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Untersuchungsergebnis.sml Download desIsabelle {Isabelle[95] Abap[161] [0]}zum Wurzelverzeichnis wechseln (* ========================================================================= *) (* SOME SAMPLE PROBLEMS TO TEST PROOF PROCEDURES *) (* Copyright (c) 2001 Joe Leslie-Hurd, distributed under the BSD License *) (* ========================================================================= *) (* ========================================================================= *) (* A type of problems. *) (* ========================================================================= *) type problem = {name : string, comments : string list, goal : Formula.quotation}; (* ========================================================================= *) (* Helper functions. *) (* ========================================================================= *) local fun mkCollection collection = "Collection: " ^ collection; fun mkProblem collection description (problem : problem) : problem = let val {name,comments,goal} = problem val comments = if List.null comments then [] else "" :: comments val comments = "Description: " ^ description :: comments val comments = mkCollection collection :: comments in {name = name, comments = comments, goal = goal} end; in fun isCollection collection {name = _, comments, goal = _} = Useful.mem (mkCollection collection) comments; fun mkProblems collection description problems = List.map (mkProblem collection description) problems; end; (* ========================================================================= *) (* The collection of problems. *) (* ========================================================================= *) val problems : problem list = (* ========================================================================= *) (* Problems without equality. *) (* ========================================================================= *) mkProblems "nonequality" "Problems without equality from various sources" [ (* ------------------------------------------------------------------------- *) (* Trivia (some of which demonstrate ex-bugs in provers). *) (* ------------------------------------------------------------------------- *) {name = "TRUE", comments = [], goal = ` T`}, {name = "SIMPLE", comments = [], goal = ` !x y. ?z. p x \/ p y ==> p z`}, {name = "CYCLIC", comments = [], goal = ` (!x. p (g (c x))) ==> ?z. p (g z)`}, {name = "MICHAEL_NORRISH_BUG", comments = [], goal = ` (!x. ?y. f y x x) ==> ?z. f z 0 0`}, {name = "RELATION_COMPOSITION", comments = [], goal = ` (!x. ?y. p x y) /\ (!x. ?y. q x y) /\ (!x y z. p x y /\ q y z ==> r x z) ==> !x. ?y. r x y`}, {name = "TOBIAS_NIPKOW", comments = [], goal = ` (!x y. p x y ==> f x = f y) /\ (!x. f (g x) = f x) /\ p (g a) (g b) ==> f a = f b`}, {name = "SLEDGEHAMMER", comments = ["From Tobias Nipkow: A ==> A takes 1 minute in sledgehammer."], goal = ` (!x y z t. x @ y = z @ t <=> ?u. x = z @ u /\ u @ y = t \/ x @ u = z /\ y = u @ t) ==> !x y z t. x @ y = z @ t <=> ?u. x = z @ u /\ u @ y = t \/ x @ u = z /\ y = u @ t`}, {name = "SPLITTING_UNSOUNDNESS", comments = ["A trivial example to illustrate a bug spotted by", "Geoff Sutcliffe in Dec 2008."], goal = ` (!x. p x /\ q x ==> F) ==> !x. p x ==> !x. q x ==> F`}, (* ------------------------------------------------------------------------- *) (* Propositional Logic. *) (* ------------------------------------------------------------------------- *) {name = "PROP_1", comments = [], goal = ` p ==> q <=> ~q ==> ~p`}, {name = "PROP_2", comments = [], goal = ` ~~p <=> p`}, {name = "PROP_3", comments = [], goal = ` ~(p ==> q) ==> q ==> p`}, {name = "PROP_4", comments = [], goal = ` ~p ==> q <=> ~q ==> p`}, {name = "PROP_5", comments = [], goal = ` (p \/ q ==> p \/ r) ==> p \/ (q ==> r)`}, {name = "PROP_6", comments = [], goal = ` p \/ ~p`}, {name = "PROP_7", comments = [], goal = ` p \/ ~~~p`}, {name = "PROP_8", comments = [], goal = ` ((p ==> q) ==> p) ==> p`}, {name = "PROP_9", comments = [], goal = ` (p \/ q) /\ (~p \/ q) /\ (p \/ ~q) ==> ~(~q \/ ~q)`}, {name = "PROP_10", comments = [], goal = ` (q ==> r) /\ (r ==> p /\ q) /\ (p ==> q /\ r) ==> (p <=> q)`}, {name = "PROP_11", comments = [], goal = ` p <=> p`}, {name = "PROP_12", comments = [], goal = ` ((p <=> q) <=> r) <=> p <=> q <=> r`}, {name = "PROP_13", comments = [], goal = ` p \/ q /\ r <=> (p \/ q) /\ (p \/ r)`}, {name = "PROP_14", comments = [], goal = ` (p <=> q) <=> (q \/ ~p) /\ (~q \/ p)`}, {name = "PROP_15", comments = [], goal = ` p ==> q <=> ~p \/ q`}, {name = "PROP_16", comments = [], goal = ` (p ==> q) \/ (q ==> p)`}, {name = "PROP_17", comments = [], goal = ` p /\ (q ==> r) ==> s <=> (~p \/ q \/ s) /\ (~p \/ ~r \/ s)`}, {name = "MATHS4_EXAMPLE", comments = [], goal = ` (a \/ ~k ==> g) /\ (g ==> q) /\ ~q ==> k`}, {name = "LOGICPROOF_1996", comments = [], goal = ` (p ==> r) /\ (~p ==> ~q) /\ (p \/ q) ==> p /\ r`}, {name = "XOR_ASSOC", comments = [], goal = ` ~(~(p <=> q) <=> r) <=> ~(p <=> ~(q <=> r))`}, {name = "ALL_3_CLAUSES", comments = [], goal = ` (p \/ q \/ r) /\ (p \/ q \/ ~r) /\ (p \/ ~q \/ r) /\ (p \/ ~q \/ ~r) /\ (~p \/ q \/ r) /\ (~p \/ q \/ ~r) /\ (~p \/ ~q \/ r) /\ (~p \/ ~q \/ ~r) ==> F`}, {name = "CLAUSE_SIMP", comments = [], goal = ` (lit ==> clause) ==> (lit \/ clause <=> clause)`}, {name = "SPLIT_NOT_IFF", comments = ["A way to split a goal that looks like ~(p <=> q)"], goal = ` ~(p <=> q) <=> (p ==> ~q) /\ (~q ==> p)`}, (* ------------------------------------------------------------------------- *) (* Monadic Predicate Logic. *) (* ------------------------------------------------------------------------- *) {name = "P18", comments = ["The drinker's principle."], goal = ` ?very_popular_guy. !whole_pub. drinks very_popular_guy ==> drinks whole_pub`}, {name = "P19", comments = [], goal = ` ?x. !y z. (p y ==> q z) ==> p x ==> q x`}, {name = "P20", comments = [], goal = ` (!x y. ?z. !w. p x /\ q y ==> r z /\ u w) /\ (!x y. p x /\ q y) ==> ?z. r z`}, {name = "P21", comments = [], goal = ` (?x. p ==> q x) /\ (?x. q x ==> p) ==> ?x. p <=> q x`}, {name = "P22", comments = [], goal = ` (!x. p <=> q x) ==> (p <=> !x. q x)`}, {name = "P23", comments = [], goal = ` (!x. p \/ q x) <=> p \/ !x. q x`}, {name = "P24", comments = [], goal = ` ~(?x. u x /\ q x) /\ (!x. p x ==> q x \/ r x) /\ ~(?x. p x ==> ?x. q x) /\ (!x. q x /\ r x ==> u x) ==> ?x. p x /\ r x`}, {name = "P25", comments = [], goal = ` (?x. p x) /\ (!x. u x ==> ~g x /\ r x) /\ (!x. p x ==> g x /\ u x) /\ ((!x. p x ==> q x) \/ ?x. q x /\ p x) ==> ?x. q x /\ p x`}, {name = "P26", comments = [], goal = ` ((?x. p x) <=> ?x. q x) /\ (!x y. p x /\ q y ==> (r x <=> u y)) ==> ((!x. p x ==> r x) <=> !x. q x ==> u x)`}, {name = "P27", comments = [], goal = ` (?x. p x /\ ~q x) /\ (!x. p x ==> r x) /\ (!x. u x /\ s x ==> p x) /\ (?x. r x /\ ~q x) ==> (!x. u x ==> ~r x) ==> !x. u x ==> ~s x`}, {name = "P28", comments = [], goal = ` (!x. p x ==> !x. q x) /\ ((!x. q x \/ r x) ==> ?x. q x /\ r x) /\ ((?x. r x) ==> !x. l x ==> m x) ==> !x. p x /\ l x ==> m x`}, {name = "P29", comments = [], goal = ` (?x. p x) /\ (?x. g x) ==> ((!x. p x ==> h x) /\ (!x. g x ==> j x) <=> !x y. p x /\ g y ==> h x /\ j y)`}, {name = "P30", comments = [], goal = ` (!x. p x \/ g x ==> ~h x) /\ (!x. (g x ==> ~u x) ==> p x /\ h x) ==> !x. u x`}, {name = "P31", comments = [], goal = ` ~(?x. p x /\ (g x \/ h x)) /\ (?x. q x /\ p x) /\ (!x. ~h x ==> j x) ==> ?x. q x /\ j x`}, {name = "P32", comments = [], goal = ` (!x. p x /\ (g x \/ h x) ==> q x) /\ (!x. q x /\ h x ==> j x) /\ (!x. r x ==> h x) ==> !x. p x /\ r x ==> j x`}, {name = "P33", comments = [], goal = ` (!x. p a /\ (p x ==> p b) ==> p c) <=> (!x. p a ==> p x \/ p c) /\ (p a ==> p b ==> p c)`}, {name = "P34", comments = ["This gives rise to 5184 clauses when naively converted to CNF!"], goal = ` ((?x. !y. p x <=> p y) <=> (?x. q x) <=> !y. q y) <=> (?x. !y. q x <=> q y) <=> (?x. p x) <=> !y. p y`}, {name = "P35", comments = [], goal = ` ?x y. p x y ==> !x y. p x y`}, (* ------------------------------------------------------------------------- *) (* Predicate logic without functions. *) (* ------------------------------------------------------------------------- *) {name = "P36", comments = [], goal = ` (!x. ?y. p x y) /\ (!x. ?y. g x y) /\ (!x y. p x y \/ g x y ==> !z. p y z \/ g y z ==> h x z) ==> !x. ?y. h x y`}, {name = "P37", comments = [], goal = ` (!z. ?w. !x. ?y. (p x z ==> p y w) /\ p y z /\ (p y w ==> ?v. q v w)) /\ (!x z. ~p x z ==> ?y. q y z) /\ ((?x y. q x y) ==> !x. r x x) ==> !x. ?y. r x y`}, {name = "P38", comments = [], goal = ` (!x. p a /\ (p x ==> ?y. p y /\ r x y) ==> ?z w. p z /\ r x w /\ r w z) <=> !x. (~p a \/ p x \/ ?z w. p z /\ r x w /\ r w z) /\ (~p a \/ ~(?y. p y /\ r x y) \/ ?z w. p z /\ r x w /\ r w z)`}, {name = "P39", comments = [], goal = ` ~?x. !y. p y x <=> ~p y y`}, {name = "P40", comments = [], goal = ` (?y. !x. p x y <=> p x x) ==> ~!x. ?y. !z. p z y <=> ~p z x`}, {name = "P41", comments = [], goal = ` (!z. ?y. !x. p x y <=> p x z /\ ~p x x) ==> ~?z. !x. p x z`}, {name = "P42", comments = [], goal = ` ~?y. !x. p x y <=> ~?z. p x z /\ p z x`}, {name = "P43", comments = [], goal = ` (!x y. q x y <=> !z. p z x <=> p z y) ==> !x y. q x y <=> q y x`}, {name = "P44", comments = [], goal = ` (!x. p x ==> (?y. g y /\ h x y) /\ ?y. g y /\ ~h x y) /\ (?x. j x /\ !y. g y ==> h x y) ==> ?x. j x /\ ~p x`}, {name = "P45", comments = [], goal = ` (!x. p x /\ (!y. g y /\ h x y ==> j x y) ==> !y. g y /\ h x y ==> r y) /\ ~(?y. l y /\ r y) /\ (?x. p x /\ (!y. h x y ==> l y) /\ !y. g y /\ h x y ==> j x y) ==> ?x. p x /\ ~?y. g y /\ h x y`}, {name = "P46", comments = [], goal = ` (!x. p x /\ (!y. p y /\ h y x ==> g y) ==> g x) /\ ((?x. p x /\ ~g x) ==> ?x. p x /\ ~g x /\ !y. p y /\ ~g y ==> j x y) /\ (!x y. p x /\ p y /\ h x y ==> ~j y x) ==> !x. p x ==> g x`}, {name = "P50", comments = [], goal = ` (!x. f0 a x \/ !y. f0 x y) ==> ?x. !y. f0 x y`}, {name = "LOGICPROOF_L10", comments = [], goal = ` !x. ?y. ~(P y x <=> ~P y y)`}, {name = "BARBER", comments = ["One resolution of the barber paradox"], goal = ` (!x. man x ==> (shaves barber x <=> ~shaves x x)) ==> ~man barber`}, (* ------------------------------------------------------------------------- *) (* Full predicate logic. *) (* ------------------------------------------------------------------------- *) {name = "LOGICPROOF_1999", comments = ["A non-theorem."], goal = ` (?x. p x /\ q x) ==> ?x. p (f x x) \/ !y. q y`}, {name = "P55", comments = ["Example from Manthey and Bry, CADE-9. [JRH]"], goal = ` lives agatha /\ lives butler /\ lives charles /\ (killed agatha agatha \/ killed butler agatha \/ killed charles agatha) /\ (!x y. killed x y ==> hates x y /\ ~richer x y) /\ (!x. hates agatha x ==> ~hates charles x) /\ (hates agatha agatha /\ hates agatha charles) /\ (!x. lives x /\ ~richer x agatha ==> hates butler x) /\ (!x. hates agatha x ==> hates butler x) /\ (!x. ~hates x agatha \/ ~hates x butler \/ ~hates x charles) ==> killed agatha agatha /\ ~killed butler agatha /\ ~killed charles agatha`}, {name = "P57", comments = [], goal = ` p (f a b) (f b c) /\ p (f b c) (f a c) /\ (!x y z. p x y /\ p y z ==> p x z) ==> p (f a b) (f a c)`}, {name = "P58", comments = ["See info-hol 1498 [JRH]"], goal = ` !x. ?v w. !y z. p x /\ q y ==> (p v \/ r w) /\ (r z ==> q v)`}, {name = "P59", comments = [], goal = ` (!x. p x <=> ~p (f x)) ==> ?x. p x /\ ~p (f x)`}, {name = "P60", comments = [], goal = ` !x. p x (f x) <=> ?y. (!z. p z y ==> p z (f x)) /\ p x y`}, (* ------------------------------------------------------------------------- *) (* From Gilmore's classic paper. *) (* ------------------------------------------------------------------------- *) {name = "GILMORE_1", comments = ["Amazingly, this still seems non-trivial... in HOL [MESON_TAC] it", "works at depth 45! [JRH]", "Confirmed (depth=45, inferences=263702, time=148s), though if", "lemmaizing is used then a lemma is discovered at depth 29 that allows", "a much quicker proof (depth=30, inferences=10039, time=5.5s)."], goal = ` ?x. !y z. (f y ==> g y <=> f x) /\ (f y ==> h y <=> g x) /\ ((f y ==> g y) ==> h y <=> h x) ==> f z /\ g z /\ h z`}, {name = "GILMORE_2", comments = ["This is not valid, according to Gilmore. [JRH]", "Confirmed: ordered resolution quickly saturates."], goal = ` ?x y. !z. (f x z <=> f z y) /\ (f z y <=> f z z) /\ (f x y <=> f y x) ==> (f x y <=> f x z)`}, {name = "GILMORE_3", comments = [], goal = ` ?x. !y z. ((f y z ==> g y ==> h x) ==> f x x) /\ ((f z x ==> g x) ==> h z) /\ f x y ==> f z z`}, {name = "GILMORE_4", comments = [], goal = ` ?x y. !z. (f x y ==> f y z /\ f z z) /\ (f x y /\ g x y ==> g x z /\ g z z)`}, {name = "GILMORE_5", comments = [], goal = ` (!x. ?y. f x y \/ f y x) /\ (!x y. f y x ==> f y y) ==> ?z. f z z`}, {name = "GILMORE_6", comments = [], goal = ` !x. ?y. (?w. !v. f w x ==> g v w /\ g w x) ==> (?w. !v. f w y ==> g v w /\ g w y) \/ !z v. ?w. g v z \/ h w y z ==> g z w`}, {name = "GILMORE_7", comments = [], goal = ` (!x. k x ==> ?y. l y /\ (f x y ==> g x y)) /\ (?z. k z /\ !w. l w ==> f z w) ==> ?v w. k v /\ l w /\ g v w`}, {name = "GILMORE_8", comments = [], goal = ` ?x. !y z. ((f y z ==> g y ==> !w. ?v. h w v x) ==> f x x) /\ ((f z x ==> g x) ==> !w. ?v. h w v z) /\ f x y ==> f z z`}, {name = "GILMORE_9", comments = ["Model elimination (in HOL):", "- With lemmaizing: (depth=18, inferences=15632, time=21s)", "- Without: gave up waiting after (depth=25, inferences=2125072, ...)"], goal = ` !x. ?y. !z. ((!w. ?v. f y w v /\ g y w /\ ~h y x) ==> (!w. ?v. f x w v /\ g z u /\ ~h x z) ==> !w. ?v. f x w v /\ g y w /\ ~h x y) /\ ((!w. ?v. f x w v /\ g y w /\ ~h x y) ==> ~(!w. ?v. f x w v /\ g z w /\ ~h x z) ==> (!w. ?v. f y w v /\ g y w /\ ~h y x) /\ !w. ?v. f z w v /\ g y w /\ ~h z y)`}, {name = "GILMORE_9a", comments = ["Translation of Gilmore procedure using separate definitions. [JRH]"], goal = ` (!x y. p x y <=> !w. ?v. f x w v /\ g y w /\ ~h x y) ==> !x. ?y. !z. (p y x ==> p x z ==> p x y) /\ (p x y ==> ~p x z ==> p y x /\ p z y)`}, {name = "BAD_CONNECTIONS", comments = ["The interesting example where connections make the proof longer. [JRH]"], goal = ` ~a /\ (a \/ b) /\ (c \/ d) /\ (~b \/ e \/ f) /\ (~c \/ ~e) /\ (~c \/ ~f) /\ (~b \/ g \/ h) /\ (~d \/ ~g) /\ (~d \/ ~h) ==> F`}, {name = "LOS", comments = ["The classic Los puzzle. (Clausal version MSC006-1 in the TPTP library.)", "Note: this is actually in the decidable AE subset, though