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(* ========================================================================= *)
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(* SOME SAMPLE PROBLEMS TO TEST PROOF PROCEDURES *)
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(* Copyright (c) 2001 Joe Hurd, distributed under the BSD License *)
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(* ========================================================================= *)
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(* ========================================================================= *)
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(* A type of problems. *)
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(* ========================================================================= *)
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type problem =
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{name : string,
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comments : string list,
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goal : Formula.quotation};
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(* ========================================================================= *)
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(* Helper functions. *)
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(* ========================================================================= *)
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local
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fun mkCollection collection = "Collection: " ^ collection;
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fun mkProblem collection description (problem : problem) : problem =
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let
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val {name,comments,goal} = problem
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val comments = if List.null comments then [] else "" :: comments
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val comments = "Description: " ^ description :: comments
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val comments = mkCollection collection :: comments
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in
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{name = name, comments = comments, goal = goal}
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end;
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in
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fun isCollection collection {name = _, comments, goal = _} =
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Useful.mem (mkCollection collection) comments;
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fun mkProblems collection description problems =
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List.map (mkProblem collection description) problems;
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end;
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(* ========================================================================= *)
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(* The collection of problems. *)
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(* ========================================================================= *)
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val problems : problem list =
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(* ========================================================================= *)
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(* Problems without equality. *)
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(* ========================================================================= *)
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mkProblems "nonequality" "Problems without equality from various sources" [
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(* ------------------------------------------------------------------------- *)
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(* Trivia (some of which demonstrate ex-bugs in provers). *)
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(* ------------------------------------------------------------------------- *)
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{name = "TRUE",
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comments = [],
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goal = `
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T`},
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{name = "SIMPLE",
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comments = [],
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goal = `
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!x y. ?z. p x \/ p y ==> p z`},
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{name = "CYCLIC",
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comments = [],
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goal = `
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(!x. p (g (c x))) ==> ?z. p (g z)`},
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{name = "MICHAEL_NORRISH_BUG",
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comments = [],
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goal = `
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(!x. ?y. f y x x) ==> ?z. f z 0 0`},
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{name = "RELATION_COMPOSITION",
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comments = [],
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goal = `
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(!x. ?y. p x y) /\ (!x. ?y. q x y) /\
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(!x y z. p x y /\ q y z ==> r x z) ==> !x. ?y. r x y`},
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{name = "TOBIAS_NIPKOW",
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comments = [],
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goal = `
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(!x y. p x y ==> f x = f y) /\ (!x. f (g x) = f x) /\ p (g a) (g b) ==>
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f a = f b`},
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{name = "SLEDGEHAMMER",
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comments = ["From Tobias Nipkow: A ==> A takes 1 minute in sledgehammer."],
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goal = `
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(!x y z t.
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x @ y = z @ t <=>
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?u. x = z @ u /\ u @ y = t \/ x @ u = z /\ y = u @ t) ==>
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!x y z t.
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x @ y = z @ t <=> ?u. x = z @ u /\ u @ y = t \/ x @ u = z /\ y = u @ t`},
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{name = "SPLITTING_UNSOUNDNESS",
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comments = ["A trivial example to illustrate a bug spotted by",
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"Geoff Sutcliffe in Dec 2008."],
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goal = `
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(!x. p x /\ q x ==> F) ==> !x. p x ==> !x. q x ==> F`},
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(* ------------------------------------------------------------------------- *)
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(* Propositional Logic. *)
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(* ------------------------------------------------------------------------- *)
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{name = "PROP_1",
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comments = [],
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goal = `
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p ==> q <=> ~q ==> ~p`},
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{name = "PROP_2",
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comments = [],
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goal = `
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~~p <=> p`},
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{name = "PROP_3",
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comments = [],
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goal = `
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~(p ==> q) ==> q ==> p`},
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{name = "PROP_4",
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comments = [],
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goal = `
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~p ==> q <=> ~q ==> p`},
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{name = "PROP_5",
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comments = [],
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goal = `
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(p \/ q ==> p \/ r) ==> p \/ (q ==> r)`},
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{name = "PROP_6",
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comments = [],
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goal = `
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p \/ ~p`},
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{name = "PROP_7",
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comments = [],
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goal = `
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p \/ ~~~p`},
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{name = "PROP_8",
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comments = [],
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goal = `
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((p ==> q) ==> p) ==> p`},
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{name = "PROP_9",
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comments = [],
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goal = `
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(p \/ q) /\ (~p \/ q) /\ (p \/ ~q) ==> ~(~q \/ ~q)`},
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{name = "PROP_10",
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comments = [],
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goal = `
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(q ==> r) /\ (r ==> p /\ q) /\ (p ==> q /\ r) ==> (p <=> q)`},
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{name = "PROP_11",
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comments = [],
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goal = `
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p <=> p`},
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{name = "PROP_12",
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comments = [],
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goal = `
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((p <=> q) <=> r) <=> p <=> q <=> r`},
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{name = "PROP_13",
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comments = [],
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goal = `
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p \/ q /\ r <=> (p \/ q) /\ (p \/ r)`},
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{name = "PROP_14",
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comments = [],
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goal = `
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(p <=> q) <=> (q \/ ~p) /\ (~q \/ p)`},
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{name = "PROP_15",
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comments = [],
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goal = `
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p ==> q <=> ~p \/ q`},
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{name = "PROP_16",
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comments = [],
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goal = `
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(p ==> q) \/ (q ==> p)`},
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{name = "PROP_17",
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comments = [],
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goal = `
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p /\ (q ==> r) ==> s <=> (~p \/ q \/ s) /\ (~p \/ ~r \/ s)`},
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{name = "MATHS4_EXAMPLE",
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comments = [],
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goal = `
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(a \/ ~k ==> g) /\ (g ==> q) /\ ~q ==> k`},
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{name = "LOGICPROOF_1996",
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comments = [],
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goal = `
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(p ==> r) /\ (~p ==> ~q) /\ (p \/ q) ==> p /\ r`},
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{name = "XOR_ASSOC",
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comments = [],
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goal = `
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~(~(p <=> q) <=> r) <=> ~(p <=> ~(q <=> r))`},
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{name = "ALL_3_CLAUSES",
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comments = [],
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goal = `
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(p \/ q \/ r) /\ (p \/ q \/ ~r) /\ (p \/ ~q \/ r) /\ (p \/ ~q \/ ~r) /\
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(~p \/ q \/ r) /\ (~p \/ q \/ ~r) /\ (~p \/ ~q \/ r) /\
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(~p \/ ~q \/ ~r) ==> F`},
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{name = "CLAUSE_SIMP",
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comments = [],
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goal = `
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(lit ==> clause) ==> (lit \/ clause <=> clause)`},
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{name = "SPLIT_NOT_IFF",
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comments = ["A way to split a goal that looks like ~(p <=> q)"],
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goal = `
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~(p <=> q) <=> (p ==> ~q) /\ (q ==> ~p)`},
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(* ------------------------------------------------------------------------- *)
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(* Monadic Predicate Logic. *)
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(* ------------------------------------------------------------------------- *)
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{name = "P18",
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comments = ["The drinker's principle."],
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goal = `
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?very_popular_guy. !whole_pub. drinks very_popular_guy ==> drinks whole_pub`},
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{name = "P19",
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comments = [],
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goal = `
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?x. !y z. (p y ==> q z) ==> p x ==> q x`},
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{name = "P20",
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comments = [],
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goal = `
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(!x y. ?z. !w. p x /\ q y ==> r z /\ u w) /\ (!x y. p x /\ q y) ==> ?z. r z`},
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{name = "P21",
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comments = [],
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goal = `
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(?x. p ==> q x) /\ (?x. q x ==> p) ==> ?x. p <=> q x`},
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{name = "P22",
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comments = [],
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goal = `
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(!x. p <=> q x) ==> (p <=> !x. q x)`},
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{name = "P23",
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comments = [],
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goal = `
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(!x. p \/ q x) <=> p \/ !x. q x`},
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{name = "P24",
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comments = [],
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goal = `
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~(?x. u x /\ q x) /\ (!x. p x ==> q x \/ r x) /\ ~(?x. p x ==> ?x. q x) /\
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(!x. q x /\ r x ==> u x) ==> ?x. p x /\ r x`},
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{name = "P25",
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comments = [],
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goal = `
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(?x. p x) /\ (!x. u x ==> ~g x /\ r x) /\ (!x. p x ==> g x /\ u x) /\
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((!x. p x ==> q x) \/ ?x. q x /\ p x) ==> ?x. q x /\ p x`},
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{name = "P26",
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comments = [],
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goal = `
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((?x. p x) <=> ?x. q x) /\ (!x y. p x /\ q y ==> (r x <=> u y)) ==>
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((!x. p x ==> r x) <=> !x. q x ==> u x)`},
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{name = "P27",
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comments = [],
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goal = `
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(?x. p x /\ ~q x) /\ (!x. p x ==> r x) /\ (!x. u x /\ s x ==> p x) /\
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(?x. r x /\ ~q x) ==> (!x. u x ==> ~r x) ==> !x. u x ==> ~s x`},
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{name = "P28",
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comments = [],
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goal = `
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(!x. p x ==> !x. q x) /\ ((!x. q x \/ r x) ==> ?x. q x /\ r x) /\
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((?x. r x) ==> !x. l x ==> m x) ==> !x. p x /\ l x ==> m x`},
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{name = "P29",
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comments = [],
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goal = `
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(?x. p x) /\ (?x. g x) ==>
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((!x. p x ==> h x) /\ (!x. g x ==> j x) <=>
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!x y. p x /\ g y ==> h x /\ j y)`},
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{name = "P30",
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comments = [],
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goal = `
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(!x. p x \/ g x ==> ~h x) /\ (!x. (g x ==> ~u x) ==> p x /\ h x) ==>
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!x. u x`},
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{name = "P31",
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comments = [],
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goal = `
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~(?x. p x /\ (g x \/ h x)) /\ (?x. q x /\ p x) /\ (!x. ~h x ==> j x) ==>
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?x. q x /\ j x`},
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{name = "P32",
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comments = [],
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goal = `
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(!x. p x /\ (g x \/ h x) ==> q x) /\ (!x. q x /\ h x ==> j x) /\
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(!x. r x ==> h x) ==> !x. p x /\ r x ==> j x`},
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{name = "P33",
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comments = [],
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goal = `
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(!x. p a /\ (p x ==> p b) ==> p c) <=>
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(!x. p a ==> p x \/ p c) /\ (p a ==> p b ==> p c)`},
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{name = "P34",
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comments =
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["This gives rise to 5184 clauses when naively converted to CNF!"],
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goal = `
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((?x. !y. p x <=> p y) <=> (?x. q x) <=> !y. q y) <=>
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(?x. !y. q x <=> q y) <=> (?x. p x) <=> !y. p y`},
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{name = "P35",
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comments = [],
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goal = `
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?x y. p x y ==> !x y. p x y`},
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(* ------------------------------------------------------------------------- *)
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(* Predicate logic without functions. *)
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(* ------------------------------------------------------------------------- *)
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{name = "P36",
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comments = [],
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goal = `
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(!x. ?y. p x y) /\ (!x. ?y. g x y) /\
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(!x y. p x y \/ g x y ==> !z. p y z \/ g y z ==> h x z) ==> !x. ?y. h x y`},
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{name = "P37",
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comments = [],
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goal = `
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(!z. ?w. !x. ?y. (p x z ==> p y w) /\ p y z /\ (p y w ==> ?v. q v w)) /\
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(!x z. ~p x z ==> ?y. q y z) /\ ((?x y. q x y) ==> !x. r x x) ==>
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!x. ?y. r x y`},
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{name = "P38",
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comments = [],
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goal = `
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(!x. p a /\ (p x ==> ?y. p y /\ r x y) ==> ?z w. p z /\ r x w /\ r w z) <=>
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!x.
