Cambridge University Press 978-0-521-89957-4 - Handbook of Practical and Automated Reasoning John Harrison Index More information

Index

(x, y) (pair), 594 ---, 618 − ( of literal), 51 -/, 617 − (set difference), 594 //, 617 C− (negation of literal set), 181 ::, 616 ∧ (and), 27 logical consequence), 40, 130 Add,14 |=M (holds in M), 130 add 0, 565 ℘ (power set), 598 add assum, 480 ⊂ (proper ), 594 add assum’, 497 → (sequent), 471 add default, 436 \ (set difference), 594 add suc, 565 ⊆ (subset), 594 adequate set (of connectives), 46 × (Cartesian product), 594 adjustcoeff, 342 → (function space), 595 AE fragment, 309 → (reduction relation), 258 aedecide, 310 →∗ (reflexive transitive closure of →), 258 affine transformation, 417 →+ (transitive closure of →), 258 affirmative negative rule,81  (provability), 246, 470, 474 afn dlo, 335 {1, 2, 3} (set enumeration), 594 al-Khwarizmi, 6 **, 618 algebra, 6 */, 617 algebra of logic, 7 +/, 617 algebraic number, 527 --, 618 algebraically closed, 352, 397

668

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 669

align,92 axiom distribimp, 477 alldepmappings, 322 axiom doubleneg, 477 allfunctions, 322 axiom eqrefl, 477 allmappings, 322 axiom exists, 477 allnonemptysubsets, 620 axiom existseq, 477 allpairs, 620 axiom funcong, 477 allpartitions, 444 axiom iffimp1, 477 allpredicates, 322 axiom iffimp2, 477 allsatvaluations,56 axiom impall, 477 allsets, 620 axiom impiff, 477 allsubsets, 620 axiom not, 477 alltuples, 322 axiom or, 477 alpha, 493 axiom predcong, 477 alpha-convert, 133 axiom true, 477 alphanumeric,17 axiomatizable, 329 alternation, 8 axiomatized, 329 analogue computer, 63 analytic tableaux, 176 backchain, 207 And,26 backjump,88 and left, 483 backjumping, 88 and pair, 484 backtrack,87 and right, 483 Backus–Naur form, 19 andcnf,78 backward deletion, 190 andcnf3,79 backward replacement, 190 anglicize premiss, 318 bag, 597 anglicize syllogism, 318 BAPA, 454 ante disj, 513 basic complex qelim, 365 antecedent, 31 basic real qelim, 375 antecedent,31 BDD, 100 antisymmetric relation, 595 Bdd, 102 apply, 621 bdd, 102 arity, 119 bdd and, 103 arrangement, 441 bdd iff, 104 arrangement, 442 bdd imp, 104 arreq, 442 bdd or, 104 ASCII, 11 bddnode, 101 askolemize, 149 bddtaut, 104 assertsign, 362 behead, 359 assignment, 131 bell, 444 assoc, 620 Bell number, 444 association list, 621 belongs, 439 associative, 593 bi-implication, 30, 39 associative–commutative, 285 biconditional, 30 associativity, 12 bijection, 596 assume, 515 bijective, 596 assumps, 511 binary, 119 assumptate, 510 binary decision diagram, 100 AST, 12 binary decision tree, 99 at once, 512 bind, 120, 605 Atom,26 Birkhoff rules, 246 atom, 25 Birkhoff’s theorem, 246 atom, 92, 318 bit, 65 atom union,32 bit-blasting, 455 atomic propositions, 25 bitlength,71 atoms,35 Blank, 558 auto tac, 512 bmeson, 296 axiom, 3, 474 BNF, 19 Axiom of Choice, 144, 598 bool interp, 125 axiom addimp, 477 Boole, 7 axiom allimp, 477 Boolean variable, 25 axiom and, 477 bottom-up method, 172