that doesn't", "yield a very efficient proof. [JRH]"], goal = ` (!x y z. p x y ==> p y z ==> p x z) /\ (!x y z. q x y ==> q y z ==> q x z) /\ (!x y. q x y ==> q y x) /\ (!x y. p x y \/ q x y) ==> (!x y. p x y) \/ !x y. q x y`}, {name = "STEAM_ROLLER", comments = [], goal = ` ((!x. p1 x ==> p0 x) /\ ?x. p1 x) /\ ((!x. p2 x ==> p0 x) /\ ?x. p2 x) /\ ((!x. p3 x ==> p0 x) /\ ?x. p3 x) /\ ((!x. p4 x ==> p0 x) /\ ?x. p4 x) /\ ((!x. p5 x ==> p0 x) /\ ?x. p5 x) /\ ((?x. q1 x) /\ !x. q1 x ==> q0 x) /\ (!x. p0 x ==> (!y. q0 y ==> r x y) \/ !y. p0 y /\ s0 y x /\ (?z. q0 z /\ r y z) ==> r x y) /\ (!x y. p3 y /\ (p5 x \/ p4 x) ==> s0 x y) /\ (!x y. p3 x /\ p2 y ==> s0 x y) /\ (!x y. p2 x /\ p1 y ==> s0 x y) /\ (!x y. p1 x /\ (p2 y \/ q1 y) ==> ~r x y) /\ (!x y. p3 x /\ p4 y ==> r x y) /\ (!x y. p3 x /\ p5 y ==> ~r x y) /\ (!x. p4 x \/ p5 x ==> ?y. q0 y /\ r x y) ==> ?x y. p0 x /\ p0 y /\ ?z. q1 z /\ r y z /\ r x y`}, {name = "MODEL_COMPLETENESS", comments = ["An incestuous example used to establish completeness", "characterization. [JRH]"], goal = ` (!w x. sentence x ==> holds w x \/ holds w (not x)) /\ (!w x. ~(holds w x /\ holds w (not x))) ==> ((!x. sentence x ==> (!w. models w s ==> holds w x) \/ !w. models w s ==> holds w (not x)) <=> !w v. models w s /\ models v s ==> !x. sentence x ==> (holds w x <=> holds v x))`} ] @ (* ========================================================================= *) (* Problems with equality. *) (* ========================================================================= *) mkProblems "equality" "Equality problems from various sources" [ (* ------------------------------------------------------------------------- *) (* Trivia (some of which demonstrate ex-bugs in the prover). *) (* ------------------------------------------------------------------------- *) {name = "REFLEXIVITY", comments = [], goal = ` c = c`}, {name = "SYMMETRY", comments = [], goal = ` !x y. x = y ==> y = x`}, {name = "TRANSITIVITY", comments = [], goal = ` !x y z. x = y /\ y = z ==> x = z`}, {name = "TRANS_SYMM", comments = [], goal = ` !x y z. x = y /\ y = z ==> z = x`}, {name = "SUBSTITUTIVITY", comments = [], goal = ` !x y. f x /\ x = y ==> f y`}, {name = "CYCLIC_SUBSTITUTION_BUG", comments = [], goal = ` !y. (!x. y = g (c x)) ==> ?z. y = g z`}, {name = "JUDITA_1", comments = [], goal = ` ~(a = b) /\ (!x. x = c) ==> F`}, {name = "JUDITA_2", comments = [], goal = ` ~(a = b) /\ (!x y. x = y) ==> F`}, {name = "JUDITA_3", comments = [], goal = ` p a /\ ~p b /\ (!x. x = c) ==> F`}, {name = "JUDITA_4", comments = [], goal = ` p a /\ ~p b /\ (!x y. x = y) ==> F`}, {name = "JUDITA_5", comments = [], goal = ` p a /\ p b /\ ~(a = b) /\ ~p c /\ (!x. x = a \/ x = d) ==> F`}, {name = "INJECTIVITY_1", comments = [], goal = ` (!x y. f x = f y ==> x = y) /\ f a = f b ==> a = b`}, {name = "INJECTIVITY_2", comments = [], goal = ` (!x y. g (f x) = g (f y) ==> x = y) /\ f a = f b ==> a = b`}, {name = "SELF_REWRITE", comments = [], goal = ` f (g (h c)) = h c /\ g (h c) = b /\ f b = a /\ (!x. ~(a = h x)) ==> F`}, {name = "EQUALITY_ORDERING", comments = ["Positive resolution saturates if equality literals are", "ordered like other literals, instead of considering their", "left and right sides."], goal = ` p a /\ q a /\ q b /\ r b /\ (~p c \/ c = a) /\ (~r c \/ c = b) /\ (!x. ~q x \/ p x \/ r x) /\ (~p c \/ ~q c) /\ (~q c \/ ~r c) /\ (c = b \/ p c \/ q c) /\ (~p b \/ c = a \/ q c) ==> F`}, (* ------------------------------------------------------------------------- *) (* Simple equality problems. *) (* ------------------------------------------------------------------------- *) {name = "P48", comments = [], goal = ` (a = b \/ c = d) /\ (a = c \/ b = d) ==> a = d \/ b = c`}, {name = "P49", comments = [], goal = ` (?x y. !z. z = x \/ z = y) /\ p a /\ p b /\ ~(a = b) ==> !x. p x`}, {name = "P51", comments = [], goal = ` (?z w. !x y. f0 x y <=> x = z /\ y = w) ==> ?z. !x. (?w. !y. f0 x y <=> y = w) <=> x = z`}, {name = "P52", comments = [], goal = ` (?z w. !x y. f0 x y <=> x = z /\ y = w) ==> ?w. !y. (?z. !x. f0 x y <=> x = z) <=> y = w`}, {name = "UNSKOLEMIZED_MELHAM", comments = ["The Melham problem after an inverse skolemization step."], goal = ` (!x y. g x = g y ==> f x = f y) ==> !y. ?w. !x. y = g x ==> w = f x`}, {name = "CONGRUENCE_CLOSURE_EXAMPLE", comments = ["The example always given for congruence closure."], goal = ` !x. f (f (f (f (f x)))) = x /\ f (f (f x)) = x ==> f x = x`}, {name = "EWD_1", comments = ["A simple example (see EWD1266a and the application to Morley's", "theorem). [JRH]"], goal = ` (!x. f x ==> g x) /\ (?x. f x) /\ (!x y. g x /\ g y ==> x = y) ==> !y. g y ==> f y`}, {name = "EWD_2", comments = [], goal = ` (!x. f (f x) = f x) /\ (!x. ?y. f y = x) ==> !x. f x = x`}, {name = "JIA", comments = ["Needs only the K combinator"], goal = ` (!x y. k . x . y = x) /\ (!v. P (v . a) a) /\ (!w. Q (w . b) b) ==> !z. ?x y. P (z . x . y) x /\ Q (z . x . y) y`}, {name = "WISHNU", comments = ["Wishnu Prasetya's example. [JRH]"], goal = ` (?x. x = f (g x) /\ !x'. x' = f (g x') ==> x = x') <=> ?y. y = g (f y) /\ !y'. y' = g (f y') ==> y = y'`}, {name = "AGATHA", comments = ["An equality version of the Agatha puzzle. [JRH]"], goal = ` (?x. lives x /\ killed x agatha) /\ (lives agatha /\ lives butler /\ lives charles) /\ (!x. lives x ==> x = agatha \/ x = butler \/ x = charles) /\ (!x y. killed x y ==> hates x y) /\ (!x y. killed x y ==> ~richer x y) /\ (!x. hates agatha x ==> ~hates charles x) /\ (!x. ~(x = butler) ==> hates agatha x) /\ (!x. ~richer x agatha ==> hates butler x) /\ (!x. hates agatha x ==> hates butler x) /\ (!x. ?y. ~hates x y) /\ ~(agatha = butler) ==> killed agatha agatha /\ ~killed butler agatha /\ ~killed charles agatha`}, {name = "DIVIDES_9_1", comments = [], goal = ` (!x y. x * y = y * x) /\ (!x y z. x * y * z = x * (y * z)) /\ (!x y. divides x y <=> ?z. y = z * x) ==> !x y z. divides x y ==> divides x (z * y)`}, {name = "DIVIDES_9_2", comments = [], goal = ` (!x y. x * y = y * x) /\ (!x y z. x * y * z = x * (y * z)) /\ (!x y. divides x y <=> ?z. z * x = y) ==> !x y z. divides x y ==> divides x (z * y)`}, {name = "XOR_COUNT_COMMUTATIVE", comments = ["The xor literal counting function in Normalize is commutative."], goal = ` (!x y. x + y = y + x) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\ pl = p1 * p2 + n1 * n2 /\ nl = p1 * n2 + n1 * p2 /\ pr = p2 * p1 + n2 * n1 /\ nr = p2 * n1 + n2 * p1 ==> pl = pr /\ nl = nr`}, {name = "XOR_COUNT_ASSOCIATIVE", comments = ["The xor literal counting function in Normalize is associative."], goal = ` (!x y. x + y = y + x) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\ px = p1 * p2 + n1 * n2 /\ nx = p1 * n2 + n1 * p2 /\ pl = px * p3 + nx * n3 /\ nl = px * n3 + nx * p3 /\ py = p2 * p3 + n2 * n3 /\ ny = p2 * n3 + n2 * p3 /\ pr = p1 * py + n1 * ny /\ nr = p1 * ny + n1 * py ==> pl = pr /\ nl = nr`}, {name = "REVERSE_REVERSE", comments = ["Proving the goal", " !l. finite l ==> rev (rev l) = l", "after first generalizing it to", " !l k. finite l /\\ finite k ==> rev (rev l @ k) = rev k @ l", "and then applying list induction."], goal = ` finite nil /\ (!h t. finite (h :: t) <=> finite t) /\ (!l1 l2. finite (l1 @ l2) <=> finite l1 /\ finite l2) /\ (!l. nil @ l = l) /\ (!h t l. (h :: t) @ l = h :: t @ l) /\ rev nil = nil /\ (!h t. rev (h :: t) = rev t @ h :: nil) /\ (!l. l @ nil = l) /\ (!l1 l2 l3. l1 @ l2 @ l3 = (l1 @ l2) @ l3) ==> (!k. finite k ==> rev (rev nil @ k) = rev k @ nil) /\ !t. finite t ==> (!k. finite k ==> rev (rev t @ k) = rev k @ t) ==> !h k. finite k ==> rev (rev (h :: t) @ k) = rev k @ h :: t`}, (* ------------------------------------------------------------------------- *) (* Group theory examples. *) (* ------------------------------------------------------------------------- *) {name = "GROUP_INVERSE_INVERSE", comments = [], goal = ` (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. i x * x = e) ==> !x. i (i x) = x`}, {name = "GROUP_RIGHT_INVERSE", comments = ["Size 18, 61814 seconds. [JRH]"], goal = ` (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. i x * x = e) ==> !x. x * i x = e`}, {name = "GROUP_RIGHT_IDENTITY", comments = [], goal = ` (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. i x * x = e) ==> !x. x * e = x`}, {name = "GROUP_IDENTITY_EQUIV", comments = [], goal = ` (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. i x * x = e) ==> !x. x * x = x <=> x = e`}, {name = "KLEIN_GROUP_COMMUTATIVE", comments = [], goal = ` (!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. x * e = x) /\ (!x. x * x = e) ==> !x y. x * y = y * x`}, {name = "DOUBLE_DISTRIB", comments = ["From a John Harrison post to hol-info on 2008-04-15"], goal = ` (!x y z. x * y * z = x * z * (y * z)) /\ (!x y z. z * (x * y) = z * x * (z * y)) ==> !a b c. a * b * (c * a) = a * c * (b * a)`}, (* ------------------------------------------------------------------------- *) (* Ring theory examples. *) (* ------------------------------------------------------------------------- *) {name = "CONWAY_2", comments = ["From a John Harrison post to hol-info on 2008-04-15"], goal = ` (!x. 0 + x = x) /\ (!x y. x + y = y + x) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x. 1 * x = x) /\ (!x. x * 1 = x) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x. 0 * x = 0) /\ (!x. x * 0 = 0) /\ (!x y z. x * (y + z) = x * y + x * z) /\ (!x y z. (x + y) * z = x * z + y * z) /\ (!x y. star (x * y) = 1 + x * star (y * x) * y) /\ (!x y. star (x + y) = star (star (x) * y) * star (x)) ==> !a. star (star (star (star (a)))) = star (star (star (a)))`}, {name = "JACOBSON_2", comments = [], goal = ` (!x. 0 + x = x) /\ (!x. x + 0 = x) /\ (!x. ~x + x = 0) /\ (!x. x + ~x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y. x + y = y + x) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x y z. x * (y + z) = x * y + x * z) /\ (!x y z. (x + y) * z = x * z + y * z) /\ (!x. x * x = x) ==> !x y. x * y = y * x`}, {name = "JACOBSON_3", comments = [], goal = ` (!x. 0 + x = x) /\ (!x. x + 0 = x) /\ (!x. ~x + x = 0) /\ (!x. x + ~x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y. x + y = y + x) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x y z. x * (y + z) = x * y + x * z) /\ (!x y z. (x + y) * z = x * z + y * z) /\ (!x. x * (x * x) = x) ==> !x y. x * y = y * x`}, (* ------------------------------------------------------------------------- *) (* Set theory examples. *) (* ------------------------------------------------------------------------- *) {name = "UNION_2_SUBSET", comments = [], goal = ` (!x y. subset x y ==> subset y x ==> x = y) /\ (!x y. (!z. member z x ==> member z y) ==> subset x y) /\ (!x y z. member z (x + y) <=> member z x \/ member z y) ==> !x y. subset (x + y) (y + x)`}, {name = "UNION_2", comments = [], goal = ` (!x y. subset x y ==> subset y x ==> x = y) /\ (!x y. (!z. member z x ==> member z y) ==> subset x y) /\ (!x y z. member z (x + y) <=> member z x \/ member z y) ==> !x y. x + y = y + x`}, {name = "UNION_3_SUBSET", comments = ["From an email from Tobias Nipkow, 4 Nov 2008.", "The Isabelle version of metis diverges on this goal"], goal = ` (!x y. subset x y ==> subset y x ==> x = y) /\ (!x y. (!z. member z x ==> member z y) ==> subset x y) /\ (!x y z. member z (x + y) <=> member z x \/ member z y) ==> !x y z. subset (x + y + z) (z + y + x)`}, {name = "UNION_3", comments = ["From an email from Tobias Nipkow, 28 Oct 2008.", "The Isabelle version of metis diverges on this goal"], goal = ` (!x y. subset x y ==> subset y x ==> x = y) /\ (!x y. (!z. member z x ==> member z y) ==> subset x y) /\ (!x y z. member z (x + y) <=> member z x \/ member z y) ==> !x y z. x + y + z = z + y + x`}] @ (* ========================================================================= *) (* Some sample problems from the TPTP archive. *) (* Note: for brevity some relation/function names have been shortened. *) (* ========================================================================= *) mkProblems "tptp" "Sample problems from the TPTP collection" [ {name = "ALG005-1", comments = [], goal = ` (!x y. x + (y + x) = x) /\ (!x y. x + (x + y) = y + (y + x)) /\ (!x y z. x + y + z = x + z + (y + z)) /\ (!x y. x * y = x + (x + y)) ==> ~(a * b * c = a * (b * c)) ==> F`}, {name = "ALG006-1", comments = [], goal = ` (!x y. x + (y + x) = x) /\ (!x y. x + (x + y) = y + (y + x)) /\ (!x y z. x + y + z = x + z + (y + z)) ==> ~(a + c + b = a + b + c) ==> F`}, {name = "BOO021-1", comments = [], goal = ` (!x y. (x + y) * y = y) /\ (!x y z. x * (y + z) = y * x + z * x) /\ (!x. x + i x = 1) /\ (!x y. x * y + y = y) /\ (!x y z. x + y * z = (y + x) * (z + x)) /\ (!x. x * i x = 0) ==> ~(b * a = a * b) ==> F`}, {name = "COL058-2", comments = [], goal = ` (!x y. r (r 0 x) y = r x (r y y)) ==> ~(r (r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) (r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) = r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) ==> F`}, {name = "COL060-3", comments = [], goal = ` (!x y z. b . x . y . z = x . (y . z)) /\ (!x y. t . x . y = y . x) ==> ~(b . (b . (t . b) . b) . t . x . y . z = y . (x . z)) ==> F`}, {name = "GEO002-4", comments = [], goal = ` (!x y z v. ~between x y z \/ ~between y v z \/ between x y v) /\ (!x y z. ~equidistant x y z z \/ x == y) /\ (!x y z v w. ~between x y z \/ ~between v z w \/ between x (outer_pasch y x v w z) v) /\ (!x y z v w. ~between x y z \/ ~between v z w \/ between w y (outer_pasch y x v w z)) /\ (!x y z v. between x y (extension x y z v)) /\ (!x y z v. equidistant x (extension y x z v) z v) /\ (!x y z v. ~(x == y) \/ ~between z v x \/ between z v y) ==> ~between a a b ==> F`}, {name = "GEO036-2", comments = [], goal = ` (!x y. equidistant x y y x) /\ (!x y z x' y' z'. ~equidistant x y z x' \/ ~equidistant x y y' z' \/ equidistant z x' y' z') /\ (!x y z. ~equidistant x y z z \/ x = y) /\ (!x y z v. between x y (extension x y z v)) /\ (!x y z v. equidistant x (extension y x z v) z v) /\ (!x y z v w x' y' z'. ~equidistant x y z v \/ ~equidistant y w v x' \/ ~equidistant x y' z z' \/ ~equidistant y y' v z' \/ ~between x y w \/ ~between z v x' \/ x = y \/ equidistant w y' x' z') /\ (!x y. ~between x y x \/ x = y) /\ (!x y z v w. ~between x y z \/ ~between v w z \/ between y (inner_pasch x y z w v) v) /\ (!x y z v w. ~between x y z \/ ~between v w z \/ between w (inner_pasch x y z w v) x) /\ ~between lower_dimension_point_1 lower_dimension_point_2 lower_dimension_point_3 /\ ~between lower_dimension_point_2 