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(~p a \/ p x \/ ?z w. p z /\ r x w /\ r w z) /\
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(~p a \/ ~(?y. p y /\ r x y) \/ ?z w. p z /\ r x w /\ r w z)`},
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{name = "P39",
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comments = [],
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goal = `
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~?x. !y. p y x <=> ~p y y`},
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{name = "P40",
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comments = [],
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goal = `
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(?y. !x. p x y <=> p x x) ==> ~!x. ?y. !z. p z y <=> ~p z x`},
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{name = "P41",
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comments = [],
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goal = `
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(!z. ?y. !x. p x y <=> p x z /\ ~p x x) ==> ~?z. !x. p x z`},
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{name = "P42",
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comments = [],
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goal = `
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~?y. !x. p x y <=> ~?z. p x z /\ p z x`},
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{name = "P43",
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comments = [],
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goal = `
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(!x y. q x y <=> !z. p z x <=> p z y) ==> !x y. q x y <=> q y x`},
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{name = "P44",
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comments = [],
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goal = `
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(!x. p x ==> (?y. g y /\ h x y) /\ ?y. g y /\ ~h x y) /\
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(?x. j x /\ !y. g y ==> h x y) ==> ?x. j x /\ ~p x`},
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{name = "P45",
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comments = [],
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goal = `
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(!x. p x /\ (!y. g y /\ h x y ==> j x y) ==> !y. g y /\ h x y ==> r y) /\
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~(?y. l y /\ r y) /\
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(?x. p x /\ (!y. h x y ==> l y) /\ !y. g y /\ h x y ==> j x y) ==>
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?x. p x /\ ~?y. g y /\ h x y`},
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{name = "P46",
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comments = [],
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goal = `
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|
397 |
(!x. p x /\ (!y. p y /\ h y x ==> g y) ==> g x) /\
|
|
398 |
((?x. p x /\ ~g x) ==> ?x. p x /\ ~g x /\ !y. p y /\ ~g y ==> j x y) /\
|
|
399 |
(!x y. p x /\ p y /\ h x y ==> ~j y x) ==> !x. p x ==> g x`},
|
|
400 |
|
|
401 |
{name = "P50",
|
|
402 |
comments = [],
|
|
403 |
goal = `
|
|
404 |
(!x. f0 a x \/ !y. f0 x y) ==> ?x. !y. f0 x y`},
|
|
405 |
|
|
406 |
{name = "LOGICPROOF_L10",
|
|
407 |
comments = [],
|
|
408 |
goal = `
|
|
409 |
!x. ?y. ~(P y x <=> ~P y y)`},
|
|
410 |
|
|
411 |
{name = "BARBER",
|
|
412 |
comments = ["One resolution of the barber paradox"],
|
|
413 |
goal = `
|
|
414 |
(!x. man x ==> (shaves barber x <=> ~shaves x x)) ==> ~man barber`},
|
|
415 |
|
|
416 |
(* ------------------------------------------------------------------------- *)
|
|
417 |
(* Full predicate logic. *)
|
|
418 |
(* ------------------------------------------------------------------------- *)
|
|
419 |
|
|
420 |
{name = "LOGICPROOF_1999",
|
|
421 |
comments = ["A non-theorem."],
|
|
422 |
goal = `
|
|
423 |
(?x. p x /\ q x) ==> ?x. p (f x x) \/ !y. q y`},
|
|
424 |
|
|
425 |
{name = "P55",
|
|
426 |
comments = ["Example from Manthey and Bry, CADE-9. [JRH]"],
|
|
427 |
goal = `
|
|
428 |
lives agatha /\ lives butler /\ lives charles /\
|
|
429 |
(killed agatha agatha \/ killed butler agatha \/ killed charles agatha) /\
|
|
430 |
(!x y. killed x y ==> hates x y /\ ~richer x y) /\
|
|
431 |
(!x. hates agatha x ==> ~hates charles x) /\
|
|
432 |
(hates agatha agatha /\ hates agatha charles) /\
|
|
433 |
(!x. lives x /\ ~richer x agatha ==> hates butler x) /\
|
|
434 |
(!x. hates agatha x ==> hates butler x) /\
|
|
435 |
(!x. ~hates x agatha \/ ~hates x butler \/ ~hates x charles) ==>
|
|
436 |
killed agatha agatha /\ ~killed butler agatha /\ ~killed charles agatha`},
|
|
437 |
|
|
438 |
{name = "P57",
|
|
439 |
comments = [],
|
|
440 |
goal = `
|
|
441 |
p (f a b) (f b c) /\ p (f b c) (f a c) /\
|
|
442 |
(!x y z. p x y /\ p y z ==> p x z) ==> p (f a b) (f a c)`},
|
|
443 |
|
|
444 |
{name = "P58",
|
|
445 |
comments = ["See info-hol 1498 [JRH]"],
|
|
446 |
goal = `
|
|
447 |
!x. ?v w. !y z. p x /\ q y ==> (p v \/ r w) /\ (r z ==> q v)`},
|
|
448 |
|
|
449 |
{name = "P59",
|
|
450 |
comments = [],
|
|
451 |
goal = `
|
|
452 |
(!x. p x <=> ~p (f x)) ==> ?x. p x /\ ~p (f x)`},
|
|
453 |
|
|
454 |
{name = "P60",
|
|
455 |
comments = [],
|
|
456 |
goal = `
|
|
457 |
!x. p x (f x) <=> ?y. (!z. p z y ==> p z (f x)) /\ p x y`},
|
|
458 |
|
|
459 |
(* ------------------------------------------------------------------------- *)
|
|
460 |
(* From Gilmore's classic paper. *)
|
|
461 |
(* ------------------------------------------------------------------------- *)
|
|
462 |
|
|
463 |
{name = "GILMORE_1",
|
|
464 |
comments =
|
|
465 |
["Amazingly, this still seems non-trivial... in HOL [MESON_TAC] it",
|
|
466 |
"works at depth 45! [JRH]",
|
|
467 |
"Confirmed (depth=45, inferences=263702, time=148s), though if",
|
|
468 |
"lemmaizing is used then a lemma is discovered at depth 29 that allows",
|
|
469 |
"a much quicker proof (depth=30, inferences=10039, time=5.5s)."],
|
|
470 |
goal = `
|
|
471 |
?x. !y z.
|
|
472 |
(f y ==> g y <=> f x) /\ (f y ==> h y <=> g x) /\
|
|
473 |
((f y ==> g y) ==> h y <=> h x) ==> f z /\ g z /\ h z`},
|
|
474 |
|
|
475 |
{name = "GILMORE_2",
|
|
476 |
comments =
|
|
477 |
["This is not valid, according to Gilmore. [JRH]",
|
|
478 |
"Confirmed: ordered resolution quickly saturates."],
|
|
479 |
goal = `
|
|
480 |
?x y. !z.
|
|
481 |
(f x z <=> f z y) /\ (f z y <=> f z z) /\ (f x y <=> f y x) ==>
|
|
482 |
(f x y <=> f x z)`},
|
|
483 |
|
|
484 |
{name = "GILMORE_3",
|
|
485 |
comments = [],
|
|
486 |
goal = `
|
|
487 |
?x. !y z.
|
|
488 |
((f y z ==> g y ==> h x) ==> f x x) /\ ((f z x ==> g x) ==> h z) /\
|
|
489 |
f x y ==> f z z`},
|
|
490 |
|
|
491 |
{name = "GILMORE_4",
|
|
492 |
comments = [],
|
|
493 |
goal = `
|
|
494 |
?x y. !z. (f x y ==> f y z /\ f z z) /\ (f x y /\ g x y ==> g x z /\ g z z)`},
|
|
495 |
|
|
496 |
{name = "GILMORE_5",
|
|
497 |
comments = [],
|
|
498 |
goal = `
|
|
499 |
(!x. ?y. f x y \/ f y x) /\ (!x y. f y x ==> f y y) ==> ?z. f z z`},
|
|
500 |
|
|
501 |
{name = "GILMORE_6",
|
|
502 |
comments = [],
|
|
503 |
goal = `
|
|
504 |
!x. ?y.
|
|
505 |
(?w. !v. f w x ==> g v w /\ g w x) ==>
|
|
506 |
(?w. !v. f w y ==> g v w /\ g w y) \/
|
|
507 |
!z v. ?w. g v z \/ h w y z ==> g z w`},
|
|
508 |
|
|
509 |
{name = "GILMORE_7",
|
|
510 |
comments = [],
|
|
511 |
goal = `
|
|
512 |
(!x. k x ==> ?y. l y /\ (f x y ==> g x y)) /\
|
|
513 |
(?z. k z /\ !w. l w ==> f z w) ==> ?v w. k v /\ l w /\ g v w`},
|
|
514 |
|
|
515 |
{name = "GILMORE_8",
|
|
516 |
comments = [],
|
|
517 |
goal = `
|
|
518 |
?x. !y z.
|
|
519 |
((f y z ==> g y ==> !w. ?v. h w v x) ==> f x x) /\
|
|
520 |
((f z x ==> g x) ==> !w. ?v. h w v z) /\ f x y ==> f z z`},
|
|
521 |
|
|
522 |
{name = "GILMORE_9",
|
|
523 |
comments =
|
|
524 |
["Model elimination (in HOL):",
|
|
525 |
"- With lemmaizing: (depth=18, inferences=15632, time=21s)",
|
|
526 |
"- Without: gave up waiting after (depth=25, inferences=2125072, ...)"],
|
|
527 |
goal = `
|
|
528 |
!x. ?y. !z.
|
|
529 |
((!w. ?v. f y w v /\ g y w /\ ~h y x) ==>
|
|
530 |
(!w. ?v. f x w v /\ g z u /\ ~h x z) ==>
|
|
531 |
!w. ?v. f x w v /\ g y w /\ ~h x y) /\
|
|
532 |
((!w. ?v. f x w v /\ g y w /\ ~h x y) ==>
|
|
533 |
~(!w. ?v. f x w v /\ g z w /\ ~h x z) ==>
|
|
534 |
(!w. ?v. f y w v /\ g y w /\ ~h y x) /\
|
|
535 |
!w. ?v. f z w v /\ g y w /\ ~h z y)`},
|
|
536 |
|
|
537 |
{name = "GILMORE_9a",
|
|
538 |
comments =
|
|
539 |
["Translation of Gilmore procedure using separate definitions. [JRH]"],
|
|
540 |
goal = `
|
|
541 |
(!x y. p x y <=> !w. ?v. f x w v /\ g y w /\ ~h x y) ==>
|
|
542 |
!x. ?y. !z.
|
|
543 |
(p y x ==> p x z ==> p x y) /\ (p x y ==> ~p x z ==> p y x /\ p z y)`},
|
|
544 |
|
|
545 |
{name = "BAD_CONNECTIONS",
|
|
546 |
comments =
|
|
547 |
["The interesting example where connections make the proof longer. [JRH]"],
|
|
548 |
goal = `
|
|
549 |
~a /\ (a \/ b) /\ (c \/ d) /\ (~b \/ e \/ f) /\ (~c \/ ~e) /\ (~c \/ ~f) /\
|
|
550 |
(~b \/ g \/ h) /\ (~d \/ ~g) /\ (~d \/ ~h) ==> F`},
|
|
551 |
|
|
552 |
{name = "LOS",
|
|
553 |
comments =
|
|
554 |
["The classic Los puzzle. (Clausal version MSC006-1 in the TPTP library.)",
|
|
555 |
"Note: this is actually in the decidable AE subset, though that doesn't",
|
|
556 |
"yield a very efficient proof. [JRH]"],
|
|
557 |
goal = `
|
|
558 |
(!x y z. p x y ==> p y z ==> p x z) /\
|
|
559 |
(!x y z. q x y ==> q y z ==> q x z) /\ (!x y. q x y ==> q y x) /\
|
|
560 |
(!x y. p x y \/ q x y) ==> (!x y. p x y) \/ !x y. q x y`},
|
|
561 |
|
|
562 |
{name = "STEAM_ROLLER",
|
|
563 |
comments = [],
|
|
564 |
goal = `
|
|
565 |
((!x. p1 x ==> p0 x) /\ ?x. p1 x) /\ ((!x. p2 x ==> p0 x) /\ ?x. p2 x) /\
|
|
566 |
((!x. p3 x ==> p0 x) /\ ?x. p3 x) /\ ((!x. p4 x ==> p0 x) /\ ?x. p4 x) /\
|
|
567 |
((!x. p5 x ==> p0 x) /\ ?x. p5 x) /\ ((?x. q1 x) /\ !x. q1 x ==> q0 x) /\
|
|
568 |
(!x.
|
|
569 |
p0 x ==>
|
|
570 |
(!y. q0 y ==> r x y) \/
|
|
571 |
!y. p0 y /\ s0 y x /\ (?z. q0 z /\ r y z) ==> r x y) /\
|
|
572 |
(!x y. p3 y /\ (p5 x \/ p4 x) ==> s0 x y) /\
|
|
573 |
(!x y. p3 x /\ p2 y ==> s0 x y) /\ (!x y. p2 x /\ p1 y ==> s0 x y) /\
|
|
574 |
(!x y. p1 x /\ (p2 y \/ q1 y) ==> ~r x y) /\
|
|
575 |
(!x y. p3 x /\ p4 y ==> r x y) /\ (!x y. p3 x /\ p5 y ==> ~r x y) /\
|
|
576 |
(!x. p4 x \/ p5 x ==> ?y. q0 y /\ r x y) ==>
|
|
577 |
?x y. p0 x /\ p0 y /\ ?z. q1 z /\ r y z /\ r x y`},
|
|
578 |
|
|
579 |
{name = "MODEL_COMPLETENESS",
|
|
580 |
comments = ["An incestuous example used to establish completeness",
|
|
581 |
"characterization. [JRH]"],
|
|
582 |
goal = `
|
|
583 |
(!w x. sentence x ==> holds w x \/ holds w (not x)) /\
|
|
584 |
(!w x. ~(holds w x /\ holds w (not x))) ==>
|
|
585 |
((!x.
|
|
586 |
sentence x ==>
|
|
587 |
(!w. models w s ==> holds w x) \/
|
|
588 |
!w. models w s ==> holds w (not x)) <=>
|
|
589 |
!w v.