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

670 Index

bound, 120 commutative, 7, 593 bound variable, 118 compactness (first-order logic), 227 bounded prove, 572 compactness (first-order with ), 242 boundednum prove, 573 compactness (propositional logic), 107 boundquant step, 571 compatible, 384, 585 bpuremeson, 296 complement edge, 100 bracket, 18 complementary literals, 51, 58 bracket, 626 complete, 247 branching quantifier, 150 complete, 279 brand, 296 complete (proof system), 247 Brand’s transformation, 289 complete (theory), 245, 329 bset, 345 complete induction, 601 Buchberger’s algorithm, 410 complete and simplify, 283 butlast, 618 completeness (first-order logic), 504 by, 511 completion, 278 byte, 65 complex qelim, 366 complits’, 498 C (complex numbers), 594 composition, 596 CAD, 367 computable function, 560 calculemus, 4 computed table (BDD), 103 calculus ratiocinator, 4 concl, 477 can, 618 conclude, 517 canonical, 256, 262 condense, 370 canonical , 152 conditional, 37 canonize, 621 Config, 559 canonizer, 448 config, 559 cardinality, 597 conflict, 87 carry,66 conflict clause, 89 carry-select adder, 68 confluent, 258 carryselect,68 congruence, 236, 249 Cartesian product, 594 congruence closure, 249 cases, 517 congruent, 594 casesplit, 374 congruent, 250 catch, 616 congruent to,71 categorical, 245 conj intro tac, 508 categoricity, 245 conjoin,67 ccsatisfiable, 253 conjths, 483 ccvalid, 253 certificate, 519 conjunct, 30 characteristic function, 597 conjunction, 8, 30 characteristic of a ring/field, 382 , 54 characteristica universalis, 4 conjuncts,30 chooselang, 437 connected relation, 595 chop list, 618 connection tableaux, 215 Church’s theorem, 564 connective, 8 Church’s thesis, 555 consequences,93 Church–Rosser, 260 consequent, 31 Church–Turing thesis, 555 consequent,31 cinterpolate, 434 conservative, 150 , 528 consider, 517 classify, 550 consistent (theory), 329 clausal, 80 Const,14 clause, 80 constructive proof, 527 Cn, 244 constructor, 612 CNF, 54 continuation, 176 cnf,61 , 39 cnnf, 332 contradiction (principle), 527 codomain, 596 contrapos, 483 coefficients, 358 , 3, 45 cofactors, 387 contrapositive, 45, 214 cofinite, 598 contrapositives, 219

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 671

convergent, 256 dest and,30 convex, 204, 447 dest def, 443 cooper, 348 dest eq, 235 Cooper’s algorithm, 341 dest iff,30 coordinate, 415 dest iffdef, 105 coordinations, 414 dest imp,31 countable, 598 dest nimp, 105 countermodel, 323 dest numeral, 337 cqelim, 365 dest or,30 Craig interpolation theorem, 427 dholds, 548 crit1, 277 dhquant, 549 critical pair, 275 diag, 537, 539 critical pairs, 277 diagonal lemma, 540 , 608 diagonalization, 537 cut, 472 diamond property, 258 cut-free, 472 Dickson’s lemma, 411 cylindrical algebraic decomposition, 367 difference logic, 349 differential algebra, 425 Davis–Putnam procedure (first-order), 162 digital computer, 63 Davis–Putnam procedure (propositional), 79 dilemma rule, 90 davisputnam, 163 Diophantine set, 580 De Morgan’s laws, 46 direction, 558 decidable (theory), 329 disequation, 593 decide finite, 322 disj elim tac, 513 decide fmp, 323 disjoint union, 613 decide monadic, 324 disjunct, 30 decision literal, 87 disjunction, 8, 30 decision problem, 308 , 54 declarative programming, 212 disjuncts,30 declarative proof , 516 distinctpairs, 620 decreasing, 619 distrib, 57, 58 Dedekind infinite, 598 divides, 593 dedmatrix, 372 divlcm, 344 Deduced,86 Dixon resultant, 425 deduction theorem, 505 DLO, 333 deepen, 177 dlobasic, 334 default parser, 20, 29, 629 DNF, 54 defcnf, 77, 78 dnf, 56, 59 defcnf3,79 do list, 618 defcnfs,78 dom, 621 definable, 531 domain, 123, 596 defined, 621 double negation, 527 definite clause, 203 downward L¨owenheim–Skolem, 227, 242 definition, 605 DP, 79 definitional CNF, 73 dp,84 defstep,76 dp loop, 163 degree, 355 dp mfn, 163 degree, 358 dp refine, 163 degree (of a polynomial), 358 dp refine loop, 164 delayed theory combination, 450 dplb,89 delconst, 374 dplbsat,89 Delta, 550 dpli,87 demodulation, 255, 297 dplisat,88 dense linear order, 333 dplitaut,88 denumerable, 598 DPLL, 85 depth, 177 dpll,85 derivability conditions (L¨ob), 577 dpllsat,85 derived rule, 479 dplltaut,85 descending chain, 601 dpltaut,89 deskol’, 498 dpsat,84 deskolcont, 503 dptaut,84