lower_dimension_point_3 lower_dimension_point_1 /\ ~between lower_dimension_point_3 lower_dimension_point_1 lower_dimension_point_2 /\ (!x y z v w. ~equidistant x y x z \/ ~equidistant v y v z \/ ~equidistant w y w z \/ between x v w \/ between v w x \/ between w x v \/ y = z) /\ (!x y z v w. ~between x y z \/ ~between v y w \/ x = y \/ between x v (euclid1 x v y w z)) /\ (!x y z v w. ~between x y z \/ ~between v y w \/ x = y \/ between x w (euclid2 x v y w z)) /\ (!x y z v w. ~between x y z \/ ~between v y w \/ x = y \/ between (euclid1 x v y w z) z (euclid2 x v y w z)) /\ (!x y z x' y' z'. ~equidistant x y x z \/ ~equidistant x x' x y' \/ ~between x y x' \/ ~between y z' x' \/ between z (continuous x y z z' x' y') y') /\ (!x y z x' y' z'. ~equidistant x y x z \/ ~equidistant x x' x y' \/ ~between x y x' \/ ~between y z' x' \/ equidistant x z' x (continuous x y z z' x' y')) /\ (!x y z v. ~(x = y) \/ ~between x z v \/ between y z v) /\ (!x y z v. ~(x = y) \/ ~between z x v \/ between z y v) /\ (!x y z v. ~(x = y) \/ ~between z v x \/ between z v y) /\ (!x y z v w. ~(x = y) \/ ~equidistant x z v w \/ equidistant y z v w) /\ (!x y z v w. ~(x = y) \/ ~equidistant z x v w \/ equidistant z y v w) /\ (!x y z v w. ~(x = y) \/ ~equidistant z v x w \/ equidistant z v y w) /\ (!x y z v w. ~(x = y) \/ ~equidistant z v w x \/ equidistant z v w y) /\ (!x y z x' y' z'. ~(x = y) \/ inner_pasch x z x' y' z' = inner_pasch y z x' y' z') /\ (!x y z x' y' z'. ~(x = y) \/ inner_pasch z x x' y' z' = inner_pasch z y x' y' z') /\ (!x y z x' y' z'. ~(x = y) \/ inner_pasch z x' x y' z' = inner_pasch z x' y y' z') /\ (!x y z x' y' z'. ~(x = y) \/ inner_pasch z x' y' x z' = inner_pasch z x' y' y z') /\ (!x y z x' y' z'. ~(x = y) \/ inner_pasch z x' y' z' x = inner_pasch z x' y' z' y) /\ (!x y z x' y' z'. ~(x = y) \/ euclid1 x z x' y' z' = euclid1 y z x' y' z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid1 z x x' y' z' = euclid1 z y x' y' z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid1 z x' x y' z' = euclid1 z x' y y' z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid1 z x' y' x z' = euclid1 z x' y' y z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid1 z x' y' z' x = euclid1 z x' y' z' y) /\ (!x y z x' y' z'. ~(x = y) \/ euclid2 x z x' y' z' = euclid2 y z x' y' z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid2 z x x' y' z' = euclid2 z y x' y' z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid2 z x' x y' z' = euclid2 z x' y y' z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid2 z x' y' x z' = euclid2 z x' y' y z') /\ (!x y z x' y' z'. ~(x = y) \/ euclid2 z x' y' z' x = euclid2 z x' y' z' y) /\ (!x y z v w. ~(x = y) \/ extension x z v w = extension y z v w) /\ (!x y z v w. ~(x = y) \/ extension z x v w = extension z y v w) /\ (!x y z v w. ~(x = y) \/ extension z v x w = extension z v y w) /\ (!x y z v w. ~(x = y) \/ extension z v w x = extension z v w y) /\ (!x y z v w x' y'. ~(x = y) \/ continuous x z v w x' y' = continuous y z v w x' y') /\ (!x y z v w x' y'. ~(x = y) \/ continuous z x v w x' y' = continuous z y v w x' y') /\ (!x y z v w x' y'. ~(x = y) \/ continuous z v x w x' y' = continuous z v y w x' y') /\ (!x y z v w x' y'. ~(x = y) \/ continuous z v w x x' y' = continuous z v w y x' y') /\ (!x y z v w x' y'. ~(x = y) \/ continuous z v w x' x y' = continuous z v w x' y y') /\ (!x y z v w x' y'. ~(x = y) \/ continuous z v w x' y' x = continuous z v w x' y' y) ==> lower_dimension_point_1 = lower_dimension_point_2 \/ lower_dimension_point_2 = lower_dimension_point_3 \/ lower_dimension_point_1 = lower_dimension_point_3 ==> F`}, {name = "GRP010-4", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. ~(x = y) \/ i x = i y) /\ (!x y z. ~(x = y) \/ x * z = y * z) /\ (!x y z. ~(x = y) \/ z * x = z * y) /\ (!x y z. x * y * z = x * (y * z)) /\ (!x. 1 * x = x) /\ (!x. i x * x = 1) /\ c * b = 1 ==> ~(b * c = 1) ==> F`}, {name = "GRP057-1", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y z v. x * i (i (i y * (i x * z)) * v * i (y * v)) = z) /\ (!x y. ~(x = y) \/ i x = i y) /\ (!x y z. ~(x = y) \/ x * z = y * z) /\ (!x y z. ~(x = y) \/ z * x = z * y) ==> ~(i a1 * a1 = i b1 * b1) \/ ~(i b2 * b2 * a2 = a2) \/ ~(a3 * b3 * c3 = a3 * (b3 * c3)) ==> F`}, {name = "GRP086-1", comments = ["Bug check: used to be unsolvable without sticky constraints"], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y z. x * (y * z * i (x * z)) = y) /\ (!x y. ~(x = y) \/ i x = i y) /\ (!x y z. ~(x = y) \/ x * z = y * z) /\ (!x y z. ~(x = y) \/ z * x = z * y) ==> (!x y. ~(i a1 * a1 = i b1 * b1) \/ ~(i b2 * b2 * a2 = a2) \/ ~(a3 * b3 * c3 = a3 * (b3 * c3)) \/ ~(x * y = y * x)) ==> F`}, {name = "GRP104-1", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y z. double_divide x (inverse (double_divide (inverse (double_divide (double_divide x y) (inverse z))) y)) = z) /\ (!x y. x * y = inverse (double_divide y x)) /\ (!x y. ~(x = y) \/ inverse x = inverse y) /\ (!x y z. ~(x = y) \/ x * z = y * z) /\ (!x y z. ~(x = y) \/ z * x = z * y) /\ (!x y z. ~(x = y) \/ double_divide x z = double_divide y z) /\ (!x y z. ~(x = y) \/ double_divide z x = double_divide z y) ==> (!x y. ~(inverse a1 * a1 = inverse b1 * b1) \/ ~(inverse b2 * b2 * a2 = a2) \/ ~(a3 * b3 * c3 = a3 * (b3 * c3)) \/ ~(x * y = y * x)) ==> F`}, {name = "GRP128-4.003", comments = [], goal = ` (!x y. ~elt x \/ ~elt y \/ product e_1 x y \/ product e_2 x y \/ product e_3 x y) /\ (!x y. ~elt x \/ ~elt y \/ product x e_1 y \/ product x e_2 y \/ product x e_3 y) /\ elt e_1 /\ elt e_2 /\ elt e_3 /\ ~(e_1 == e_2) /\ ~(e_1 == e_3) /\ ~(e_2 == e_1) /\ ~(e_2 == e_3) /\ ~(e_3 == e_1) /\ ~(e_3 == e_2) /\ (!x y. ~elt x \/ ~elt y \/ product x y e_1 \/ product x y e_2 \/ product x y e_3) /\ (!x y z v. ~product x y z \/ ~product x y v \/ z == v) /\ (!x y z v. ~product x y z \/ ~product x v z \/ y == v) /\ (!x y z v. ~product x y z \/ ~product v y z \/ x == v) ==> (!x y z v. product x y z \/ ~product x z v \/ ~product z y v) /\ (!x y z v. product x y z \/ ~product v x z \/ ~product v y x) /\ (!x y z v. ~product x y z \/ ~product z y v \/ product x z v) ==> F`}, {name = "HEN006-3", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. ~(x <= y) \/ x / y = 0) /\ (!x y. ~(x / y = 0) \/ x <= y) /\ (!x y. x / y <= x) /\ (!x y z. x / y / (z / y) <= x / z / y) /\ (!x. 0 <= x) /\ (!x y. ~(x <= y) \/ ~(y <= x) \/ x = y) /\ (!x. x <= 1) /\ (!x y z. ~(x = y) \/ x / z = y / z) /\ (!x y z. ~(x = y) \/ z / x = z / y) /\ (!x y z. ~(x = y) \/ ~(x <= z) \/ y <= z) /\ (!x y z. ~(x = y) \/ ~(z <= x) \/ z <= y) /\ a / b <= d ==> ~(a / d <= b) ==> F`}, {name = "LAT005-3", comments = ["SAM's lemma"], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x. meet x x = x) /\ (!x. join x x = x) /\ (!x y. meet x (join x y) = x) /\ (!x y. join x (meet x y) = x) /\ (!x y. meet x y = meet y x) /\ (!x y. join x y = join y x) /\ (!x y z. meet (meet x y) z = meet x (meet y z)) /\ (!x y z. join (join x y) z = join x (join y z)) /\ (!x y z. ~(x = y) \/ join x z = join y z) /\ (!x y z. ~(x = y) \/ join z x = join z y) /\ (!x y z. ~(x = y) \/ meet x z = meet y z) /\ (!x y z. ~(x = y) \/ meet z x = meet z y) /\ (!x. meet x 0 = 0) /\ (!x. join x 0 = x) /\ (!x. meet x 1 = x) /\ (!x. join x 1 = 1) /\ (!x y z. ~(meet x