|
|
590 |
models w s /\ models v s ==>
|
|
591 |
!x. sentence x ==> (holds w x <=> holds v x))`}
|
|
592 |
|
|
593 |
] @
|
|
594 |
|
|
595 |
(* ========================================================================= *)
|
|
596 |
(* Problems with equality. *)
|
|
597 |
(* ========================================================================= *)
|
|
598 |
|
|
599 |
mkProblems "equality" "Equality problems from various sources" [
|
|
600 |
|
|
601 |
(* ------------------------------------------------------------------------- *)
|
|
602 |
(* Trivia (some of which demonstrate ex-bugs in the prover). *)
|
|
603 |
(* ------------------------------------------------------------------------- *)
|
|
604 |
|
|
605 |
{name = "REFLEXIVITY",
|
|
606 |
comments = [],
|
|
607 |
goal = `
|
|
608 |
c = c`},
|
|
609 |
|
|
610 |
{name = "SYMMETRY",
|
|
611 |
comments = [],
|
|
612 |
goal = `
|
|
613 |
!x y. x = y ==> y = x`},
|
|
614 |
|
|
615 |
{name = "TRANSITIVITY",
|
|
616 |
comments = [],
|
|
617 |
goal = `
|
|
618 |
!x y z. x = y /\ y = z ==> x = z`},
|
|
619 |
|
|
620 |
{name = "TRANS_SYMM",
|
|
621 |
comments = [],
|
|
622 |
goal = `
|
|
623 |
!x y z. x = y /\ y = z ==> z = x`},
|
|
624 |
|
|
625 |
{name = "SUBSTITUTIVITY",
|
|
626 |
comments = [],
|
|
627 |
goal = `
|
|
628 |
!x y. f x /\ x = y ==> f y`},
|
|
629 |
|
|
630 |
{name = "CYCLIC_SUBSTITUTION_BUG",
|
|
631 |
comments = [],
|
|
632 |
goal = `
|
|
633 |
!y. (!x. y = g (c x)) ==> ?z. y = g z`},
|
|
634 |
|
|
635 |
{name = "JUDITA_1",
|
|
636 |
comments = [],
|
|
637 |
goal = `
|
|
638 |
~(a = b) /\ (!x. x = c) ==> F`},
|
|
639 |
|
|
640 |
{name = "JUDITA_2",
|
|
641 |
comments = [],
|
|
642 |
goal = `
|
|
643 |
~(a = b) /\ (!x y. x = y) ==> F`},
|
|
644 |
|
|
645 |
{name = "JUDITA_3",
|
|
646 |
comments = [],
|
|
647 |
goal = `
|
|
648 |
p a /\ ~p b /\ (!x. x = c) ==> F`},
|
|
649 |
|
|
650 |
{name = "JUDITA_4",
|
|
651 |
comments = [],
|
|
652 |
goal = `
|
|
653 |
p a /\ ~p b /\ (!x y. x = y) ==> F`},
|
|
654 |
|
|
655 |
{name = "JUDITA_5",
|
|
656 |
comments = [],
|
|
657 |
goal = `
|
|
658 |
p a /\ p b /\ ~(a = b) /\ ~p c /\ (!x. x = a \/ x = d) ==> F`},
|
|
659 |
|
|
660 |
{name = "INJECTIVITY_1",
|
|
661 |
comments = [],
|
|
662 |
goal = `
|
|
663 |
(!x y. f x = f y ==> x = y) /\ f a = f b ==> a = b`},
|
|
664 |
|
|
665 |
{name = "INJECTIVITY_2",
|
|
666 |
comments = [],
|
|
667 |
goal = `
|
|
668 |
(!x y. g (f x) = g (f y) ==> x = y) /\ f a = f b ==> a = b`},
|
|
669 |
|
|
670 |
{name = "SELF_REWRITE",
|
|
671 |
comments = [],
|
|
672 |
goal = `
|
|
673 |
f (g (h c)) = h c /\ g (h c) = b /\ f b = a /\ (!x. ~(a = h x)) ==> F`},
|
|
674 |
|
|
675 |
{name = "EQUALITY_ORDERING",
|
|
676 |
comments = ["Positive resolution saturates if equality literals are",
|
|
677 |
"ordered like other literals, instead of considering their",
|
|
678 |
"left and right sides."],
|
|
679 |
goal = `
|
|
680 |
p a /\ q a /\ q b /\ r b /\ (~p c \/ c = a) /\ (~r c \/ c = b) /\
|
|
681 |
(!x. ~q x \/ p x \/ r x) /\ (~p c \/ ~q c) /\ (~q c \/ ~r c) /\
|
|
682 |
(c = b \/ p c \/ q c) /\ (~p b \/ c = a \/ q c) ==> F`},
|
|
683 |
|
|
684 |
(* ------------------------------------------------------------------------- *)
|
|
685 |
(* Simple equality problems. *)
|
|
686 |
(* ------------------------------------------------------------------------- *)
|
|
687 |
|
|
688 |
{name = "P48",
|
|
689 |
comments = [],
|
|
690 |
goal = `
|
|
691 |
(a = b \/ c = d) /\ (a = c \/ b = d) ==> a = d \/ b = c`},
|
|
692 |
|
|
693 |
{name = "P49",
|
|
694 |
comments = [],
|
|
695 |
goal = `
|
|
696 |
(?x y. !z. z = x \/ z = y) /\ p a /\ p b /\ ~(a = b) ==> !x. p x`},
|
|
697 |
|
|
698 |
{name = "P51",
|
|
699 |
comments = [],
|
|
700 |
goal = `
|
|
701 |
(?z w. !x y. f0 x y <=> x = z /\ y = w) ==>
|
|
702 |
?z. !x. (?w. !y. f0 x y <=> y = w) <=> x = z`},
|
|
703 |
|
|
704 |
{name = "P52",
|
|
705 |
comments = [],
|
|
706 |
goal = `
|
|
707 |
(?z w. !x y. f0 x y <=> x = z /\ y = w) ==>
|
|
708 |
?w. !y. (?z. !x. f0 x y <=> x = z) <=> y = w`},
|
|
709 |
|
|
710 |
{name = "UNSKOLEMIZED_MELHAM",
|
|
711 |
comments = ["The Melham problem after an inverse skolemization step."],
|
|
712 |
goal = `
|
|
713 |
(!x y. g x = g y ==> f x = f y) ==> !y. ?w. !x. y = g x ==> w = f x`},
|
|
714 |
|
|
715 |
{name = "CONGRUENCE_CLOSURE_EXAMPLE",
|
|
716 |
comments = ["The example always given for congruence closure."],
|
|
717 |
goal = `
|
|
718 |
!x. f (f (f (f (f x)))) = x /\ f (f (f x)) = x ==> f x = x`},
|
|
719 |
|
|
720 |
{name = "EWD_1",
|
|
721 |
comments =
|
|
722 |
["A simple example (see EWD1266a and the application to Morley's",
|
|
723 |
"theorem). [JRH]"],
|
|
724 |
goal = `
|
|
725 |
(!x. f x ==> g x) /\ (?x. f x) /\ (!x y. g x /\ g y ==> x = y) ==>
|
|
726 |
!y. g y ==> f y`},
|
|
727 |
|
|
728 |
{name = "EWD_2",
|
|
729 |
comments = [],
|
|
730 |
goal = `
|
|
731 |
(!x. f (f x) = f x) /\ (!x. ?y. f y = x) ==> !x. f x = x`},
|
|
732 |
|
|
733 |
{name = "JIA",
|
|
734 |
comments = ["Needs only the K combinator"],
|
|
735 |
goal = `
|
|
736 |
(!x y. k . x . y = x) /\ (!v. P (v . a) a) /\ (!w. Q (w . b) b) ==>
|
|
737 |
!z. ?x y. P (z . x . y) x /\ Q (z . x . y) y`},
|
|
738 |
|
|
739 |
{name = "WISHNU",
|
|
740 |
comments = ["Wishnu Prasetya's example. [JRH]"],
|
|
741 |
goal = `
|
|
742 |
(?x. x = f (g x) /\ !x'. x' = f (g x') ==> x = x') <=>
|
|
743 |
?y. y = g (f y) /\ !y'. y' = g (f y') ==> y = y'`},
|
|
744 |
|
|
745 |
{name = "AGATHA",
|
|
746 |
comments = ["An equality version of the Agatha puzzle. [JRH]"],
|
|
747 |
goal = `
|
|
748 |
(?x. lives x /\ killed x agatha) /\
|
|
749 |
(lives agatha /\ lives butler /\ lives charles) /\
|
|
750 |
(!x. lives x ==> x = agatha \/ x = butler \/ x = charles) /\
|
|
751 |
(!x y. killed x y ==> hates x y) /\ (!x y. killed x y ==> ~richer x y) /\
|
|
752 |
(!x. hates agatha x ==> ~hates charles x) /\
|
|
753 |
(!x. ~(x = butler) ==> hates agatha x) /\
|
|
754 |
(!x. ~richer x agatha ==> hates butler x) /\
|
|
755 |
(!x. hates agatha x ==> hates butler x) /\ (!x. ?y. ~hates x y) /\
|
|
756 |
~(agatha = butler) ==>
|
|
757 |
killed agatha agatha /\ ~killed butler agatha /\ ~killed charles agatha`},
|
|
758 |
|
|
759 |
{name = "DIVIDES_9_1",
|
|
760 |
comments = [],
|
|
761 |
goal = `
|
|
762 |
(!x y. x * y = y * x) /\ (!x y z. x * y * z = x * (y * z)) /\
|
|
763 |
(!x y. divides x y <=> ?z. y = z * x) ==>
|
|
764 |
!x y z. divides x y ==> divides x (z * y)`},
|
|
765 |
|
|
766 |
{name = "DIVIDES_9_2",
|
|
767 |
comments = [],
|
|
768 |
goal = `
|
|
769 |
(!x y. x * y = y * x) /\ (!x y z. x * y * z = x * (y * z)) /\
|
|
770 |
(!x y. divides x y <=> ?z. z * x = y) ==>
|
|
771 |
!x y z. divides x y ==> divides x (z * y)`},
|
|
772 |
|
|
773 |
{name = "XOR_COUNT_COMMUTATIVE",
|
|
774 |
comments = ["The xor literal counting function in Normalize is commutative."],
|
|
775 |
goal = `
|
|
776 |
(!x y. x + y = y + x) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
777 |
(!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\
|
|
778 |
pl = p1 * p2 + n1 * n2 /\ nl = p1 * n2 + n1 * p2 /\
|
|
779 |
pr = p2 * p1 + n2 * n1 /\ nr = p2 * n1 + n2 * p1 ==> pl = pr /\ nl = nr`},
|
|
780 |
|
|
781 |
{name = "XOR_COUNT_ASSOCIATIVE",
|
|
782 |
comments = ["The xor literal counting function in Normalize is associative."],
|
|
783 |
goal = `
|
|
784 |
(!x y. x + y = y + x) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
785 |
(!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\
|
|
786 |
px = p1 * p2 + n1 * n2 /\ nx = p1 * n2 + n1 * p2 /\
|
|
787 |
pl = px * p3 + nx * n3 /\ nl = px * n3 + nx * p3 /\
|
|
788 |
py = p2 * p3 + n2 * n3 /\ ny = p2 * n3 + n2 * p3 /\
|
|
789 |
pr = p1 * py + n1 * ny /\ nr = p1 * ny + n1 * py ==> pl = pr /\ nl = nr`},
|
|
790 |
|
|
791 |
{name = "REVERSE_REVERSE",
|
|
792 |
comments = ["Proving the goal",
|
|
793 |
" !l. finite l ==> rev (rev l) = l",
|
|
794 |
"after first generalizing it to",
|
|
795 |
" !l k. finite l /\\ finite k ==> rev (rev l @ k) = rev k @ l",
|
|
796 |
"and then applying list induction."],
|
|
797 |
goal = `
|
|
798 |
finite nil /\ (!h t. finite (h :: t) <=> finite t) /\
|
|
799 |
(!l1 l2. finite (l1 @ l2) <=> finite l1 /\ finite l2) /\
|
|
800 |
(!l. nil @ l = l) /\ (!h t l. (h :: t) @ l = h :: t @ l) /\
|
|
801 |
rev nil = nil /\ (!h t. rev (h :: t) = rev t @ h :: nil) /\
|
|
802 |
(!l. l @ nil = l) /\ (!l1 l2 l3. l1 @ l2 @ l3 = (l1 @ l2) @ l3) ==>
|
|
803 |
(!k. finite k ==> rev (rev nil @ k) = rev k @ nil) /\
|
|
804 |
!t.