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

672 Index

drinker’s principle, 129 exists left, 491 dtermval, 548 exists left th, 491 dual, 48 expand2, 222 dual,48 expand connective, 485 duality, 27, 49 expand le, 569 dyadic, 119 expand lt, 569 expand nle, 569 EA fragment, 311 expand nlt, 569 eager, 450 expand node, 102 earlier, 619 explode, 618 ebddtaut, 106 expression,14 edge (of graph), 62 extensionality, 594 einterpolate, 435 extract thm, 508 el, 618 elementarily equivalent, 245 fa,66 elementary equivalence, 245 factor, 183 elementary theory, 329 factoring, 183 elim bex, 570 ,26 elim skolemvar, 502 Fibonacci sequence, 609 eliminate connective, 485 field, 394 eliminate connective’, 498 filter, 619 emerge, 251 find, 619 emeson, 296 find count,85 emodify, 295 find nestnonvar, 294 empty set, 594 find nvsubterm, 295 end itlist, 618 findasubset, 445 Entscheidungsproblem, 555 findsign, 362 enumerated type, 612 findsubset, 445 eq sym, 489 finite model property, 321 eq trans, 489 finite partial function, 621 equal, 221 first, 618 equalitize, 240 first-order logic, 118 equality axioms, 235 firstassum, 511 equate, 621 fix, 517 equate2,92 fixpoint lemma, 540 equatecons,95 flat, 292 equated, 621 flattening, 292 equateset,96 Fn, 119 equisatisfiable, 73 fol, 119 equivalence, 30 Forall,26 equivalence classes, 595 forall, 618 equivalence relation, 595 forall2, 618 equivalence axioms, 240 forall intro tac, 509 equivalent, 621 form match, 500 equivalid, 73 formal degree (of a polynomial), 359 essentially undecidable, 585 formal verification, 107, 521 Euclid, 3 formalizing mathematics, 521 eval,32 formlcm, 341 evalc, 348 formula, 25, 118 ex falso quodlibet, 482 formula,26 ex falso, 482 formulaclass, 550 ex falso’, 498 forward deletion, 190 exception, 615 FPF, 621 excluded middle (law), 527 fpf, 621 , 48, 66 free, 120 exec, 560 free logic, 123 existential quantifier, 120 free word problem, 380 Exists,26 free in, 476 exists, 618 Frege proof system, 470 exists elim tac, 513 full adder, 66 exists intro tac, 510 fullunify, 170