y = x) \/ meet y (join x z) = join x (meet z y)) /\ (!x y. ~complement x y \/ meet x y = 0) /\ (!x y. ~complement x y \/ join x y = 1) /\ (!x y. ~(meet x y = 0) \/ ~(join x y = 1) \/ complement x y) /\ (!x y z. ~(x = y) \/ ~complement x z \/ complement y z) /\ (!x y z. ~(x = y) \/ ~complement z x \/ complement z y) /\ complement r1 (join a b) /\ complement r2 (meet a b) ==> ~(r1 = meet (join r1 (meet r2 b)) (join r1 (meet r2 a))) ==> F`}, {name = "LCL009-1", comments = [], goal = ` (!x y. ~p (x - y) \/ ~p x \/ p y) /\ (!x y z. p (x - y - (z - y - (x - z)))) ==> ~p (a - b - c - (a - (b - c))) ==> F`}, {name = "LCL087-1", comments = ["Solved quickly by resolution when NOT tracking term-ordering constraints."], goal = ` (!x y. ~p (implies x y) \/ ~p x \/ p y) /\ (!x y z v w. p (implies (implies (implies x y) (implies z v)) (implies w (implies (implies v x) (implies z x))))) ==> ~p (implies a (implies b a)) ==> F`}, {name = "LCL107-1", comments = ["A very small problem that's tricky to prove."], goal = ` (!x y. ~p (x - y) \/ ~p x \/ p y) /\ (!x y z v w x' y'. p (x - y - z - (v - w - (x' - w - (x' - v) - y')) - (z - (y - x - y')))) ==> ~p (a - b - c - (e - b - (a - e - c))) ==> F`}, {name = "LCL187-1", comments = [], goal = ` (!x. axiom (or (not (or x x)) x)) /\ (!x y. axiom (or (not x) (or y x))) /\ (!x y. axiom (or (not (or x y)) (or y x))) /\ (!x y z. axiom (or (not (or x (or y z))) (or y (or x z)))) /\ (!x y z. axiom (or (not (or (not x) y)) (or (not (or z x)) (or z y)))) /\ (!x. theorem x \/ ~axiom x) /\ (!x y. theorem x \/ ~axiom (or (not y) x) \/ ~theorem y) /\ (!x y z. theorem (or (not x) y) \/ ~axiom (or (not x) z) \/ ~theorem (or (not z) y)) ==> ~theorem (or (not p) (or (not (not p)) q)) ==> F`}, {name = "LDA007-3", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y z. f x (f y z) = f (f x y) (f x z)) /\ (!x y z. ~(x = y) \/ f x z = f y z) /\ (!x y z. ~(x = y) \/ f z x = f z y) /\ tt = f t t /\ ts = f t s /\ tt_ts = f tt ts /\ tk = f t k /\ tsk = f ts k ==> ~(f t tsk = f tt_ts tk) ==> F`}, {name = "NUM001-1", comments = [], goal = ` (!x. x == x) /\ (!x y z. ~(x == y) \/ ~(y == z) \/ x == z) /\ (!x y. x + y == y + x) /\ (!x y z. x + (y + z) == x + y + z) /\ (!x y. x + y - y == x) /\ (!x y. x == x + y - y) /\ (!x y z. x - y + z == x + z - y) /\ (!x y z. x + y - z == x - z + y) /\ (!x y z v. ~(x == y) \/ ~(z == x + v) \/ z == y + v) /\ (!x y z v. ~(x == y) \/ ~(z == v + x) \/ z == v + y) /\ (!x y z v. ~(x == y) \/ ~(z == x - v) \/ z == y - v) /\ (!x y z v. ~(x == y) \/ ~(z == v - x) \/ z == v - y) ==> ~(a + b + c == a + (b + c)) ==> F`}, {name = "NUM014-1", comments = [], goal = ` (!x. product x x (square x)) /\ (!x y z. ~product x y z \/ product y x z) /\ (!x y z. ~product x y z \/ divides x z) /\ (!x y z v. ~prime x \/ ~product y z v \/ ~divides x v \/ divides x y \/ divides x z) /\ prime a /\ product a (square c) (square b) ==> ~divides a b ==> F`}, {name = "PUZ001-1", comments = [], goal = ` lives agatha /\ lives butler /\ lives charles /\ (!x y. ~killed x y \/ ~richer x y) /\ (!x. ~hates agatha x \/ ~hates charles x) /\ (!x. ~hates x agatha \/ ~hates x butler \/ ~hates x charles) /\ hates agatha agatha /\ hates agatha charles /\ (!x y. ~killed x y \/ hates x y) /\ (!x. ~hates agatha x \/ hates butler x) /\ (!x. ~lives x \/ richer x agatha \/ hates butler x) ==> killed butler agatha \/ killed charles agatha ==> F`}, {name = "PUZ011-1", comments = ["Curiosity: solved trivially by meson without cache cutting, but not with."], goal = ` ocean atlantic /\ ocean indian /\ borders atlantic brazil /\ borders atlantic uruguay /\ borders atlantic venesuela /\ borders atlantic zaire /\ borders atlantic nigeria /\ borders atlantic angola /\ borders indian india /\ borders indian pakistan /\ borders indian iran /\ borders indian somalia /\ borders indian kenya /\ borders indian tanzania /\ south_american brazil /\ south_american uruguay /\ south_american venesuela /\ african zaire /\ african nigeria /\ african angola /\ african somalia /\ african kenya /\ african tanzania /\ asian india /\ asian pakistan /\ asian iran ==> (!x y z. ~ocean x \/ ~borders x y \/ ~african y \/ ~borders x z \/ ~asian z) ==> F`}, {name = "PUZ020-1", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. ~(x = y) \/ statement_by x = statement_by y) /\ (!x. ~person x \/ knight x \/ knave x) /\ (!x. ~person x \/ ~knight x \/ ~knave x) /\ (!x y. ~says x y \/ a_truth y \/ ~a_truth y) /\ (!x y. ~says x y \/ ~(x = y)) /\ (!x y. ~says x y \/ y = statement_by x) /\ (!x y. ~person x \/ ~(x = statement_by y)) /\ (!x. ~person x \/ ~a_truth (statement_by x) \/ knight x) /\ (!x. ~person x \/ a_truth (statement_by x) \/ knave x) /\ (!x y. ~(x = y) \/ ~knight x \/ knight y) /\ (!x y. ~(x = y) \/ ~knave x \/ knave y) /\ (!x y. ~(x = y) \/ ~person x \/ person y) /\ (!x y z. ~(x = y) \/ ~says x z \/ says y z) /\ (!x y z. ~(x = y) \/ ~says z x \/ says z y) /\ (!x y. ~(x = y) \/ ~a_truth x \/ a_truth y) /\ (!x y. ~knight x \/ ~says x y \/ a_truth y) /\ (!x y. ~knave x \/ ~says x y \/ ~a_truth y) /\ person husband /\ person wife /\ ~(husband = wife) /\ says husband (statement_by husband) /\ (~a_truth (statement_by husband) \/ ~knight husband \/ knight wife) /\ (a_truth (statement_by husband) \/ ~knight husband) /\ (a_truth (statement_by husband) \/ knight wife) /\ (~knight wife \/ a_truth (statement_by husband)) ==> ~knight husband ==> F`}, {name = "ROB002-1", comments = [], goal = ` (!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\ (!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. x + y = y + x) /\ (!x y z. x + y + z = x + (y + z)) /\ (!x y. negate (negate (x + y) + negate (x + negate y)) = x) /\ (!x y z. ~(x = y) \/ x + z = y + z) /\ (!x y z. ~(x = y) \/ z + x = z + y) /\ (!x y. ~(x = y) \/ negate x = negate y) /\ (!x. negate (negate x) = x) ==> ~(negate (a + negate b) + negate (negate a + negate b) = b) ==> F`} ] @ (* ========================================================================= *) (* Some problems from HOL. *) (* ========================================================================= *) mkProblems "hol" "HOL subgoals sent to MESON_TAC" [ {name = "Coder_4_0", comments = [], goal = ` (!x y. x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ {truth} /\ (!x y z. ~(APPEND . x . y = APPEND . z . y) \/ x = z) /\ (!x y z. ~(APPEND . x . y = APPEND . x . z) \/ y = z) ==> {wf_encoder . p . e} /\ (!x. e . x = f . x \/ ~{p . x}) /\ {wf_encoder . p . f} /\ {p . q} /\ {p . q'} /\ APPEND . (f . q) . r = APPEND . (f . q') . r' /\ q = q' /\ ~(r' = r) ==> F`}, {name = "DeepSyntax_47", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\ (!x y. ~(x = y) \/ int_neg x = int_neg y) /\ (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ eval_form y v \/ ~eval_form x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z v. int_lt (int_add x y) (int_add z v) \/ ~int_lt x z \/ ~int_lt y v) /\ (!x. int_add x (int_of_num 0) = x) /\ (!x. int_add x (int_neg x) = int_of_num 0) /\ (!x y z. int_add x (int_add y z) = int_add (int_add x y) z) ==> int_lt (int_neg d) (int_of_num 0) /\ eval_form g x /\ int_lt (int_add x d) i /\ ~int_lt x i ==> F`}, {name = "divides_9", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x y. x * y = y * x) /\ (!x