|
|
805 |
finite t ==> (!k. finite k ==> rev (rev t @ k) = rev k @ t) ==>
|
|
806 |
!h k. finite k ==> rev (rev (h :: t) @ k) = rev k @ h :: t`},
|
|
807 |
|
|
808 |
(* ------------------------------------------------------------------------- *)
|
|
809 |
(* Group theory examples. *)
|
|
810 |
(* ------------------------------------------------------------------------- *)
|
|
811 |
|
|
812 |
{name = "GROUP_INVERSE_INVERSE",
|
|
813 |
comments = [],
|
|
814 |
goal = `
|
|
815 |
(!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\
|
|
816 |
(!x. i x * x = e) ==> !x. i (i x) = x`},
|
|
817 |
|
|
818 |
{name = "GROUP_RIGHT_INVERSE",
|
|
819 |
comments = ["Size 18, 61814 seconds. [JRH]"],
|
|
820 |
goal = `
|
|
821 |
(!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\
|
|
822 |
(!x. i x * x = e) ==> !x. x * i x = e`},
|
|
823 |
|
|
824 |
{name = "GROUP_RIGHT_IDENTITY",
|
|
825 |
comments = [],
|
|
826 |
goal = `
|
|
827 |
(!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\
|
|
828 |
(!x. i x * x = e) ==> !x. x * e = x`},
|
|
829 |
|
|
830 |
{name = "GROUP_IDENTITY_EQUIV",
|
|
831 |
comments = [],
|
|
832 |
goal = `
|
|
833 |
(!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\
|
|
834 |
(!x. i x * x = e) ==> !x. x * x = x <=> x = e`},
|
|
835 |
|
|
836 |
{name = "KLEIN_GROUP_COMMUTATIVE",
|
|
837 |
comments = [],
|
|
838 |
goal = `
|
|
839 |
(!x y z. x * (y * z) = x * y * z) /\ (!x. e * x = x) /\ (!x. x * e = x) /\
|
|
840 |
(!x. x * x = e) ==> !x y. x * y = y * x`},
|
|
841 |
|
|
842 |
{name = "DOUBLE_DISTRIB",
|
|
843 |
comments = ["From a John Harrison post to hol-info on 2008-04-15"],
|
|
844 |
goal = `
|
|
845 |
(!x y z. x * y * z = x * z * (y * z)) /\
|
|
846 |
(!x y z. z * (x * y) = z * x * (z * y)) ==>
|
|
847 |
!a b c. a * b * (c * a) = a * c * (b * a)`},
|
|
848 |
|
|
849 |
(* ------------------------------------------------------------------------- *)
|
|
850 |
(* Ring theory examples. *)
|
|
851 |
(* ------------------------------------------------------------------------- *)
|
|
852 |
|
|
853 |
{name = "CONWAY_2",
|
|
854 |
comments = ["From a John Harrison post to hol-info on 2008-04-15"],
|
|
855 |
goal = `
|
|
856 |
(!x. 0 + x = x) /\ (!x y. x + y = y + x) /\
|
|
857 |
(!x y z. x + (y + z) = x + y + z) /\ (!x. 1 * x = x) /\ (!x. x * 1 = x) /\
|
|
858 |
(!x y z. x * (y * z) = x * y * z) /\ (!x. 0 * x = 0) /\ (!x. x * 0 = 0) /\
|
|
859 |
(!x y z. x * (y + z) = x * y + x * z) /\
|
|
860 |
(!x y z. (x + y) * z = x * z + y * z) /\
|
|
861 |
(!x y. star (x * y) = 1 + x * star (y * x) * y) /\
|
|
862 |
(!x y. star (x + y) = star (star (x) * y) * star (x)) ==>
|
|
863 |
!a. star (star (star (star (a)))) = star (star (star (a)))`},
|
|
864 |
|
|
865 |
{name = "JACOBSON_2",
|
|
866 |
comments = [],
|
|
867 |
goal = `
|
|
868 |
(!x. 0 + x = x) /\ (!x. x + 0 = x) /\ (!x. ~x + x = 0) /\
|
|
869 |
(!x. x + ~x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
870 |
(!x y. x + y = y + x) /\ (!x y z. x * (y * z) = x * y * z) /\
|
|
871 |
(!x y z. x * (y + z) = x * y + x * z) /\
|
|
872 |
(!x y z. (x + y) * z = x * z + y * z) /\ (!x. x * x = x) ==>
|
|
873 |
!x y. x * y = y * x`},
|
|
874 |
|
|
875 |
{name = "JACOBSON_3",
|
|
876 |
comments = [],
|
|
877 |
goal = `
|
|
878 |
(!x. 0 + x = x) /\ (!x. x + 0 = x) /\ (!x. ~x + x = 0) /\
|
|
879 |
(!x. x + ~x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
880 |
(!x y. x + y = y + x) /\ (!x y z. x * (y * z) = x * y * z) /\
|
|
881 |
(!x y z. x * (y + z) = x * y + x * z) /\
|
|
882 |
(!x y z. (x + y) * z = x * z + y * z) /\ (!x. x * (x * x) = x) ==>
|
|
883 |
!x y. x * y = y * x`},
|
|
884 |
|
|
885 |
(* ------------------------------------------------------------------------- *)
|
|
886 |
(* Set theory examples. *)
|
|
887 |
(* ------------------------------------------------------------------------- *)
|
|
888 |
|
|
889 |
{name = "UNION_2_SUBSET",
|
|
890 |
comments = [],
|
|
891 |
goal = `
|
|
892 |
(!x y. subset x y ==> subset y x ==> x = y) /\
|
|
893 |
(!x y. (!z. member z x ==> member z y) ==> subset x y) /\
|
|
894 |
(!x y z. member z (x + y) <=> member z x \/ member z y) ==>
|
|
895 |
!x y. subset (x + y) (y + x)`},
|
|
896 |
|
|
897 |
{name = "UNION_2",
|
|
898 |
comments = [],
|
|
899 |
goal = `
|
|
900 |
(!x y. subset x y ==> subset y x ==> x = y) /\
|
|
901 |
(!x y. (!z. member z x ==> member z y) ==> subset x y) /\
|
|
902 |
(!x y z. member z (x + y) <=> member z x \/ member z y) ==>
|
|
903 |
!x y. x + y = y + x`},
|
|
904 |
|
|
905 |
{name = "UNION_3_SUBSET",
|
|
906 |
comments = ["From an email from Tobias Nipkow, 4 Nov 2008.",
|
|
907 |
"The Isabelle version of metis diverges on this goal"],
|
|
908 |
goal = `
|
|
909 |
(!x y. subset x y ==> subset y x ==> x = y) /\
|
|
910 |
(!x y. (!z. member z x ==> member z y) ==> subset x y) /\
|
|
911 |
(!x y z. member z (x + y) <=> member z x \/ member z y) ==>
|
|
912 |
!x y z. subset (x + y + z) (z + y + x)`},
|
|
913 |
|
|
914 |
{name = "UNION_3",
|
|
915 |
comments = ["From an email from Tobias Nipkow, 28 Oct 2008.",
|
|
916 |
"The Isabelle version of metis diverges on this goal"],
|
|
917 |
goal = `
|
|
918 |
(!x y. subset x y ==> subset y x ==> x = y) /\
|
|
919 |
(!x y. (!z. member z x ==> member z y) ==> subset x y) /\
|
|
920 |
(!x y z. member z (x + y) <=> member z x \/ member z y) ==>
|
|
921 |
!x y z. x + y + z = z + y + x`}] @
|
|
922 |
|
|
923 |
(* ========================================================================= *)
|
|
924 |
(* Some sample problems from the TPTP archive. *)
|
|
925 |
(* Note: for brevity some relation/function names have been shortened. *)
|
|
926 |
(* ========================================================================= *)
|
|
927 |
|
|
928 |
mkProblems "tptp" "Sample problems from the TPTP collection"
|
|
929 |
|
|
930 |
[
|
|
931 |
|
|
932 |
{name = "ALG005-1",
|
|
933 |
comments = [],
|
|
934 |
goal = `
|
|
935 |
(!x y. x + (y + x) = x) /\ (!x y. x + (x + y) = y + (y + x)) /\
|
|
936 |
(!x y z. x + y + z = x + z + (y + z)) /\ (!x y. x * y = x + (x + y)) ==>
|
|
937 |
~(a * b * c = a * (b * c)) ==> F`},
|
|
938 |
|
|
939 |
{name = "ALG006-1",
|
|
940 |
comments = [],
|
|
941 |
goal = `
|
|
942 |
(!x y. x + (y + x) = x) /\ (!x y. x + (x + y) = y + (y + x)) /\
|
|
943 |
(!x y z. x + y + z = x + z + (y + z)) ==> ~(a + c + b = a + b + c) ==> F`},
|
|
944 |
|
|
945 |
{name = "BOO021-1",
|
|
946 |
comments = [],
|
|
947 |
goal = `
|
|
948 |
(!x y. (x + y) * y = y) /\ (!x y z. x * (y + z) = y * x + z * x) /\
|
|
949 |
(!x. x + i x = 1) /\ (!x y. x * y + y = y) /\
|
|
950 |
(!x y z. x + y * z = (y + x) * (z + x)) /\ (!x. x * i x = 0) ==>
|
|
951 |
~(b * a = a * b) ==> F`},
|
|
952 |
|
|
953 |
{name = "COL058-2",
|
|
954 |
comments = [],
|
|
955 |
goal = `
|
|
956 |
(!x y. r (r 0 x) y = r x (r y y)) ==>
|
|
957 |
~(r (r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0)))
|
|
958 |
(r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) =
|
|
959 |
r (r 0 (r (r 0 (r 0 0)) (r 0 (r 0 0)))) (r 0 (r 0 0))) ==> F`},
|
|
960 |
|
|
961 |
{name = "COL060-3",
|
|
962 |
comments = [],
|
|
963 |
goal = `
|
|
964 |
(!x y z. b . x . y . z = x . (y . z)) /\ (!x y. t . x . y = y . x) ==>
|
|
965 |
~(b . (b . (t . b) . b) . t . x . y . z = y . (x . z)) ==> F`},
|
|
966 |
|
|
967 |
{name = "GEO002-4",
|
|
968 |
comments = [],
|
|
969 |
goal = `
|
|
970 |
(!x y z v. ~between x y z \/ ~between y v z \/ between x y v) /\
|
|
971 |
(!x y z. ~equidistant x y z z \/ x == y) /\
|
|
972 |
(!x y z v w.
|
|
973 |
~between x y z \/ ~between v z w \/
|
|
974 |
between x (outer_pasch y x v w z) v) /\
|
|
975 |
(!x y z v w.
|
|
976 |
~between x y z \/ ~between v z w \/
|
|
977 |
between w y (outer_pasch y x v w z)) /\
|
|
978 |
(!x y z v. between x y (extension x y z v)) /\
|
|
979 |
(!x y z v. equidistant x (extension y x z v) z v) /\
|
|
980 |
(!x y z v. ~(x == y) \/ ~between z v x \/ between z v y) ==>
|
|
981 |
~between a a b ==> F`},
|
|
982 |
|
|
983 |
{name = "GEO036-2",
|
|
984 |
comments = [],
|
|
985 |
goal = `
|
|
986 |
(!x y. equidistant x y y x) /\
|
|
987 |
(!x y z x' y' z'.
|
|
988 |
~equidistant x y z x' \/ ~equidistant x y y' z' \/
|
|
989 |
equidistant z x' y' z') /\ (!x y z. ~equidistant x y z z \/ x = y) /\
|
|
990 |
(!x y z v. between x y (extension x y z v)) /\
|
|
991 |
(!x y z v. equidistant x (extension y x z v) z v) /\
|
|
992 |
(!x y z v w x' y' z'.
|
|
993 |
~equidistant x y z v \/ ~equidistant y w v x' \/
|
|
994 |
~equidistant x y' z z' \/ ~equidistant y y' v z' \/ ~between x y w \/
|
|
995 |
~between z v x' \/ x = y \/ equidistant w y' x' z') /\
|
|
996 |
(!x y. ~between x y x \/ x = y) /\
|
|
997 |
(!x y z v w.
|
|
998 |
~between x y z \/ ~between v w z \/
|
|
999 |
between y (inner_pasch x y z w v) v) /\
|
|
1000 |
(!x y z v w.
|
|
1001 |
~between x y z \/ ~between v w z \/
|
|
1002 |
between w (inner_pasch x y z w v) x) /\
|
|
1003 |
~between lower_dimension_point_1 lower_dimension_point_2
|
|
1004 |
lower_dimension_point_3 /\
|
|
1005 |
~between lower_dimension_point_2 lower_dimension_point_3
|
|
1006 |
lower_dimension_point_1 /\
|
|
1007 |
~between lower_dimension_point_3 lower_dimension_point_1
|
|
1008 |
lower_dimension_point_2 /\
|
|
1009 |
(!x y z v w.
|
|
1010 |
~equidistant x y x z \/ ~equidistant v y v z \/ ~equidistant w y w z \/
|
|
1011 |
between x v w \/ between v w x \/ between w x v \/ y = z) /\
|
|
1012 |
(!x y z v w.
|
|
1013 |
~between x y z \/ ~between v y w \/ x = y \/
|
|
1014 |
between x v (euclid1 x v y w z)) /\
|
|
1015 |
(!x y z v w.
|
|
1016 |
~between x y z \/ ~between v y w \/ x = y \/
|
|
1017 |
between x w (euclid2 x v y w z)) /\
|
|
1018 |
(!x y z v w.
|
|
1019 |
~between x y z \/ ~between v y w \/ x = y \/
|
|
1020 |
between (euclid1 x v y w z) z (euclid2 x v y w z)) /\
|
|
1021 |
(!x y z x' y' z'.
|
|
1022 |
~equidistant x y x z \/ ~equidistant x x' x y' \/ ~between x y x' \/
|
|
1023 |
~between y z' x' \/ between z (continuous x y z z' x' y') y') /\
|
|
1024 |
(!x y z x' y' z'.
|
|
1025 |
~equidistant x y x z \/ ~equidistant x x' x y' \/ ~between x y x' \/
|
|
1026 |
~between y z' x' \/ equidistant x z' x (continuous x y z z' x' y')) /\
|
|
1027 |
(!x y z v. ~(x = y) \/ ~between x z v \/ between y z v) /\
|
|
1028 |
(!x y z v. ~(x = y) \/ ~between z x v \/ between z y v) /\
|
|
1029 |
(!x y z v. ~(x = y) \/ ~between z v x \/ between z v y) /\
|
|
1030 |
(!x y z v w. ~(x = y) \/ ~equidistant x z v w \/ equidistant y z v w) /\
|
|
1031 |
(!x y z v w. ~(x = y) \/ ~equidistant z x v w \/ equidistant z y v w) /\
|
|
1032 |
(!x y z v w. ~(x = y) \/ ~equidistant z v x w \/ equidistant z v y w) /\
|
|
1033 |
(!x y z v w. ~(x = y) \/ ~equidistant z v w x \/ equidistant z v w y) /\
|
|
1034 |
(!x y z x' y' z'.
|
|
1035 |
~(x = y) \/ inner_pasch x z x' y' z' = inner_pasch y z x' y' z') /\
|
|
1036 |
(!x y z x' y' z'.
|
|
1037 |
~(x = y) \/ inner_pasch z x x' y' z' = inner_pasch z y x' y' z') /\
|
|
1038 |
(!x y z x' y' z'.
|
|
1039 |
~(x = y) \/ inner_pasch z x' x y' z' = inner_pasch z x' y y' z') /\
|
|
1040 |
(!x y z x' y' z'.
|
|
1041 |
~(x = y) \/ inner_pasch z x' y' x z' = inner_pasch z x' y' y z') /\
|
|
1042 |
(!x y z x' y' z'.
|
|
1043 |
~(x = y) \/ inner_pasch z x' y' z' x = inner_pasch z x' y' z' y) /\
|
|
1044 |
(!x y z x' y' z'.
|
|
1045 |
~(x = y) \/ euclid1 x z x' y' z' = euclid1 y z x' y' z') /\
|
|
1046 |
(!x y z x' y' z'.