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 673

fully solved, 169 halfsum,66 func, 621 handle, 616 funcs, 146 Hauptsatz, 472 function, 595 have, 516 function congruence, 239 hd, 618 functional programming, 603 head, 358 functional reflexive axioms, 298 head monomial, 402 functions, 146 headconst, 359 Fundamental Theorem of Algebra, 352 herbfuns, 156 funpow, 612 herbloop, 159 fv, 127 Herbrand base, 203 fvt, 127 Herbrand domain, 155 Herbrand interpretation, 156 gate, 64 Herbrand skeleton problem, 584 GCD, 593 Herbrand universe, 155 gcd num, 618 Herbrand’s theorem, 157 GCH, 578 Hermite normal form, 352 gen, 477 higher-order logic, 122 gen right, 490 Hilbert proof system, 470 gen right alpha, 509 Hilbert’s tenth problem, 580 gen right th, 490 Hilbert’s programme, 526 generalization (inference rule), 474 Hobbes, 3 generalize, 131 holds, 125 generalized continuum hypothesis, 578 holds in, 129 generic, 612 homo, 438 genimp, 490 homogenization, 437 gform, 533 homogenize, 438 gilmore, 160 homol, 438 Gilmore procedure, 158 homomorphism, 242 gilmore loop, 160 homot, 438 given clause, 184 Horn clause, 203 given clause algorithm, 185 Horner form, 352 global method, 172 hornify, 208 gnumeral, 537 hornprove, 208 goal, 506 hyperresolution, 201 Goals, 507 goals, 507 icongruence, 490 G¨odel’sfirst theorem, 546, 554 Id, 387 G¨odel’ssecond theorem, 576 ideal (in a ring), 387 grammar, 19 identity, 611 graph, 62 identity function, 596, 611 graph, 621 IF logic, 228 greatest common divisor, 593 iff, 26, 593 grobner, 410 Iff,26 Gr¨obner basis, 409 iff def, 485 grobner decide, 413 iff imp1, 482 grobner trivial, 412 iff imp2, 482 groebner, 412 image, 596 ground formula, 127 image, 620 ground term, 127 Imp,26 groundterms, 159 imp add assum, 480 groundtuples, 159 imp add concl, 481 group, 241 imp antisym, 482 grpform, 378 imp contr, 486 grpterm, 377 imp false conseqs, 486 gterm, 533 imp false rule, 486 Guessed,86 imp false rule’, 497 imp front, 487 ha,66 imp front’, 497 half adder, 66 imp front th, 487 halfcarry,66 imp insert, 480

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

674 Index

imp intro tac, 510 invariant under translation, 416 imp mono th, 483 inverse image, 597 imp refl, 478 inverse method, 172, 472 imp swap, 481 , 64 imp swap2, 481 irredundant,93 imp swap th, 481 irreflexive relation, 595 imp trans, 480 is const name, 628 imp trans2, 482 is constant, 358 imp trans chain, 482 is eq, 235 imp trans th, 481 is nonvar, 294 imp true rule, 486 is numeral, 337 imp true rule’, 497 is undefined, 621 imp truefalse, 483 isomorphic, 245 imp unduplicate, 479 isotropic, 423 imperative programming, 603 ispec, 493 implication, 37 istriv, 167 implies, 26 isubst, 492 implode, 618 iterative deepening, 177, 208 inclusive logic, 123 itlist, 618 inclusive or, 48 itlist2, 618 incorporate, 191 increasing, 619 jmodify, 509 independence-friendly logic, 228 joinable, 258 independent set of formulas, 116 justify, 511 index, 619 indicator function, 597 killer transformation, 289 inductive definitions, 598 Knuth–Bendix completion, 278 , 593 Kolmogorov complexity, 588 inequation, 593 Kruskal’s theorem, 303 inference rule, 246 inferisign, 371 langpartition, 439 inferpsign, 370 language, 119 init sgns, 365 last, 618 injection, 596 lazy, 450 injective, 596 LCF, 469 input resolution, 233 lcffol, 503 input tape, 560 lcfptab, 487 insert, 620 lcfrefute, 501 insertat, 618 lcftab, 499 instantiate, 594 lcftaut, 488 instantiation, 131 LCM (integers), 593 Int, 617 LCM (monomials), 405 int lang, 436 lcm num, 618 integer, 594 le 0, 569 integer qelim, 348 le def, 565 integral domain, 390 le suc, 569 intensional, 579 learning, 89 inter,97 least, 602 interpolant, 427 least Herbrand model, 203 interpolate, 434 least significant bit, 69 interpretation, 10, 123 Left, 558 interpreter, 604 Leibniz, 4 interreduce, 283 lemma tac, 512 intersect, 620 length, 618 intersection, 594 lerules, 210 introduce connective, 570 lex,18 intuitionism, 528 lexical analysis, 16 , 528 lexicographic, 602 invariant, 415 lexicographic path order, 266 invariant under rotation, 416 lexord, 267 invariant under scaling, 417 lexwhile,17