y. ~divides x y \/ y = gv1556 x y * x) /\ (!x y z. divides x y \/ ~(y = z * x)) ==> divides gv1546 gv1547 /\ ~divides gv1546 (gv1547 * gv1548) ==> F`}, {name = "Encode_28", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ MOD x z = MOD y v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ (!x y. ~(x = y) \/ BIT2 x = BIT2 y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ EXP x z = EXP y v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ DIV x z = DIV y v) /\ (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. x * y = y * x) /\ (!x y z. MOD (MOD x (y * z)) y = MOD x y \/ ~(0 < y) \/ ~(0 < z)) ==> (!x. MOD x (NUMERAL (BIT2 ZERO)) = 0 \/ MOD x (NUMERAL (BIT2 ZERO)) = NUMERAL (BIT1 ZERO)) /\ MOD (DIV x (NUMERAL (BIT2 ZERO)) * NUMERAL (BIT2 ZERO) + MOD x (NUMERAL (BIT2 ZERO))) (NUMERAL (BIT2 ZERO) * EXP (NUMERAL (BIT2 ZERO)) m) = MOD (DIV y (NUMERAL (BIT2 ZERO)) * NUMERAL (BIT2 ZERO) + MOD y (NUMERAL (BIT2 ZERO))) (NUMERAL (BIT2 ZERO) * EXP (NUMERAL (BIT2 ZERO)) m) /\ 0 < EXP (NUMERAL (BIT2 ZERO)) m /\ 0 < NUMERAL (BIT2 ZERO) /\ (!x y. ~(MOD (x * NUMERAL (BIT2 ZERO) + MOD (x) (NUMERAL (BIT2 ZERO))) (NUMERAL (BIT2 ZERO)) = MOD (y * NUMERAL (BIT2 ZERO) + MOD (y) (NUMERAL (BIT2 ZERO))) (NUMERAL (BIT2 ZERO)))) ==> F`}, {name = "euclid_4", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x y. ~divides x y \/ y = x * gv5371 x y) /\ (!x y z. divides x y \/ ~(y = x * z)) ==> divides gv5316 gv5317 /\ divides gv5315 gv5316 /\ ~divides gv5315 gv5317 ==> F`}, {name = "euclid_8", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x y. ~divides x y \/ y = x * gv7050 x y) /\ (!x y z. divides x y \/ ~(y = x * z)) ==> divides gv7000 gv7001 /\ ~divides gv7000 (gv7002 * gv7001) ==> F`}, {name = "extra_arith_6", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. ~(SUC x * y = SUC x * z) \/ y = z) /\ (!x y z. SUC x * y = SUC x * z \/ ~(y = z)) /\ (!x y z. x * (y * z) = x * y * z) /\ (!x y. x * y = y * x) ==> SUC n * b = q * (SUC n * a) /\ 0 < SUC n /\ ~(b = q * a) ==> F`}, {name = "extra_real_5", comments = [], goal = ` ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ {truth} ==> (!x y z v. {real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/ ~{P . z} \/ ~{real_lte . (gv6327 . v) . v}) /\ (!x y z. ~{real_lt . x . (sup . P)} \/ {P . (gv6327 . x)} \/ ~{P . y} \/ ~{real_lte . (gv6327 . z) . z}) /\ (!x y z. ~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv6327 . x)} \/ ~{P . y} \/ ~{real_lte . (gv6327 . z) . z}) /\ (!x y z. ~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv6327 . x)} \/ ~{P . y} \/ {P . (gv6327 . z)}) /\ (!x y z. ~{real_lt . x . (sup . P)} \/ {P . (gv6327 . x)} \/ ~{P . y} \/ {P . (gv6327 . z)}) /\ (!x y z v. {real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/ ~{P . z} \/ {P . (gv6327 . v)}) /\ {P . x} /\ (!x. {real_lte . x . z} \/ ~{P . x}) /\ (!x. {real_lt . (gv6328 . x) . (gv6329 . x)} \/ {real_lt . (gv6328 . x) . x}) /\ (!x. {P . (gv6329 . x)} \/ {real_lt . (gv6328 . x) . x}) /\ (!x y. ~{real_lt . (gv6328 . x) . y} \/ ~{P . y} \/ ~{real_lt . (gv6328 . x) . x}) ==> F`}, {name = "gcd_19", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\ (!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\ (!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\ (!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\ (!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==> ~(x + z <= x) /\ divides c (d * SUC x) /\ divides c (d * SUC (x + z)) /\ ~divides c (d * z) ==> F`}, {name = "gcd_20", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\ (!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\ (!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\ (!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\ (!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==> y <= y + z /\ divides c (d * SUC (y + z)) /\ divides c (d * SUC y) /\ ~divides c (d * z) ==> F`}, {name = "gcd_21", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\ (!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\ (!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\ (!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\ (!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\ (!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==> divides c (d * SUC y) /\ y <= y + z /\ divides c (d * SUC (y + z)) /\ ~divides c (d * z) ==> F`}, {name = "int_arith_6", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\ (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. int_mul x y = int_mul y x) /\ (!x. ~int_lt x x) /\ (!x y z. ~(int_mul x y = int_mul z y) \/ y = int_of_num 0 \/ x = z) /\ (!x y z. int_mul x y = int_mul z y \/ ~(y = int_of_num 0)) /\ (!x y z. int_mul x y = int_mul z y \/ ~(x = z)) ==> int_lt (int_of_num 0) gv1085 /\ int_mul gv1085 gv1086 = int_mul gv1085 gv1087 /\ ~(gv1086 = gv1087) ==> F`}, {name = "int_arith_139", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\ (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x. int_add (int_of_num 0) x = x) /\ (!x y z v. int_le (int_add x y) (int_add z v) \/ ~int_le x z \/ ~int_le y v) /\ (!x y z. int_add x (int_add y z) = int_add (int_add x y) z) /\ (!x y. int_add x y = int_add y x) /\ (!x y z. ~int_le (int_add x y) (int_add x z) \/ int_le y z) /\ (!x y z. int_le (int_add x y) (int_add x z) \/ ~int_le y z) ==> int_le x y /\ int_le (int_of_num 0) (int_add c x) /\ ~int_le (int_of_num 0) (int_add c y) ==> F`}, {name = "llist_69", comments = [], goal = ` (!x y. ~(x = y) \/ LTL x = LTL y) /\ (!x y. ~(x = y) \/ SOME x = SOME y) /\ (!x y. ~(x = y) \/ LHD x = LHD y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ LCONS x z = LCONS y v) /\ (!x y. ~(x = y) \/ g x = g y) /\ (!x y. ~(x = y) \/ THE x = THE y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ LNTH z x = LNTH v y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ LDROP z x = LDROP v y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y z. ~(x = LCONS y z) \/ LTL x = SOME z) /\ (!x y z. ~(x = LCONS y z) \/ LHD x = SOME y) /\ (!x y z. x = LCONS y z \/ ~(LHD x = SOME y) \/ ~(LTL x = SOME z)) ==> LTL (g (LCONS LNIL t)) = SOME (g (LCONS (THE (LTL (THE (LNTH n t)))) (THE (LDROP (SUC n) t)))) /\ LHD (g (LCONS LNIL t)) = SOME (THE (LHD (THE (LNTH n t)))) /\ LHD (g t) = SOME (THE (LHD (THE (LNTH n t)))) /\ LTL (g t) = SOME (g (LCONS (THE (LTL (THE (LNTH n t)))) (THE (LDROP (SUC n) t)))) /\ ~(g (LCONS LNIL t) = g t) ==> F`}, {name = "MachineTransition_0", comments = [], goal = ` (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ {truth} /\ Eq = equality /\ (!x y z. ~{Next . x . y . z} \/ {x . (pair . (gv940 . x . y . z) . z)}) /\ (!x y z. ~{Next . x . y . z} \/ {y . (gv940 . x . y . z)}) /\ (!x y z v. {Next . x . y . z} \/ ~{y . v} \/ ~{x . (pair . v . z)}) /\ (!x y z. ~{Prev . x . y . z} \/ {y . (gv935 . x . y . z)}) /\ (!x y z. ~{Prev . x . y . z} \/ {x . (pair . z . (gv935 . x . y . z))}) /\ (!x y z v. {Prev . x . y . z} \/ ~{x . (pair . z . v)} \/ ~{y . v}) ==> {Next . gv914 . (Eq . gv915) . gv916} /\ ~{Prev . gv914 . (Eq . gv916) . gv915} ==> F`}, {name = "MachineTransition_2_1", comments = [], goal = ` (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ {truth} /\ (!x y z. ReachIn . x . (Next . x . y) . z = ReachIn . x . y . (SUC . z)) /\ (!x y z. ~{Next . x . y . z} \/ {x . (pair . (gv5488 . x . y . z) . z)}) /\ (!x y z. ~{Next . x . y . z} \/ {y . (gv5488 . x . y . z)}) /\ (!x y z v. {Next . x . y . z} \/ ~{y . v} \/ ~{x . (pair . v . z)}) /\ (!x y z. ReachIn . x . y . (SUC . z) = Next . x . (ReachIn . x . y . z)) /\ (!x y. ReachIn . x . y . 