|
|
1047 |
~(x = y) \/ euclid1 z x x' y' z' = euclid1 z y x' y' z') /\
|
|
1048 |
(!x y z x' y' z'.
|
|
1049 |
~(x = y) \/ euclid1 z x' x y' z' = euclid1 z x' y y' z') /\
|
|
1050 |
(!x y z x' y' z'.
|
|
1051 |
~(x = y) \/ euclid1 z x' y' x z' = euclid1 z x' y' y z') /\
|
|
1052 |
(!x y z x' y' z'.
|
|
1053 |
~(x = y) \/ euclid1 z x' y' z' x = euclid1 z x' y' z' y) /\
|
|
1054 |
(!x y z x' y' z'.
|
|
1055 |
~(x = y) \/ euclid2 x z x' y' z' = euclid2 y z x' y' z') /\
|
|
1056 |
(!x y z x' y' z'.
|
|
1057 |
~(x = y) \/ euclid2 z x x' y' z' = euclid2 z y x' y' z') /\
|
|
1058 |
(!x y z x' y' z'.
|
|
1059 |
~(x = y) \/ euclid2 z x' x y' z' = euclid2 z x' y y' z') /\
|
|
1060 |
(!x y z x' y' z'.
|
|
1061 |
~(x = y) \/ euclid2 z x' y' x z' = euclid2 z x' y' y z') /\
|
|
1062 |
(!x y z x' y' z'.
|
|
1063 |
~(x = y) \/ euclid2 z x' y' z' x = euclid2 z x' y' z' y) /\
|
|
1064 |
(!x y z v w. ~(x = y) \/ extension x z v w = extension y z v w) /\
|
|
1065 |
(!x y z v w. ~(x = y) \/ extension z x v w = extension z y v w) /\
|
|
1066 |
(!x y z v w. ~(x = y) \/ extension z v x w = extension z v y w) /\
|
|
1067 |
(!x y z v w. ~(x = y) \/ extension z v w x = extension z v w y) /\
|
|
1068 |
(!x y z v w x' y'.
|
|
1069 |
~(x = y) \/ continuous x z v w x' y' = continuous y z v w x' y') /\
|
|
1070 |
(!x y z v w x' y'.
|
|
1071 |
~(x = y) \/ continuous z x v w x' y' = continuous z y v w x' y') /\
|
|
1072 |
(!x y z v w x' y'.
|
|
1073 |
~(x = y) \/ continuous z v x w x' y' = continuous z v y w x' y') /\
|
|
1074 |
(!x y z v w x' y'.
|
|
1075 |
~(x = y) \/ continuous z v w x x' y' = continuous z v w y x' y') /\
|
|
1076 |
(!x y z v w x' y'.
|
|
1077 |
~(x = y) \/ continuous z v w x' x y' = continuous z v w x' y y') /\
|
|
1078 |
(!x y z v w x' y'.
|
|
1079 |
~(x = y) \/ continuous z v w x' y' x = continuous z v w x' y' y) ==>
|
|
1080 |
lower_dimension_point_1 = lower_dimension_point_2 \/
|
|
1081 |
lower_dimension_point_2 = lower_dimension_point_3 \/
|
|
1082 |
lower_dimension_point_1 = lower_dimension_point_3 ==> F`},
|
|
1083 |
|
|
1084 |
{name = "GRP010-4",
|
|
1085 |
comments = [],
|
|
1086 |
goal = `
|
|
1087 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1088 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. ~(x = y) \/ i x = i y) /\
|
|
1089 |
(!x y z. ~(x = y) \/ x * z = y * z) /\
|
|
1090 |
(!x y z. ~(x = y) \/ z * x = z * y) /\ (!x y z. x * y * z = x * (y * z)) /\
|
|
1091 |
(!x. 1 * x = x) /\ (!x. i x * x = 1) /\ c * b = 1 ==> ~(b * c = 1) ==> F`},
|
|
1092 |
|
|
1093 |
{name = "GRP057-1",
|
|
1094 |
comments = [],
|
|
1095 |
goal = `
|
|
1096 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1097 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\
|
|
1098 |
(!x y z v. x * i (i (i y * (i x * z)) * v * i (y * v)) = z) /\
|
|
1099 |
(!x y. ~(x = y) \/ i x = i y) /\ (!x y z. ~(x = y) \/ x * z = y * z) /\
|
|
1100 |
(!x y z. ~(x = y) \/ z * x = z * y) ==>
|
|
1101 |
~(i a1 * a1 = i b1 * b1) \/ ~(i b2 * b2 * a2 = a2) \/
|
|
1102 |
~(a3 * b3 * c3 = a3 * (b3 * c3)) ==> F`},
|
|
1103 |
|
|
1104 |
{name = "GRP086-1",
|
|
1105 |
comments = ["Bug check: used to be unsolvable without sticky constraints"],
|
|
1106 |
goal = `
|
|
1107 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1108 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\
|
|
1109 |
(!x y z. x * (y * z * i (x * z)) = y) /\ (!x y. ~(x = y) \/ i x = i y) /\
|
|
1110 |
(!x y z. ~(x = y) \/ x * z = y * z) /\
|
|
1111 |
(!x y z. ~(x = y) \/ z * x = z * y) ==>
|
|
1112 |
(!x y.
|
|
1113 |
~(i a1 * a1 = i b1 * b1) \/ ~(i b2 * b2 * a2 = a2) \/
|
|
1114 |
~(a3 * b3 * c3 = a3 * (b3 * c3)) \/ ~(x * y = y * x)) ==> F`},
|
|
1115 |
|
|
1116 |
{name = "GRP104-1",
|
|
1117 |
comments = [],
|
|
1118 |
goal = `
|
|
1119 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1120 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\
|
|
1121 |
(!x y z.
|
|
1122 |
double_divide x
|
|
1123 |
(inverse
|
|
1124 |
(double_divide
|
|
1125 |
(inverse (double_divide (double_divide x y) (inverse z))) y)) =
|
|
1126 |
z) /\ (!x y. x * y = inverse (double_divide y x)) /\
|
|
1127 |
(!x y. ~(x = y) \/ inverse x = inverse y) /\
|
|
1128 |
(!x y z. ~(x = y) \/ x * z = y * z) /\
|
|
1129 |
(!x y z. ~(x = y) \/ z * x = z * y) /\
|
|
1130 |
(!x y z. ~(x = y) \/ double_divide x z = double_divide y z) /\
|
|
1131 |
(!x y z. ~(x = y) \/ double_divide z x = double_divide z y) ==>
|
|
1132 |
(!x y.
|
|
1133 |
~(inverse a1 * a1 = inverse b1 * b1) \/ ~(inverse b2 * b2 * a2 = a2) \/
|
|
1134 |
~(a3 * b3 * c3 = a3 * (b3 * c3)) \/ ~(x * y = y * x)) ==> F`},
|
|
1135 |
|
|
1136 |
{name = "GRP128-4.003",
|
|
1137 |
comments = [],
|
|
1138 |
goal = `
|
|
1139 |
(!x y.
|
|
1140 |
~elt x \/ ~elt y \/ product e_1 x y \/ product e_2 x y \/
|
|
1141 |
product e_3 x y) /\
|
|
1142 |
(!x y.
|
|
1143 |
~elt x \/ ~elt y \/ product x e_1 y \/ product x e_2 y \/
|
|
1144 |
product x e_3 y) /\ elt e_1 /\ elt e_2 /\ elt e_3 /\ ~(e_1 == e_2) /\
|
|
1145 |
~(e_1 == e_3) /\ ~(e_2 == e_1) /\ ~(e_2 == e_3) /\ ~(e_3 == e_1) /\
|
|
1146 |
~(e_3 == e_2) /\
|
|
1147 |
(!x y.
|
|
1148 |
~elt x \/ ~elt y \/ product x y e_1 \/ product x y e_2 \/
|
|
1149 |
product x y e_3) /\
|
|
1150 |
(!x y z v. ~product x y z \/ ~product x y v \/ z == v) /\
|
|
1151 |
(!x y z v. ~product x y z \/ ~product x v z \/ y == v) /\
|
|
1152 |
(!x y z v. ~product x y z \/ ~product v y z \/ x == v) ==>
|
|
1153 |
(!x y z v. product x y z \/ ~product x z v \/ ~product z y v) /\
|
|
1154 |
(!x y z v. product x y z \/ ~product v x z \/ ~product v y x) /\
|
|
1155 |
(!x y z v. ~product x y z \/ ~product z y v \/ product x z v) ==> F`},
|
|
1156 |
|
|
1157 |
{name = "HEN006-3",
|
|
1158 |
comments = [],
|
|
1159 |
goal = `
|
|
1160 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1161 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\
|
|
1162 |
(!x y. ~(x <= y) \/ x / y = 0) /\ (!x y. ~(x / y = 0) \/ x <= y) /\
|
|
1163 |
(!x y. x / y <= x) /\ (!x y z. x / y / (z / y) <= x / z / y) /\
|
|
1164 |
(!x. 0 <= x) /\ (!x y. ~(x <= y) \/ ~(y <= x) \/ x = y) /\ (!x. x <= 1) /\
|
|
1165 |
(!x y z. ~(x = y) \/ x / z = y / z) /\
|
|
1166 |
(!x y z. ~(x = y) \/ z / x = z / y) /\
|
|
1167 |
(!x y z. ~(x = y) \/ ~(x <= z) \/ y <= z) /\
|
|
1168 |
(!x y z. ~(x = y) \/ ~(z <= x) \/ z <= y) /\ a / b <= d ==>
|
|
1169 |
~(a / d <= b) ==> F`},
|
|
1170 |
|
|
1171 |
{name = "LAT005-3",
|
|
1172 |
comments = ["SAM's lemma"],
|
|
1173 |
goal = `
|
|
1174 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1175 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x. meet x x = x) /\
|
|
1176 |
(!x. join x x = x) /\ (!x y. meet x (join x y) = x) /\
|
|
1177 |
(!x y. join x (meet x y) = x) /\ (!x y. meet x y = meet y x) /\
|
|
1178 |
(!x y. join x y = join y x) /\
|
|
1179 |
(!x y z. meet (meet x y) z = meet x (meet y z)) /\
|
|
1180 |
(!x y z. join (join x y) z = join x (join y z)) /\
|
|
1181 |
(!x y z. ~(x = y) \/ join x z = join y z) /\
|
|
1182 |
(!x y z. ~(x = y) \/ join z x = join z y) /\
|
|
1183 |
(!x y z. ~(x = y) \/ meet x z = meet y z) /\
|
|
1184 |
(!x y z. ~(x = y) \/ meet z x = meet z y) /\ (!x. meet x 0 = 0) /\
|
|
1185 |
(!x. join x 0 = x) /\ (!x. meet x 1 = x) /\ (!x. join x 1 = 1) /\
|
|
1186 |
(!x y z. ~(meet x y = x) \/ meet y (join x z) = join x (meet z y)) /\
|
|
1187 |
(!x y. ~complement x y \/ meet x y = 0) /\
|
|
1188 |
(!x y. ~complement x y \/ join x y = 1) /\
|
|
1189 |
(!x y. ~(meet x y = 0) \/ ~(join x y = 1) \/ complement x y) /\
|
|
1190 |
(!x y z. ~(x = y) \/ ~complement x z \/ complement y z) /\
|
|
1191 |
(!x y z. ~(x = y) \/ ~complement z x \/ complement z y) /\
|
|
1192 |
complement r1 (join a b) /\ complement r2 (meet a b) ==>
|
|
1193 |
~(r1 = meet (join r1 (meet r2 b)) (join r1 (meet r2 a))) ==> F`},
|
|
1194 |
|
|
1195 |
{name = "LCL009-1",
|
|
1196 |
comments = [],
|
|
1197 |
goal = `
|
|
1198 |
(!x y. ~p (x - y) \/ ~p x \/ p y) /\
|
|
1199 |
(!x y z. p (x - y - (z - y - (x - z)))) ==>
|
|
1200 |
~p (a - b - c - (a - (b - c))) ==> F`},
|
|
1201 |
|
|
1202 |
{name = "LCL087-1",
|
|
1203 |
comments =
|
|
1204 |
["Solved quickly by resolution when NOT tracking term-ordering constraints."],
|
|
1205 |
goal = `
|
|
1206 |
(!x y. ~p (implies x y) \/ ~p x \/ p y) /\
|
|
1207 |
(!x y z v w.
|
|
1208 |
p
|
|
1209 |
(implies (implies (implies x y) (implies z v))
|
|
1210 |
(implies w (implies (implies v x) (implies z x))))) ==>
|
|
1211 |
~p (implies a (implies b a)) ==> F`},
|
|
1212 |
|
|
1213 |
{name = "LCL107-1",
|
|
1214 |
comments = ["A very small problem that's tricky to prove."],
|
|
1215 |
goal = `
|
|
1216 |
(!x y. ~p (x - y) \/ ~p x \/ p y) /\
|
|
1217 |
(!x y z v w x' y'.
|
|
1218 |
p
|
|
1219 |
(x - y - z - (v - w - (x' - w - (x' - v) - y')) -
|
|
1220 |
(z - (y - x - y')))) ==> ~p (a - b - c - (e - b - (a - e - c))) ==> F`},
|
|
1221 |
|
|
1222 |
{name = "LCL187-1",
|
|
1223 |
comments = [],
|
|
1224 |
goal = `
|
|
1225 |
(!x. axiom (or (not (or x x)) x)) /\ (!x y. axiom (or (not x) (or y x))) /\
|
|
1226 |
(!x y. axiom (or (not (or x y)) (or y x))) /\
|
|
1227 |
(!x y z. axiom (or (not (or x (or y z))) (or y (or x z)))) /\
|
|
1228 |
(!x y z. axiom (or (not (or (not x) y)) (or (not (or z x)) (or z y)))) /\
|
|
1229 |
(!x. theorem x \/ ~axiom x) /\
|
|
1230 |
(!x y. theorem x \/ ~axiom (or (not y) x) \/ ~theorem y) /\
|
|
1231 |
(!x y z.