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 675

lfn dlo, 333 material implication, 38 lhs, 235 matrix, 139 lift qelim, 331 matrix, 375 lifting lemma, 181 max varindex,76 limited meson, 323 maximize, 619 limmeson, 323 mdiv, 384 linear, 336 measure function, 602 linear (pattern), 263 mem, 620 linear order, 333, 595 member, 594 linear resolution, 195 MESON, 214 linear add, 339 meson, 220 linear cmul, 338 metalanguage, 11 linear mul, 339 metalogic, 11 linear neg, 339 metamathematics, 11, 529 linear sub, 339 metatheorem, 11 linform, 340 mexpand, 220, 222 linrep, 347 mexpands, 223 lint, 340 MGU, 166 list conj,55 mgu, 183 list disj,55 middle term (syllogism), 317 listcases, 276 minimal, 601 listify, 437 minimal Herbrand model, 203 literal, 51 minimize, 619 local method, 172 miniscope, 314 , 64 miniscoping, 313 Logic of Computable Functions, 469 minusinf, 344 logic programming, 209 mk adder test,69 , 8 mk and,30 logical consequence, 43 mk bdd, 103 , 1 mk defcnf,76 logical reasoning, 1 mk eq, 235 logical validity, 129 mk exists,30 logically equivalent, 44, 129 mk forall,30 logically valid, 39, 129 mk iff,30 logisticos, 4 mk imp,30 logos, 4 mk index,67 look, 558 mk index2,67 lookup unique, 102 mk lits,56 Lo´ s–Vaught test, 304 mk node, 102 L¨owenheim–Skolem, 227, 242, 243 mk numeral, 337 lowest common multiple, 593 mk or,30 LPO, 266 mk skol, 501 lpo ge, 267 mkatom, 340 lpo gt, 267 mkbdd, 104 LSB, 69 mkbdde, 106 lt 0, 569 mkbdds, 106 lt def, 565 mkprop,75 lt suc, 569 ML, 11, 469 LUSH resolution, 213 mlcm, 384 mmul, 384 maincnf,75 Mod, 328 make parser,20 mod interp, 125 many-sorted logic, 229 model, 130 map, 618 model complete, 451 map2, 618 model elimination, 213 mapf, 621 model evolution, 173 mapfilter, 618 modify E, 295 match literals, 187 modify S, 290 matches,17 modify T, 291 matching, 186 modulo, 594 material conditional, 38 , 473

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

676 Index

modusponens, 477 nextin, 624 monadic, 119, 315 NNF, 51 monic, 359, 362 nnf, 52, 141 monic, 359 noetherian, 258 monoid, 398 non, 618 monoid ring, 399 non-chronological backjumping, 88 monomial, 383 Nonzero, 362 monotone, 599 normal form, 49, 258 monotonic, 599 normal interpretation, 236 monotonic truth-function, 53 normal model, 236 morder lt, 384 normalize and orient, 278 most general unifier, 166 Not,26 move, 559 note, 516 mpoly add, 385 NP-complete, 72 mpoly const, 385 nullary, 119 mpoly div, 386 Nullstellensatz (Hilbert), 394 mpoly inv, 386 Nullstellensatz (real), 453 mpoly mmul, 385 num cases, 565 mpoly mul, 386 num lecases, 569 mpoly neg, 385 num of string, 617 mpoly pow, 386 number, 532 mpoly sub, 385 numeral, 530 mpoly var, 385 numeral1, 338 mpolyatom, 386 numeral2, 338 mpolynate, 386 numeric,17 Mul,14 mul 0, 565 o-minimal, 451 mul suc, 565 object language, 11 multidegree, 384 object logic, 11 multiplexer, 64 Objective CAML, 603 multiplier,70 OCaml, 11, 603 multiset, 597 occurs check, 212 multishunt, 515 occurs in, 476 mutual recursion, 610 octet, 65 mux,68 offline, 450 offset,68 N (natural numbers), 594 Omega test, 451 n-ary, 119 onallvaluations,35 n-easy, 91 onatoms,31 n-hard, 91 One, 558 narrowing, 255 one–one, 596 natural deduction, 470 one-to-one correspondence, 596 natural number, 594 one literal rule,81 natural qelim, 350 online, 450 negate,51 operations, 348 negatef, 479 opp, 550 negating, 51 Or,26 negation, 51 orcnf,78 negation, logical, 8 order, 595 negation normal form, 51 order, 103 Negative, 362 originate, 416 negative,51 our thesis, 517 negative literal, 51 output tape, 560 negative resolution, 198 overatoms,31 negativef, 479 overlapc, 301 nelop, 444 overlapl, 301 nelop1, 444, 446 overlaps, 276 nelop refute, 444, 446 overloading, 612 Nelson–Oppen combination method, 435 nenf,53 P,28 Newman’s lemma, 260 P1-resolution, 196

