not F)
 472   FALSE_ELIM,
 473
 474   //================================================= Quantifiers rules
 475   // ======== Witness intro
 476   // Children: (P:F[t])
 477   // Arguments: (t)
 478   // ----------------------------------------
 479   // Conclusion: (= t (witness ((x T)) F[x]))
 480   // where x is a BOUND_VARIABLE unique to the pair F,t.
 481   WITNESS_INTRO,
 482   // ======== Exists intro
 483   // Children: (P:F[t])
 484   // Arguments: (t)
 485   // ----------------------------------------
 486   // Conclusion: (exists ((x T)) F[x])
 487   // where x is a BOUND_VARIABLE unique to the pair F,t.
 488   EXISTS_INTRO,
 489   // ======== Skolemize
 490   // Children: (P:(exists ((x1 T1) ... (xn Tn)) F))
 491   // Arguments: none
 492   // ----------------------------------------
 493   // Conclusion: F*sigma
 494   // sigma maps x1 ... xn to their representative skolems obtained by
 495   // SkolemManager::mkSkolemize, returned in the skolems argument of that
 496   // method.
 497   SKOLEMIZE,
 498   // ======== Instantiate
 499   // Children: (P:(forall ((x1 T1) ... (xn Tn)) F))
 500   // Arguments: (t1 ... tn)
 501   // ----------------------------------------
 502   // Conclusion: F*sigma
 503   // sigma maps x1 ... xn to t1 ... tn.
 504   INSTANTIATE,
 505
 506   // ======== Adding Inequalities
 507   // Note: an ArithLiteral is a term of the form (>< poly const)
 508   // where
 509   //   >< is >=, >, ==, <, <=, or not(== ...).
 510   //   poly is a polynomial
 511   //   const is a rational constant
 512
 513   // Children: (P1:l1, ..., Pn:ln)
 514   //           where each li is an ArithLiteral
 515   //           not(= ...) is dis-allowed!
 516   //
 517   // Arguments: (k1, ..., kn), non-zero reals
 518   // ---------------------
 519   // Conclusion: (>< (* k t1) (* k t2))
 520   //    where >< is the fusion of the combination of the ><i, (flipping each it
 521   //    its ki is negative). >< is always one of <, <=
 522   //    NB: this implies that lower bounds must have negative ki,
 523   //                      and upper bounds must have positive ki.
 524   //    t1 is the sum of the polynomials.
 525   //    t2 is the sum of the constants.
 526   ARITH_SCALE_SUM_UPPER_BOUNDS,
 527
 528   // ======== Tightening Strict Integer Upper Bounds
 529   // Children: (P:(< i c))
 530   //         where i has integer type.
 531   // Arguments: none
 532   // ---------------------
 533   // Conclusion: (<= i greatestIntLessThan(c)})
 534   INT_TIGHT_UB,
 535
 536   // ======== Tightening Strict Integer Lower Bounds
 537   // Children: (P:(> i c))
 538   //         where i has integer type.
 539   // Arguments: none
 540   // ---------------------
 541   // Conclusion: (>= i leastIntGreaterThan(c)})
 542   INT_TIGHT_LB,
 543
 544   // ======== Trichotomy of the reals
 545   // Children: (A B)
 546   // Arguments: (C)
 547   // ---------------------
 548   // Conclusion: (C),
 549   //                 where (not A) (not B) and C
 550   //                   are (> x c) (< x c) and (= x c)
 551   //                   in some order
 552   //                 note that "not" here denotes arithmetic negation, flipping
 553   //                 >= to <, etc.
 554   ARITH_TRICHOTOMY,
 555
 556   // ======== Arithmetic operator elimination
 557   // Children: none
 558   // Arguments: (t)
 559   // ---------------------
 560   // Conclusion: arith::OperatorElim::getAxiomFor(t)
 561   ARITH_OP_ELIM_AXIOM,
 562
 563   // ======== Int Trust
 564   // Children: (P1 ... Pn)
 565   // Arguments: (Q)
 566   // ---------------------
 567   // Conclusion: (Q)
 568   INT_TRUST,
 569
 570   //================================================= Unknown rule
 571   UNKNOWN,
 572 };
 573
 574 /**
 575  * Converts a proof rule to a string. Note: This function is also used in
 576  * `safe_print()`. Changing this function name or signature will result in
 577  * `safe_print()` printing "<unsupported>" instead of the proper strings for
 578  * the enum values.
 579  *
 580  * @param id The proof rule
 581  * @return The name of the proof rule
 582  */
 583 const char* toString(PfRule id);
 584
 585 /**
 586  * Writes a proof rule name to a stream.
 587  *
 588  * @param out The stream to write to
 589  * @param id The proof rule to write to the stream
 590  * @return The stream
 591  */
 592 std::ostream& operator<<(std::ostream& out, PfRule id);
 593
 594 }  // namespace CVC4
 595
 596 #endif /* CVC4__EXPR__PROOF_RULE_H */