|
|
1232 |
theorem (or (not x) y) \/ ~axiom (or (not x) z) \/
|
|
1233 |
~theorem (or (not z) y)) ==>
|
|
1234 |
~theorem (or (not p) (or (not (not p)) q)) ==> F`},
|
|
1235 |
|
|
1236 |
{name = "LDA007-3",
|
|
1237 |
comments = [],
|
|
1238 |
goal = `
|
|
1239 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1240 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\
|
|
1241 |
(!x y z. f x (f y z) = f (f x y) (f x z)) /\
|
|
1242 |
(!x y z. ~(x = y) \/ f x z = f y z) /\
|
|
1243 |
(!x y z. ~(x = y) \/ f z x = f z y) /\ tt = f t t /\ ts = f t s /\
|
|
1244 |
tt_ts = f tt ts /\ tk = f t k /\ tsk = f ts k ==>
|
|
1245 |
~(f t tsk = f tt_ts tk) ==> F`},
|
|
1246 |
|
|
1247 |
{name = "NUM001-1",
|
|
1248 |
comments = [],
|
|
1249 |
goal = `
|
|
1250 |
(!x. x == x) /\ (!x y z. ~(x == y) \/ ~(y == z) \/ x == z) /\
|
|
1251 |
(!x y. x + y == y + x) /\ (!x y z. x + (y + z) == x + y + z) /\
|
|
1252 |
(!x y. x + y - y == x) /\ (!x y. x == x + y - y) /\
|
|
1253 |
(!x y z. x - y + z == x + z - y) /\ (!x y z. x + y - z == x - z + y) /\
|
|
1254 |
(!x y z v. ~(x == y) \/ ~(z == x + v) \/ z == y + v) /\
|
|
1255 |
(!x y z v. ~(x == y) \/ ~(z == v + x) \/ z == v + y) /\
|
|
1256 |
(!x y z v. ~(x == y) \/ ~(z == x - v) \/ z == y - v) /\
|
|
1257 |
(!x y z v. ~(x == y) \/ ~(z == v - x) \/ z == v - y) ==>
|
|
1258 |
~(a + b + c == a + (b + c)) ==> F`},
|
|
1259 |
|
|
1260 |
{name = "NUM014-1",
|
|
1261 |
comments = [],
|
|
1262 |
goal = `
|
|
1263 |
(!x. product x x (square x)) /\
|
|
1264 |
(!x y z. ~product x y z \/ product y x z) /\
|
|
1265 |
(!x y z. ~product x y z \/ divides x z) /\
|
|
1266 |
(!x y z v.
|
|
1267 |
~prime x \/ ~product y z v \/ ~divides x v \/ divides x y \/
|
|
1268 |
divides x z) /\ prime a /\ product a (square c) (square b) ==>
|
|
1269 |
~divides a b ==> F`},
|
|
1270 |
|
|
1271 |
{name = "PUZ001-1",
|
|
1272 |
comments = [],
|
|
1273 |
goal = `
|
|
1274 |
lives agatha /\ lives butler /\ lives charles /\
|
|
1275 |
(!x y. ~killed x y \/ ~richer x y) /\
|
|
1276 |
(!x. ~hates agatha x \/ ~hates charles x) /\
|
|
1277 |
(!x. ~hates x agatha \/ ~hates x butler \/ ~hates x charles) /\
|
|
1278 |
hates agatha agatha /\ hates agatha charles /\
|
|
1279 |
(!x y. ~killed x y \/ hates x y) /\
|
|
1280 |
(!x. ~hates agatha x \/ hates butler x) /\
|
|
1281 |
(!x. ~lives x \/ richer x agatha \/ hates butler x) ==>
|
|
1282 |
killed butler agatha \/ killed charles agatha ==> F`},
|
|
1283 |
|
|
1284 |
{name = "PUZ011-1",
|
|
1285 |
comments =
|
|
1286 |
["Curiosity: solved trivially by meson without cache cutting, but not with."],
|
|
1287 |
goal = `
|
|
1288 |
ocean atlantic /\ ocean indian /\ borders atlantic brazil /\
|
|
1289 |
borders atlantic uruguay /\ borders atlantic venesuela /\
|
|
1290 |
borders atlantic zaire /\ borders atlantic nigeria /\
|
|
1291 |
borders atlantic angola /\ borders indian india /\
|
|
1292 |
borders indian pakistan /\ borders indian iran /\ borders indian somalia /\
|
|
1293 |
borders indian kenya /\ borders indian tanzania /\ south_american brazil /\
|
|
1294 |
south_american uruguay /\ south_american venesuela /\ african zaire /\
|
|
1295 |
african nigeria /\ african angola /\ african somalia /\ african kenya /\
|
|
1296 |
african tanzania /\ asian india /\ asian pakistan /\ asian iran ==>
|
|
1297 |
(!x y z.
|
|
1298 |
~ocean x \/ ~borders x y \/ ~african y \/ ~borders x z \/ ~asian z) ==>
|
|
1299 |
F`},
|
|
1300 |
|
|
1301 |
{name = "PUZ020-1",
|
|
1302 |
comments = [],
|
|
1303 |
goal = `
|
|
1304 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1305 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\
|
|
1306 |
(!x y. ~(x = y) \/ statement_by x = statement_by y) /\
|
|
1307 |
(!x. ~person x \/ knight x \/ knave x) /\
|
|
1308 |
(!x. ~person x \/ ~knight x \/ ~knave x) /\
|
|
1309 |
(!x y. ~says x y \/ a_truth y \/ ~a_truth y) /\
|
|
1310 |
(!x y. ~says x y \/ ~(x = y)) /\ (!x y. ~says x y \/ y = statement_by x) /\
|
|
1311 |
(!x y. ~person x \/ ~(x = statement_by y)) /\
|
|
1312 |
(!x. ~person x \/ ~a_truth (statement_by x) \/ knight x) /\
|
|
1313 |
(!x. ~person x \/ a_truth (statement_by x) \/ knave x) /\
|
|
1314 |
(!x y. ~(x = y) \/ ~knight x \/ knight y) /\
|
|
1315 |
(!x y. ~(x = y) \/ ~knave x \/ knave y) /\
|
|
1316 |
(!x y. ~(x = y) \/ ~person x \/ person y) /\
|
|
1317 |
(!x y z. ~(x = y) \/ ~says x z \/ says y z) /\
|
|
1318 |
(!x y z. ~(x = y) \/ ~says z x \/ says z y) /\
|
|
1319 |
(!x y. ~(x = y) \/ ~a_truth x \/ a_truth y) /\
|
|
1320 |
(!x y. ~knight x \/ ~says x y \/ a_truth y) /\
|
|
1321 |
(!x y. ~knave x \/ ~says x y \/ ~a_truth y) /\ person husband /\
|
|
1322 |
person wife /\ ~(husband = wife) /\ says husband (statement_by husband) /\
|
|
1323 |
(~a_truth (statement_by husband) \/ ~knight husband \/ knight wife) /\
|
|
1324 |
(a_truth (statement_by husband) \/ ~knight husband) /\
|
|
1325 |
(a_truth (statement_by husband) \/ knight wife) /\
|
|
1326 |
(~knight wife \/ a_truth (statement_by husband)) ==> ~knight husband ==> F`},
|
|
1327 |
|
|
1328 |
{name = "ROB002-1",
|
|
1329 |
comments = [],
|
|
1330 |
goal = `
|
|
1331 |
(!x. x = x) /\ (!x y. ~(x = y) \/ y = x) /\
|
|
1332 |
(!x y z. ~(x = y) \/ ~(y = z) \/ x = z) /\ (!x y. x + y = y + x) /\
|
|
1333 |
(!x y z. x + y + z = x + (y + z)) /\
|
|
1334 |
(!x y. negate (negate (x + y) + negate (x + negate y)) = x) /\
|
|
1335 |
(!x y z. ~(x = y) \/ x + z = y + z) /\
|
|
1336 |
(!x y z. ~(x = y) \/ z + x = z + y) /\
|
|
1337 |
(!x y. ~(x = y) \/ negate x = negate y) /\ (!x. negate (negate x) = x) ==>
|
|
1338 |
~(negate (a + negate b) + negate (negate a + negate b) = b) ==> F`}
|
|
1339 |
|
|
1340 |
] @
|
|
1341 |
|
|
1342 |
(* ========================================================================= *)
|
|
1343 |
(* Some problems from HOL. *)
|
|
1344 |
(* ========================================================================= *)
|
|
1345 |
|
|
1346 |
mkProblems "hol" "HOL subgoals sent to MESON_TAC" [
|
|
1347 |
|
|
1348 |
{name = "Coder_4_0",
|
|
1349 |
comments = [],
|
|
1350 |
goal = `
|
|
1351 |
(!x y.
|
|
1352 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1353 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1354 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1355 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\ ~{existential . (K . falsity)} /\
|
|
1356 |
{existential . (K . truth)} /\ ~{universal . (K . falsity)} /\
|
|
1357 |
{universal . (K . truth)} /\ ~{falsity} /\ {truth} /\
|
|
1358 |
(!x y z. ~(APPEND . x . y = APPEND . z . y) \/ x = z) /\
|
|
1359 |
(!x y z. ~(APPEND . x . y = APPEND . x . z) \/ y = z) ==>
|
|
1360 |
{wf_encoder . p . e} /\ (!x. e . x = f . x \/ ~{p . x}) /\
|
|
1361 |
{wf_encoder . p . f} /\ {p . q} /\ {p . q'} /\
|
|
1362 |
APPEND . (f . q) . r = APPEND . (f . q') . r' /\ q = q' /\ ~(r' = r) ==> F`},
|
|
1363 |
|
|
1364 |
{name = "DeepSyntax_47",
|
|
1365 |
comments = [],
|
|
1366 |
goal = `
|
|
1367 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\
|
|
1368 |
(!x y. ~(x = y) \/ int_neg x = int_neg y) /\
|
|
1369 |
(!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\
|
|
1370 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\
|
|
1371 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ eval_form y v \/ ~eval_form x z) /\
|
|
1372 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1373 |
(!x y z v.
|
|
1374 |
int_lt (int_add x y) (int_add z v) \/ ~int_lt x z \/ ~int_lt y v) /\
|
|
1375 |
(!x. int_add x (int_of_num 0) = x) /\
|
|
1376 |
(!x. int_add x (int_neg x) = int_of_num 0) /\
|
|
1377 |
(!x y z. int_add x (int_add y z) = int_add (int_add x y) z) ==>
|
|
1378 |
int_lt (int_neg d) (int_of_num 0) /\ eval_form g x /\
|
|
1379 |
int_lt (int_add x d) i /\ ~int_lt x i ==> F`},
|
|
1380 |
|
|
1381 |
{name = "divides_9",
|
|
1382 |
comments = [],
|
|
1383 |
goal = `
|
|
1384 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1385 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\
|
|
1386 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1387 |
(!x y z. x * (y * z) = x * y * z) /\ (!x y. x * y = y * x) /\
|
|
1388 |
(!x y. ~divides x y \/ y = gv1556 x y * x) /\
|
|
1389 |
(!x y z. divides x y \/ ~(y = z * x)) ==>
|
|
1390 |
divides gv1546 gv1547 /\ ~divides gv1546 (gv1547 * gv1548) ==> F`},
|
|
1391 |
|
|
1392 |
{name = "Encode_28",
|
|
1393 |
comments = [],
|
|
1394 |
goal = `
|
|
1395 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ MOD x z = MOD y v) /\
|
|
1396 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1397 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\
|
|
1398 |
(!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\
|
|
1399 |
(!x y. ~(x = y) \/ BIT2 x = BIT2 y) /\
|
|
1400 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ EXP x z = EXP y v) /\
|
|
1401 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ DIV x z = DIV y v) /\
|
|
1402 |
(!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\
|
|
1403 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\
|
|
1404 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1405 |
(!x y. x * y = y * x) /\
|
|
1406 |
(!x y z. MOD (MOD x (y * z)) y = MOD x y \/ ~(0 < y) \/ ~(0 < z)) ==>
|
|
1407 |
(!x.
|
|
1408 |
MOD x (NUMERAL (BIT2 ZERO)) = 0 \/
|
|
1409 |
MOD x (NUMERAL (BIT2 ZERO)) = NUMERAL (BIT1 ZERO)) /\
|
|
1410 |
MOD
|
|
1411 |
(DIV x (NUMERAL (BIT2 ZERO)) * NUMERAL (BIT2 ZERO) +
|
|
1412 |
MOD x (NUMERAL (BIT2 ZERO)))
|
|
1413 |
(NUMERAL (BIT2 ZERO) * EXP (NUMERAL (BIT2 ZERO)) m) =
|
|
1414 |
MOD
|
|
1415 |
(DIV y (NUMERAL (BIT2 ZERO)) * NUMERAL (BIT2 ZERO) +
|
|
1416 |
MOD y (NUMERAL (BIT2 ZERO)))
|
|
1417 |
(NUMERAL (BIT2 ZERO) * EXP (NUMERAL (BIT2 ZERO)) m) /\
|
|
1418 |
0 < EXP (NUMERAL (BIT2 ZERO)) m /\ 0 < NUMERAL (BIT2 ZERO) /\
|
|
1419 |
(!x y.