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 677

PA, 546 polyatom, 355 pair, 594 polymorphic, 612 pair, 532 polymorphism, 612 papply, 624 polynate, 354 para clauses, 302 polynomial, 352, 355 paraloop, 302 posineq, 341 parametric polymorphism, 612 Positive, 362 paramodulate, 301 positive,51 paramodulation, 297 positive literal, 51 paramodulation, 302 positive resolution, 196 parentheses, 18 power set, 598 parse, 629 pprove, 420 parse atom, 20, 628 prawitz, 175 parse atomic formula, 625 Prawitz procedure, 175 parse atomic term, 628 prawitz loop, 175 parse bracketed, 624 precedence, 12 parse expression,19 predecessors, 252 parse formula, 625 predicate, 119 parse ginfix, 623 predicate (syllogism), 317 parse infix atom, 628 predicate logic, 118 parse left infix, 624 predicate congruence, 240 parse list, 624 predicates, 239 parse product,19 premiss A, 318 parse prop formula,29 premiss E, 318 parse propvar,29 premiss I, 318 parse quant, 625 premiss O, 318 parse right infix, 624 prenex, 143 parse term, 628 , 139 parserule, 209 preorder, 595 parset, 628 Presburger arithmetic, 337 parsing, 12, 16 presolution, 198 partial CAD, 367 presolve clauses, 198 partial computable function, 560 prettyprinting, 12 partial evaluation, 609 prime,71 partial function, 596 prime form, 549 partial order, 595 primitive recursive function, 536 partition, 619, 621 primitive rule (LCF), 475 pdivide, 360 Principia Mathematica, 473 pdivide pos, 373 print atom, 630 Peano arithmetic, 546 print bdd, 102 pholds, 151 print exp,22 Pi, 550 print fargs, 629 pinterpolate, 428 print fol formula, 630 planar, 110 print formula, 627 pname,28 print goal, 507 PNF, 139 print infix term, 630 pnf, 144 print int, 617 poly add, 353 print prop formula,29 poly cmul, 359 print propvar,29 poly diff, 368 print qformula, 627 poly diffn, 368 print string, 617 poly div, 354 print term, 629 poly ladd, 353 print thm, 477 poly lmul, 354 print truthtable,36 poly mul, 354 printert, 630 poly neg, 353 procedural, 209 poly nondiv, 364 procedural proof, 515 poly nonzero, 363 product, 613 poly pow, 354 Prolog, 209 poly sub, 353 prolog, 211 poly var, 354 Prolog technology theorem prover, 214