0 = y) ==> {ReachIn . R . (Next . R . B) . gv5278 . state} /\ (!x. ~{ReachIn . R . B . gv5278 . x} \/ ~{R . (pair . x . state)}) ==> F`}, {name = "MachineTransition_52", comments = [], goal = ` (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\ (!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\ (!x y. ~{x} \/ ~(x = y) \/ {y}) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\ ~{existential . (K . falsity)} /\ {existential . (K . truth)} /\ ~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\ {truth} /\ (!x y z. {(<=) . (gv7028 . x . y . z) . (add . x . (NUMERAL . (BIT1 . ZERO)))} \/ z . (add . x . (NUMERAL . (BIT1 . ZERO))) = y . (add . x . (NUMERAL . (BIT1 . ZERO)))) /\ (!x y z. ~(x . (gv7028 . y . z . x) = z . (gv7028 . y . z . x)) \/ x . (add . y . (NUMERAL . (BIT1 . ZERO))) = z . (add . y . (NUMERAL . (BIT1 . ZERO)))) /\ (!x y z v. ~(x . (gv7028 . y . z . x) = z . (gv7028 . y . z . x)) \/ x . v = z . v \/ ~{(<=) . v . y}) /\ (!x y z v. {(<=) . (gv7028 . x . y . z) . (add . x . (NUMERAL . (BIT1 . ZERO)))} \/ z . v = y . v \/ ~{(<=) . v . x}) /\ (!x y z v. ~{(<=) . x . (add . y . (NUMERAL . (BIT1 . ZERO)))} \/ z . x = v . x \/ ~(z . (gv7027 . y . v . z) = v . (gv7027 . y . v . z)) \/ ~(z . (add . y . (NUMERAL . (BIT1 . ZERO))) = v . (add . y . (NUMERAL . (BIT1 . ZERO))))) /\ (!x y z v. ~{(<=) . x . (add . y . (NUMERAL . (BIT1 . ZERO)))} \/ z . x = v . x \/ {(<=) . (gv7027 . y . v . z) . y} \/ ~(z . (add . y . (NUMERAL . (BIT1 . ZERO))) = v . (add . y . (NUMERAL . (BIT1 . ZERO))))) ==> ({FinPath . (pair . R . s) . f2 . n} \/ ~{FinPath . (pair . R . s) . f1 . n} \/ ~(f1 . gv7034 = f2 . gv7034)) /\ (~{FinPath . (pair . R . s) . f2 . n} \/ {FinPath . (pair . R . s) . f1 . n} \/ ~(f1 . gv7034 = f2 . gv7034)) /\ (~{FinPath . (pair . R . s) . f2 . n} \/ {FinPath . (pair . R . s) . f1 . n} \/ {(<=) . gv7034 . n}) /\ ({FinPath . (pair . R . s) . f2 . n} \/ ~{FinPath . (pair . R . s) . f1 . n} \/ {(<=) . gv7034 . n}) /\ (!x. f1 . x = f2 . x \/ ~{(<=) . x . (add . n . (NUMERAL . (BIT1 . ZERO)))}) /\ {FinPath . (pair . R . s) . f2 . n} /\ {R . (pair . (f2 . n) . (f2 . (add . n . (NUMERAL . (BIT1 . ZERO)))))} /\ ~{FinPath . (pair . R . s) . f1 . n} ==> F`}, {name = "measure_138", comments = [], goal = ` (!x y z. ~SUBSET x y \/ IN z y \/ ~IN z x) /\ (!x y. SUBSET x y \/ IN (gv122874 x y) x) /\ (!x y. SUBSET x y \/ ~IN (gv122874 x y) y) /\ (!x. sigma_algebra (sigma x)) /\ (!x y z. ~IN x (INTER y z) \/ IN x z) /\ (!x y z. ~IN x (INTER y z) \/ IN x y) /\ (!x y z. IN x (INTER y z) \/ ~IN x y \/ ~IN x z) /\ (!x y. ~sigma_algebra x \/ IN (BIGUNION y) x \/ ~countable y \/ ~SUBSET y x) /\ (!x y. ~sigma_algebra x \/ IN (COMPL y) x \/ ~IN y x) /\ (!x. ~sigma_algebra x \/ IN EMPTY x) /\ (!x. sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/ SUBSET (gv122852 x) x) /\ (!x. sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/ SUBSET (gv122852 x) x) /\ (!x. sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/ countable (gv122852 x)) /\ (!x. sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/ countable (gv122852 x)) /\ (!x. sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/ ~IN (BIGUNION (gv122852 x)) x) /\ (!x. sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/ ~IN (BIGUNION (gv122852 x)) x) ==> SUBSET c (INTER p (sigma a)) /\ (!x. IN (BIGUNION x) p \/ ~countable x \/ ~SUBSET x (INTER p (sigma a))) /\ SUBSET a p /\ IN EMPTY p /\ (!x. IN (COMPL x) p \/ ~IN x (INTER p (sigma a))) /\ countable c /\ ~IN (BIGUNION c) (sigma a) ==> F`}, {name = "Omega_13", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\ (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ evalupper y v \/ ~evalupper x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. ~int_le x y \/ int_lt x y \/ x = y) /\ (!x y. int_le x y \/ ~int_lt x y) /\ (!x y. int_le x y \/ ~(x = y)) /\ (!x y z. int_lt x y \/ ~int_lt (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ ~(0 < z)) /\ (!x y z. ~int_lt x y \/ int_lt (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ ~(0 < z)) /\ (!x y z. int_lt x y \/ ~int_le x z \/ ~int_lt z y) ==> (!x y. evalupper x uppers \/ ~evalupper y uppers \/ ~int_lt x y) /\ int_le (int_mul (int_of_num p_1) x) p_2 /\ evalupper x uppers /\ int_lt y x /\ 0 < p_1 /\ ~int_le (int_mul (int_of_num p_1) y) p_2 ==> F`}, {name = "Omega_71", comments = [], goal = ` (!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\ (!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\ (!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\ (!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ (x, z) = (y, v)) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ evallower y v \/ ~evallower x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ rshadow_row v y \/ ~rshadow_row z x) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ dark_shadow_cond_row v y \/ ~dark_shadow_cond_row z x) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ EVERY y v \/ ~EVERY x z) /\ (!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\ (!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\ (!x y. int_mul x y = int_mul y x) /\ (!x y z. int_mul x (int_mul y z) = int_mul (int_mul x y) z) /\ (!x y z. int_le x y \/ ~int_le (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ ~(0 < z)) /\ (!x y z. ~int_le x y \/ int_le (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/ ~(0 < z)) ==> (!x y z. evallower (gv6249 x y z) lowers \/ ~(0 < y) \/ ~evallower x lowers \/ ~rshadow_row (y, z) lowers \/ ~dark_shadow_cond_row (y, z) lowers) /\ (!x y z. int_le (int_mul (int_of_num x) (gv6249 y x z)) z \/ ~(0 < x) \/ ~evallower y lowers \/ ~rshadow_row (x, z) lowers \/ ~dark_shadow_cond_row (x, z) lowers) /\ 0 < c /\ int_le R (int_mul (int_of_num d) x) /\ evallower x lowers /\ 0 < d /\ EVERY fst_nzero lowers /\ int_le (int_mul (int_of_num c) R) (int_mul (int_of_num d) L) /\ rshadow_row (c, L) lowers /\ dark_shadow_cond_row (c, L) lowers /\ (!x. ~int_lt (int_mul (int_of_num d) L) (int_mul (int_of_num (c * d)) (int_add x (int_of_num (NUMERAL (BIT1 ZERO))))) \/ ~int_lt (int_mul (int_of_num (c * d)) x) (int_mul (int_of_num c) R)) /\ int_le (int_mul (int_of_num c) y) L /\ evallower y lowers /\ int_le (int_mul (int_of_num (c * d)) y) (int_mul (int_of_num d) L) /\ int_le (int_mul (int_of_num c) R) (int_mul (int_of_num (c * d)) x) /\ int_lt (int_mul (int_of_num (c * d)) y) (int_mul (int_of_num c) R) /\ 0 < c * d /\ int_le (int_mul (int_of_num c) R) (int_mul (int_of_num (c * d)) j) /\ int_le (int_mul (int_of_num (c * d)) j) (int_mul (int_of_num d) L) /\ int_le (int_mul (int_mul (int_of_num c) (int_of_num d)) j) (int_mul (int_of_num d) L) /\ ~int_le (int_mul (int_of_num c) j) L ==> F`}, {name = "pred_set_1", comments = ["Small problem that's hard for ordered resolution"], goal = ` (!x y z. ~(x <= y) \/ p z y \/ ~p z x) /\ (!x y. x <= y \/ ~p (a x y) y) /\ --> -------------------- --> maximum size reached --> -------------------- [ Dauer der Verarbeitung: 0.216 Sekunden ] |