|
|
1420 |
~(MOD (x * NUMERAL (BIT2 ZERO) + MOD (x) (NUMERAL (BIT2 ZERO)))
|
|
1421 |
(NUMERAL (BIT2 ZERO)) =
|
|
1422 |
MOD (y * NUMERAL (BIT2 ZERO) + MOD (y) (NUMERAL (BIT2 ZERO)))
|
|
1423 |
(NUMERAL (BIT2 ZERO)))) ==> F`},
|
|
1424 |
|
|
1425 |
{name = "euclid_4",
|
|
1426 |
comments = [],
|
|
1427 |
goal = `
|
|
1428 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1429 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\
|
|
1430 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1431 |
(!x y z. x * (y * z) = x * y * z) /\
|
|
1432 |
(!x y. ~divides x y \/ y = x * gv5371 x y) /\
|
|
1433 |
(!x y z. divides x y \/ ~(y = x * z)) ==>
|
|
1434 |
divides gv5316 gv5317 /\ divides gv5315 gv5316 /\
|
|
1435 |
~divides gv5315 gv5317 ==> F`},
|
|
1436 |
|
|
1437 |
{name = "euclid_8",
|
|
1438 |
comments = [],
|
|
1439 |
goal = `
|
|
1440 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1441 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\
|
|
1442 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1443 |
(!x y. x * y = y * x) /\ (!x y z. x * (y * z) = x * y * z) /\
|
|
1444 |
(!x y. ~divides x y \/ y = x * gv7050 x y) /\
|
|
1445 |
(!x y z. divides x y \/ ~(y = x * z)) ==>
|
|
1446 |
divides gv7000 gv7001 /\ ~divides gv7000 (gv7002 * gv7001) ==> F`},
|
|
1447 |
|
|
1448 |
{name = "extra_arith_6",
|
|
1449 |
comments = [],
|
|
1450 |
goal = `
|
|
1451 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1452 |
(!x y. ~(x = y) \/ SUC x = SUC y) /\
|
|
1453 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\
|
|
1454 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1455 |
(!x y z. ~(SUC x * y = SUC x * z) \/ y = z) /\
|
|
1456 |
(!x y z. SUC x * y = SUC x * z \/ ~(y = z)) /\
|
|
1457 |
(!x y z. x * (y * z) = x * y * z) /\ (!x y. x * y = y * x) ==>
|
|
1458 |
SUC n * b = q * (SUC n * a) /\ 0 < SUC n /\ ~(b = q * a) ==> F`},
|
|
1459 |
|
|
1460 |
{name = "extra_real_5",
|
|
1461 |
comments = [],
|
|
1462 |
goal = `
|
|
1463 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1464 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1465 |
{truth} ==>
|
|
1466 |
(!x y z v.
|
|
1467 |
{real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/
|
|
1468 |
~{P . z} \/ ~{real_lte . (gv6327 . v) . v}) /\
|
|
1469 |
(!x y z.
|
|
1470 |
~{real_lt . x . (sup . P)} \/ {P . (gv6327 . x)} \/ ~{P . y} \/
|
|
1471 |
~{real_lte . (gv6327 . z) . z}) /\
|
|
1472 |
(!x y z.
|
|
1473 |
~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv6327 . x)} \/
|
|
1474 |
~{P . y} \/ ~{real_lte . (gv6327 . z) . z}) /\
|
|
1475 |
(!x y z.
|
|
1476 |
~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv6327 . x)} \/
|
|
1477 |
~{P . y} \/ {P . (gv6327 . z)}) /\
|
|
1478 |
(!x y z.
|
|
1479 |
~{real_lt . x . (sup . P)} \/ {P . (gv6327 . x)} \/ ~{P . y} \/
|
|
1480 |
{P . (gv6327 . z)}) /\
|
|
1481 |
(!x y z v.
|
|
1482 |
{real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/
|
|
1483 |
~{P . z} \/ {P . (gv6327 . v)}) /\ {P . x} /\
|
|
1484 |
(!x. {real_lte . x . z} \/ ~{P . x}) /\
|
|
1485 |
(!x.
|
|
1486 |
{real_lt . (gv6328 . x) . (gv6329 . x)} \/
|
|
1487 |
{real_lt . (gv6328 . x) . x}) /\
|
|
1488 |
(!x. {P . (gv6329 . x)} \/ {real_lt . (gv6328 . x) . x}) /\
|
|
1489 |
(!x y.
|
|
1490 |
~{real_lt . (gv6328 . x) . y} \/ ~{P . y} \/
|
|
1491 |
~{real_lt . (gv6328 . x) . x}) ==> F`},
|
|
1492 |
|
|
1493 |
{name = "gcd_19",
|
|
1494 |
comments = [],
|
|
1495 |
goal = `
|
|
1496 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\
|
|
1497 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1498 |
(!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\
|
|
1499 |
(!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\
|
|
1500 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\
|
|
1501 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\
|
|
1502 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1503 |
(!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\
|
|
1504 |
(!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\
|
|
1505 |
(!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\
|
|
1506 |
(!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\
|
|
1507 |
(!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
1508 |
(!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==>
|
|
1509 |
~(x + z <= x) /\ divides c (d * SUC x) /\ divides c (d * SUC (x + z)) /\
|
|
1510 |
~divides c (d * z) ==> F`},
|
|
1511 |
|
|
1512 |
{name = "gcd_20",
|
|
1513 |
comments = [],
|
|
1514 |
goal = `
|
|
1515 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\
|
|
1516 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1517 |
(!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\
|
|
1518 |
(!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\
|
|
1519 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\
|
|
1520 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\
|
|
1521 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1522 |
(!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\
|
|
1523 |
(!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\
|
|
1524 |
(!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\
|
|
1525 |
(!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\
|
|
1526 |
(!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
1527 |
(!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==>
|
|
1528 |
y <= y + z /\ divides c (d * SUC (y + z)) /\ divides c (d * SUC y) /\
|
|
1529 |
~divides c (d * z) ==> F`},
|
|
1530 |
|
|
1531 |
{name = "gcd_21",
|
|
1532 |
comments = [],
|
|
1533 |
goal = `
|
|
1534 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x + z = y + v) /\
|
|
1535 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1536 |
(!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\
|
|
1537 |
(!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\ (!x y. ~(x = y) \/ SUC x = SUC y) /\
|
|
1538 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ divides y v \/ ~divides x z) /\
|
|
1539 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y <= v \/ ~(x <= z)) /\
|
|
1540 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1541 |
(!x y z. x * (y + z) = x * y + x * z) /\ (!x y. x + SUC y = SUC (x + y)) /\
|
|
1542 |
(!x y. SUC x + y = SUC (x + y)) /\ (!x. x + 0 = x) /\ (!x. 0 + x = x) /\
|
|
1543 |
(!x y. x * SUC y = x + x * y) /\ (!x y. SUC x * y = x * y + y) /\
|
|
1544 |
(!x. x * NUMERAL (BIT1 ZERO) = x) /\ (!x. NUMERAL (BIT1 ZERO) * x = x) /\
|
|
1545 |
(!x. x * 0 = 0) /\ (!x. 0 * x = 0) /\ (!x y z. x + (y + z) = x + y + z) /\
|
|
1546 |
(!x y z. divides x y \/ ~divides x z \/ ~divides x (z + y)) ==>
|
|
1547 |
divides c (d * SUC y) /\ y <= y + z /\ divides c (d * SUC (y + z)) /\
|
|
1548 |
~divides c (d * z) ==> F`},
|
|
1549 |
|
|
1550 |
{name = "int_arith_6",
|
|
1551 |
comments = [],
|
|
1552 |
goal = `
|
|
1553 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\
|
|
1554 |
(!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\
|
|
1555 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\
|
|
1556 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1557 |
(!x y. int_mul x y = int_mul y x) /\ (!x. ~int_lt x x) /\
|
|
1558 |
(!x y z. ~(int_mul x y = int_mul z y) \/ y = int_of_num 0 \/ x = z) /\
|
|
1559 |
(!x y z. int_mul x y = int_mul z y \/ ~(y = int_of_num 0)) /\
|
|
1560 |
(!x y z. int_mul x y = int_mul z y \/ ~(x = z)) ==>
|
|
1561 |
int_lt (int_of_num 0) gv1085 /\
|
|
1562 |
int_mul gv1085 gv1086 = int_mul gv1085 gv1087 /\ ~(gv1086 = gv1087) ==> F`},
|
|
1563 |
|
|
1564 |
{name = "int_arith_139",
|
|
1565 |
comments = [],
|
|
1566 |
goal = `
|
|
1567 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\
|
|
1568 |
(!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\
|
|
1569 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\
|
|
1570 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1571 |
(!x. int_add (int_of_num 0) x = x) /\
|
|
1572 |
(!x y z v.
|
|
1573 |
int_le (int_add x y) (int_add z v) \/ ~int_le x z \/ ~int_le y v) /\
|
|
1574 |
(!x y z. int_add x (int_add y z) = int_add (int_add x y) z) /\
|
|
1575 |
(!x y. int_add x y = int_add y x) /\
|
|
1576 |
(!x y z. ~int_le (int_add x y) (int_add x z) \/ int_le y z) /\
|
|
1577 |
(!x y z. int_le (int_add x y) (int_add x z) \/ ~int_le y z) ==>
|
|
1578 |
int_le x y /\ int_le (int_of_num 0) (int_add c x) /\
|
|
1579 |
~int_le (int_of_num 0) (int_add c y) ==> F`},
|
|
1580 |
|
|
1581 |
{name = "llist_69",
|
|
1582 |
comments = [],
|
|
1583 |
goal = `
|
|
1584 |
(!x y. ~(x = y) \/ LTL x = LTL y) /\ (!x y. ~(x = y) \/ SOME x = SOME y) /\
|
|
1585 |
(!x y. ~(x = y) \/ LHD x = LHD y) /\
|
|
1586 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ LCONS x z = LCONS y v) /\
|
|
1587 |
(!x y. ~(x = y) \/ g x = g y) /\ (!x y. ~(x = y) \/ THE x = THE y) /\
|
|
1588 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ LNTH z x = LNTH v y) /\
|
|
1589 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ LDROP z x = LDROP v y) /\
|
|
1590 |
(!x y. ~(x = y) \/ SUC x = SUC y) /\
|
|
1591 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1592 |
(!x y z. ~(x = LCONS y z) \/ LTL x = SOME z) /\
|
|
1593 |
(!x y z. ~(x = LCONS y z) \/ LHD x = SOME y) /\
|
|
1594 |
(!x y z. x = LCONS y z \/ ~(LHD x = SOME y) \/ ~(LTL x = SOME z)) ==>
|
|
1595 |
LTL (g (LCONS LNIL t)) =
|
|
1596 |
SOME (g (LCONS (THE (LTL (THE (LNTH n t)))) (THE (LDROP (SUC n) t)))) /\
|
|
1597 |
LHD (g (LCONS LNIL t)) = SOME (THE (LHD (THE (LNTH n t)))) /\
|
|
1598 |
LHD (g t) = SOME (THE (LHD (THE (LNTH n t)))) /\
|
|
1599 |
LTL (g t) =
|
|
1600 |
SOME (g (LCONS (THE (LTL (THE (LNTH n t)))) (THE (LDROP (SUC n) t)))) /\
|
|
1601 |
~(g (LCONS LNIL t) = g t) ==> F`},
|
|
1602 |
|
|
1603 |
{name = "MachineTransition_0",
|
|
1604 |
comments = [],
|
|
1605 |
goal = `
|
|
1606 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1607 |
(!x y.
|
|
1608 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1609 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1610 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1611 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\
|
|
1612 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\
|
|
1613 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1614 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1615 |
{truth} /\ Eq = equality /\
|
|
1616 |
(!x y z. ~{Next . x . y . z} \/ {x . (pair . (gv940 . x . y . z) . z)}) /\
|
|
1617 |
(!x y z. ~{Next . x . y . z} \/ {y . (gv940 . x . y . z)}) /\
|
|
1618 |
(!x y z v. {Next . x . y . z} \/ ~{y . v} \/ ~{x . (pair . v . z)}) /\
|
|
1619 |
(!x y z. ~{Prev . x . y . z} \/ {y . (gv935 . x . y . z)}) /\
|
|
1620 |
(!x y z. ~{Prev . x . y . z} \/ {x . (pair . z . (gv935 . x . y . z))}) /\
|
|
1621 |
(!x y z v. {Prev . x . y . z} \/ ~{x . (pair . z . v)} \/ ~{y . v}) ==>
|
|
1622 |
{Next . gv914 . (Eq . gv915) . gv916} /\
|
|
1623 |
~{Prev . gv914 . (Eq . gv916) . gv915} ==> F`},
|
|
1624 |
|
|
1625 |
{name = "MachineTransition_2_1",
|
|
1626 |
comments = [],
|
|
1627 |
goal = `
|
|
1628 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1629 |
(!x y.
|
|
1630 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1631 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1632 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1633 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\
|
|
1634 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\
|
|
1635 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1636 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1637 |
{truth} /\
|
|
1638 |
(!x y z. ReachIn . x . (Next . x . y) . z = ReachIn . x . y . (SUC . z)) /\
|
|
1639 |
(!x y z. ~{Next . x . y . z} \/ {x . (pair . (gv5488 . x . y . z) . z)}) /\
|
|
1640 |
(!x y z. ~{Next . x . y . z} \/ {y . (gv5488 . x . y . z)}) /\
|
|
1641 |
(!x y z v. {Next . x . y . z} \/ ~{y . v} \/ ~{x . (pair . v . z)}) /\
|
|
1642 |
(!x y z. ReachIn . x . y . (SUC . z) = Next . x . (ReachIn . x . y . z)) /\
|
|
1643 |
(!x y. ReachIn . x . y . 0 = y) ==>
|
|
1644 |
{ReachIn . R . (Next . R . B) . gv5278 . state} /\
|
|
1645 |
(!x. ~{ReachIn . R . B . gv5278 . x} \/ ~{R . (pair . x . state)}) ==> F`},
|
|
1646 |
|
|
1647 |
{name = "MachineTransition_52",
|
|
1648 |
comments = [],
|
|
1649 |
goal = `
|
|
1650 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1651 |
(!x y.