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

678 Index

proof, 512 recursive learning, 112 proof planning, 465 recursive type, 614 proof rule, 246 recursively enumerable, 563 , 520 redeqs, 443 prop,28 reduce, 404 proper subset, 594 reduce1, 404 proposition, 7, 8 reduceb, 404 propositional logic, 25 reduced ordered binary decision diagram, 100 propositional variable, 25 reduced ring, 459 prove, 508 reduction order, 264 pseudo-division, 360 reflection, 519 psimplify,50 reflexive relation, 595 psimplify1,50 reflexive transitive closure, 600 psubset, 620 refutation complete, 182 psubst,41 rel signs, 369 PTTP, 214 relation, 119, 595 pullq, 143 relative computability, 588 pullquants, 143 relativization of quantifiers, 350 punctuation,17 relativize, 350 pure literal rule, 81 relevance,95 pure paramodulation, 302 rename, 183 pure presolution, 198 rename form, 225 pure resolution, 185, 201 rename term, 225 purecnf,60 renamepair, 276 purednf,58 renamerule, 207 puremeson, 220 repeat, 618 purification, 437 replace, 191, 295 pushquant, 314 replacement, 191 replacet, 295 Q (rational numbers), 594 replicate, 618 QBF, 112 resloop, 185, 192 qdiag, 539 resolution, 83, 172, 179 qed, 517 resolution, 185 qelim, 331 resolution (first-order), 180 quadrance, 414 resolution blowup,83 quantforms, 500 resolution rule,83 quantified Boolean formulas, 112 resolve clauses, 184 quantifier, 38, 118 resolve on,83 quantifier elimination, 330 resolvent, 83, 182 quantifier-free formula, 146 resolvents, 184 quelim dlo, 335 restart, 90 quine, 539 restore iffdef, 105 rev, 618 R (real numbers), 594 reverseq,97 R, 119 rewrite, 263 r.e., 563 rewrite order, 264 rabinowitsch, 412 rewrite rule, 255 Rabinowitsch trick, 396 rewrite1, 262 raise, 615 rewriting, 255 ramsey,63 rhs, 235 Ramsey’s theorem, 62 Right, 558 ran, 621 right doubleneg, 482 range, 596 right exists, 509 rawdnf,57 right imp trans, 565 read–eval–print loop, 605 right mp, 481, 565 real lang, 436 right spec, 565 real qelim, 375 right sym, 565 real qelim’, 379 right trans, 565 recursive, 604, 609 ring, 381 recursive (computable), 564 ripplecarry,67 recursive descent parsing, 19 ripplecarry0,67

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 679

ripplecarry1,68 signature, 119 rippleshift,70 simpcnf,60 rob eq, 569 simpcont, 501 rob ne, 570 simpdnf,59 rob nen, 569 simpleprolog, 210 Robbins conjecture, 285, 304 simplification order, 271 ROBDD, 100 simplify, 15, 140 robeval, 566 simplify1, 15, 140 robinson, 565 singleton set, 594 robinson consequences, 567 skolem, 149 robinson thm, 567 Skolem constant, 146 robop, 566 Skolem function, 146 root (of a polynomial), 356 Skolem normal form, 231 rule induction, 600 skolem2, 149 run, 559 skolemfuns, 500 Russell’s paradox, 470 Skolemization, 144 skolemize, 149 S-polynomial, 408 skolemizes, 226 SAM, 467 Skolemizing, 145 SAM’s lemma, 467 skolems, 226 SAT, 72 SL resolution, 213 satisfiability modulo theories, 449 SLD resolution, 213 satisfiable, 39, 129 small model property, 324 satisfiable,41 SMT, 449 satisfies, 39, 129 SN, 258 saturate,98 so, 516 saturate upto,98 solve, 169 scaling, 417 solver, 448 scanning, 16 sort, 229 schema, 474 sort, 619 scope, 120 sort defs, 106 second-order logic, 122, 145 sound, 247 secondary parser, 629 sound (proof system), 247 semantic resolution, 199 space,17 semantical incompleteness, 545 spec, 494 semantics, 10 spec’, 498 semantics of first-order logic, 123 specialize, 149 semicomputable, 563 spectrum, 327 semidecidable, 563 split, 356 semidefinite programming, 454 split sign, 373 sentence, 128 split trichotomy, 373 separate, 313 split zero, 363 separation logic, 349 splits,98 sequent, 471 splittab, 178 sequent calculus, 470 spoly, 408 set, 594 spread, 414 set of support, 200 stably infinite, 440 set eq, 620 stal intersect,97 set goal, 508 stalmarck,98 setify, 620 static, 606 shearing, 417 static typing, 607 Sheffer function, 113 statically typed, 607 Sheffer stroke, 48 status, 278 sholds, 552 Stay, 558 Shostak combination method, 447 string of exp,21 shunt, 484 strip quant, 626 Sigma, 550 strong typing, 611 sigma bound, 552 strongly normalizing, 258 sigma elim, 570 strongly typed, 611 sigma prove, 572 strongly undecidable, 585 sign, 362 structural induction, 615

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

680 Index

St˚almarck’s method, 90 top-down method, 172 subalpha, 492 toplevel, 604 subcnf,78 topterms, 432 subject (syllogism), 317 toptermt, 432 subset, 594 torsion-free, 389 subset, 620 total order, 595 subspec, 491 TPTP, 230 subst, 133 trail, 86 substitution (formulas), 133 trailmix,86 substitution (terms), 131 transcendental, 527 substq, 134 transitive closure, 600 subsumes, 59 transitive relation, 595 subsumes clause, 187 triangular set, 419 subsumption, 59, 186 triangulate, 421 subterms, 249 trigger,94 subtract, 620 triggers,93 suc 0 l, 569 triplet, 92 suc 0 r, 569 triplicate,92 suc inj, 565 trivial,59 suc inj false, 569 True,26 suitable iffdef, 105 truefalse,96 sum,66 truth, 483 superposition, 303 truth-table, 33 surjective, 596 truth-value, 32 swap, 362 tryapplyd, 621 syllogism, 317 tryapplyl, 621 symbol, 558 trydps, 443 symbolic,17 tryfind, 619 symmetric relation, 595 tsubst, 131 syntax, 10 Turing machine, 555 type, 606 tab, 177 type variable, 612 tableau, 176 tableaux, 176 uinterpolate, 432 tabrefute, 177 unary, 119 tac proof, 508 unassigned,86 tactic, 506 uncountable, 598 tail recursive, 611 undef, 618 take, 517 undefinability of truth, 530 Tape, 558 undefine, 621 tape, 558 undefined, 621 Tarski’s theorem, 530 unequal, 621 Tarski–Seidenberg, 367 unfailing completion, 286 , 39 unifiable, 183 tautology,41 unification, 164 term, 118 unifier, 165 term, 119 uniform word problem, 380 term (syllogism), 317 unify, 168 term match, 186 unify and apply, 170 terminating, 258 unify complements, 174 termination ordering, 264 unify complementsf, 494 termsize, 265 unify literals, 174 termval, 125 unify refute, 174 testform, 369 uninterpreted functions, 425 Th, 328 union, 594 theory, 328, 329 union, 620 thm, 477 unions, 620 thread, 103 uniq, 619 throw, 615 unique table (BDD), 102 time, 617 unit clause, 80 tl, 618 unit deletion, 232

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89957-4 - Handbook of Practical Logic and Automated Reasoning John Harrison Index More information

Index 681

unit propagation, 80 w.r.t., 593 unit resolution, 233 wang, 316 unit propagate,87 weak Nullstellensatz, 396 unit subpropagate,86 weakly confluent, 258 unitycoeff, 342 weakly normalizing, 258 universal algebra, 303 weight, 267 universal quantifier, 120 well-formed formula, 25 unsat core, 90 wellfounded, 601 unsatisfiable, 39 wellfounded induction, 601 unsatisfiable,41 wellorder, 601 unshunt, 484 wellordered, 601 unzip, 618 wff, 25 upward L¨owenheim–Skolem, 227, 243 witness, 206 urinterpolate, 429 WN, 258 use laterimp, 496 word problem, 380 using, 511 write, 559 UTVPI, 349 wu, 422 Wu’s method, 419 vacuous quantifier, 139 valid, 39, 129 XOR, 66 valmod, 618 valuation, 32, 123 Z (integers), 594 Var, 14, 119 Zermelo–Fraenkel set theory, 578 var, 127 Zero, 362 variant, 133 zero, 337 veref, 551 zero (of a polynomial), 356 verefboundquant, 552 zero saturate,96 verification, 107, 521 zero saturate and check,96 vertex (of a graph), 62 ZF set theory, 578 vertices (of a graph), 62 zip, 618

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