|
|
1652 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1653 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1654 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1655 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\
|
|
1656 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\
|
|
1657 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1658 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1659 |
{truth} /\
|
|
1660 |
(!x y z.
|
|
1661 |
{(<=) . (gv7028 . x . y . z) . (add . x . (NUMERAL . (BIT1 . ZERO)))} \/
|
|
1662 |
z . (add . x . (NUMERAL . (BIT1 . ZERO))) =
|
|
1663 |
y . (add . x . (NUMERAL . (BIT1 . ZERO)))) /\
|
|
1664 |
(!x y z.
|
|
1665 |
~(x . (gv7028 . y . z . x) = z . (gv7028 . y . z . x)) \/
|
|
1666 |
x . (add . y . (NUMERAL . (BIT1 . ZERO))) =
|
|
1667 |
z . (add . y . (NUMERAL . (BIT1 . ZERO)))) /\
|
|
1668 |
(!x y z v.
|
|
1669 |
~(x . (gv7028 . y . z . x) = z . (gv7028 . y . z . x)) \/
|
|
1670 |
x . v = z . v \/ ~{(<=) . v . y}) /\
|
|
1671 |
(!x y z v.
|
|
1672 |
{(<=) . (gv7028 . x . y . z) . (add . x . (NUMERAL . (BIT1 . ZERO)))} \/
|
|
1673 |
z . v = y . v \/ ~{(<=) . v . x}) /\
|
|
1674 |
(!x y z v.
|
|
1675 |
~{(<=) . x . (add . y . (NUMERAL . (BIT1 . ZERO)))} \/ z . x = v . x \/
|
|
1676 |
~(z . (gv7027 . y . v . z) = v . (gv7027 . y . v . z)) \/
|
|
1677 |
~(z . (add . y . (NUMERAL . (BIT1 . ZERO))) =
|
|
1678 |
v . (add . y . (NUMERAL . (BIT1 . ZERO))))) /\
|
|
1679 |
(!x y z v.
|
|
1680 |
~{(<=) . x . (add . y . (NUMERAL . (BIT1 . ZERO)))} \/ z . x = v . x \/
|
|
1681 |
{(<=) . (gv7027 . y . v . z) . y} \/
|
|
1682 |
~(z . (add . y . (NUMERAL . (BIT1 . ZERO))) =
|
|
1683 |
v . (add . y . (NUMERAL . (BIT1 . ZERO))))) ==>
|
|
1684 |
({FinPath . (pair . R . s) . f2 . n} \/
|
|
1685 |
~{FinPath . (pair . R . s) . f1 . n} \/ ~(f1 . gv7034 = f2 . gv7034)) /\
|
|
1686 |
(~{FinPath . (pair . R . s) . f2 . n} \/
|
|
1687 |
{FinPath . (pair . R . s) . f1 . n} \/ ~(f1 . gv7034 = f2 . gv7034)) /\
|
|
1688 |
(~{FinPath . (pair . R . s) . f2 . n} \/
|
|
1689 |
{FinPath . (pair . R . s) . f1 . n} \/ {(<=) . gv7034 . n}) /\
|
|
1690 |
({FinPath . (pair . R . s) . f2 . n} \/
|
|
1691 |
~{FinPath . (pair . R . s) . f1 . n} \/ {(<=) . gv7034 . n}) /\
|
|
1692 |
(!x.
|
|
1693 |
f1 . x = f2 . x \/
|
|
1694 |
~{(<=) . x . (add . n . (NUMERAL . (BIT1 . ZERO)))}) /\
|
|
1695 |
{FinPath . (pair . R . s) . f2 . n} /\
|
|
1696 |
{R . (pair . (f2 . n) . (f2 . (add . n . (NUMERAL . (BIT1 . ZERO)))))} /\
|
|
1697 |
~{FinPath . (pair . R . s) . f1 . n} ==> F`},
|
|
1698 |
|
|
1699 |
{name = "measure_138",
|
|
1700 |
comments = [],
|
|
1701 |
goal = `
|
|
1702 |
(!x y z. ~SUBSET x y \/ IN z y \/ ~IN z x) /\
|
|
1703 |
(!x y. SUBSET x y \/ IN (gv122874 x y) x) /\
|
|
1704 |
(!x y. SUBSET x y \/ ~IN (gv122874 x y) y) /\
|
|
1705 |
(!x. sigma_algebra (sigma x)) /\ (!x y z. ~IN x (INTER y z) \/ IN x z) /\
|
|
1706 |
(!x y z. ~IN x (INTER y z) \/ IN x y) /\
|
|
1707 |
(!x y z. IN x (INTER y z) \/ ~IN x y \/ ~IN x z) /\
|
|
1708 |
(!x y.
|
|
1709 |
~sigma_algebra x \/ IN (BIGUNION y) x \/ ~countable y \/ ~SUBSET y x) /\
|
|
1710 |
(!x y. ~sigma_algebra x \/ IN (COMPL y) x \/ ~IN y x) /\
|
|
1711 |
(!x. ~sigma_algebra x \/ IN EMPTY x) /\
|
|
1712 |
(!x.
|
|
1713 |
sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/
|
|
1714 |
SUBSET (gv122852 x) x) /\
|
|
1715 |
(!x.
|
|
1716 |
sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/
|
|
1717 |
SUBSET (gv122852 x) x) /\
|
|
1718 |
(!x.
|
|
1719 |
sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/
|
|
1720 |
countable (gv122852 x)) /\
|
|
1721 |
(!x.
|
|
1722 |
sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/
|
|
1723 |
countable (gv122852 x)) /\
|
|
1724 |
(!x.
|
|
1725 |
sigma_algebra x \/ ~IN EMPTY x \/ IN (gv122851 x) x \/
|
|
1726 |
~IN (BIGUNION (gv122852 x)) x) /\
|
|
1727 |
(!x.
|
|
1728 |
sigma_algebra x \/ ~IN EMPTY x \/ ~IN (COMPL (gv122851 x)) x \/
|
|
1729 |
~IN (BIGUNION (gv122852 x)) x) ==>
|
|
1730 |
SUBSET c (INTER p (sigma a)) /\
|
|
1731 |
(!x. IN (BIGUNION x) p \/ ~countable x \/ ~SUBSET x (INTER p (sigma a))) /\
|
|
1732 |
SUBSET a p /\ IN EMPTY p /\
|
|
1733 |
(!x. IN (COMPL x) p \/ ~IN x (INTER p (sigma a))) /\ countable c /\
|
|
1734 |
~IN (BIGUNION c) (sigma a) ==> F`},
|
|
1735 |
|
|
1736 |
{name = "Omega_13",
|
|
1737 |
comments = [],
|
|
1738 |
goal = `
|
|
1739 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\
|
|
1740 |
(!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\
|
|
1741 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ evalupper y v \/ ~evalupper x z) /\
|
|
1742 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\
|
|
1743 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\
|
|
1744 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\
|
|
1745 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1746 |
(!x y. ~int_le x y \/ int_lt x y \/ x = y) /\
|
|
1747 |
(!x y. int_le x y \/ ~int_lt x y) /\ (!x y. int_le x y \/ ~(x = y)) /\
|
|
1748 |
(!x y z.
|
|
1749 |
int_lt x y \/
|
|
1750 |
~int_lt (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/
|
|
1751 |
~(0 < z)) /\
|
|
1752 |
(!x y z.
|
|
1753 |
~int_lt x y \/
|
|
1754 |
int_lt (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/
|
|
1755 |
~(0 < z)) /\ (!x y z. int_lt x y \/ ~int_le x z \/ ~int_lt z y) ==>
|
|
1756 |
(!x y. evalupper x uppers \/ ~evalupper y uppers \/ ~int_lt x y) /\
|
|
1757 |
int_le (int_mul (int_of_num p_1) x) p_2 /\ evalupper x uppers /\
|
|
1758 |
int_lt y x /\ 0 < p_1 /\ ~int_le (int_mul (int_of_num p_1) y) p_2 ==> F`},
|
|
1759 |
|
|
1760 |
{name = "Omega_71",
|
|
1761 |
comments = [],
|
|
1762 |
goal = `
|
|
1763 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_mul x z = int_mul y v) /\
|
|
1764 |
(!x y. ~(x = y) \/ int_of_num x = int_of_num y) /\
|
|
1765 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1766 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_add x z = int_add y v) /\
|
|
1767 |
(!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\
|
|
1768 |
(!x y. ~(x = y) \/ BIT1 x = BIT1 y) /\
|
|
1769 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ (x, z) = (y, v)) /\
|
|
1770 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ evallower y v \/ ~evallower x z) /\
|
|
1771 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\
|
|
1772 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ rshadow_row v y \/ ~rshadow_row z x) /\
|
|
1773 |
(!x y z v.
|
|
1774 |
~(x = y) \/ ~(z = v) \/ dark_shadow_cond_row v y \/
|
|
1775 |
~dark_shadow_cond_row z x) /\
|
|
1776 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_le y v \/ ~int_le x z) /\
|
|
1777 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ EVERY y v \/ ~EVERY x z) /\
|
|
1778 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ int_lt y v \/ ~int_lt x z) /\
|
|
1779 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1780 |
(!x y. int_mul x y = int_mul y x) /\
|
|
1781 |
(!x y z. int_mul x (int_mul y z) = int_mul (int_mul x y) z) /\
|
|
1782 |
(!x y z.
|
|
1783 |
int_le x y \/
|
|
1784 |
~int_le (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/
|
|
1785 |
~(0 < z)) /\
|
|
1786 |
(!x y z.
|
|
1787 |
~int_le x y \/
|
|
1788 |
int_le (int_mul (int_of_num z) x) (int_mul (int_of_num z) y) \/
|
|
1789 |
~(0 < z)) ==>
|
|
1790 |
(!x y z.
|
|
1791 |
evallower (gv6249 x y z) lowers \/ ~(0 < y) \/ ~evallower x lowers \/
|
|
1792 |
~rshadow_row (y, z) lowers \/ ~dark_shadow_cond_row (y, z) lowers) /\
|
|
1793 |
(!x y z.
|
|
1794 |
int_le (int_mul (int_of_num x) (gv6249 y x z)) z \/ ~(0 < x) \/
|
|
1795 |
~evallower y lowers \/ ~rshadow_row (x, z) lowers \/
|
|
1796 |
~dark_shadow_cond_row (x, z) lowers) /\ 0 < c /\
|
|
1797 |
int_le R (int_mul (int_of_num d) x) /\ evallower x lowers /\ 0 < d /\
|
|
1798 |
EVERY fst_nzero lowers /\
|
|
1799 |
int_le (int_mul (int_of_num c) R) (int_mul (int_of_num d) L) /\
|
|
1800 |
rshadow_row (c, L) lowers /\ dark_shadow_cond_row (c, L) lowers /\
|
|
1801 |
(!x.
|
|
1802 |
~int_lt (int_mul (int_of_num d) L)
|
|
1803 |
(int_mul (int_of_num (c * d))
|
|
1804 |
(int_add x (int_of_num (NUMERAL (BIT1 ZERO))))) \/
|
|
1805 |
~int_lt (int_mul (int_of_num (c * d)) x) (int_mul (int_of_num c) R)) /\
|
|
1806 |
int_le (int_mul (int_of_num c) y) L /\ evallower y lowers /\
|
|
1807 |
int_le (int_mul (int_of_num (c * d)) y) (int_mul (int_of_num d) L) /\
|
|
1808 |
int_le (int_mul (int_of_num c) R) (int_mul (int_of_num (c * d)) x) /\
|
|
1809 |
int_lt (int_mul (int_of_num (c * d)) y) (int_mul (int_of_num c) R) /\
|
|
1810 |
0 < c * d /\
|
|
1811 |
int_le (int_mul (int_of_num c) R) (int_mul (int_of_num (c * d)) j) /\
|
|
1812 |
int_le (int_mul (int_of_num (c * d)) j) (int_mul (int_of_num d) L) /\
|
|
1813 |
int_le (int_mul (int_mul (int_of_num c) (int_of_num d)) j)
|
|
1814 |
(int_mul (int_of_num d) L) /\ ~int_le (int_mul (int_of_num c) j) L ==> F`},
|
|
1815 |
|
|
1816 |
{name = "pred_set_1",
|
|
1817 |
comments = ["Small problem that's hard for ordered resolution"],
|
|
1818 |
goal = `
|
|
1819 |
(!x y z. ~(x <= y) \/ p z y \/ ~p z x) /\ (!x y. x <= y \/ ~p (a x y) y) /\
|
|
1820 |
(!x y. x <= y \/ p (a x y) x) /\ (!x y z. ~p x (y * z) \/ p x z) /\
|
|
1821 |
(!x y z. ~p x (y * z) \/ p x y) /\
|
|
1822 |
(!x y z. p x (y * z) \/ ~p x y \/ ~p x z) ==>
|
|
1823 |
b <= c /\ b <= d /\ ~(b <= c * d) ==> F`},
|
|
1824 |
|
|
1825 |
{name = "pred_set_54_1",
|
|
1826 |
comments = [],
|
|
1827 |
goal = `
|
|
1828 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1829 |
(!x y.
|
|
1830 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1831 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1832 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1833 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\
|
|
1834 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\
|
|
1835 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1836 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1837 |
{truth} /\
|
|
1838 |
(!x y z. ~{IN . x . (INSERT . y . z)} \/ x = y \/ {IN . x . z}) /\
|
|
1839 |
(!x y z. {IN . x . (INSERT . y . z)} \/ ~(x = y)) /\
|
|
1840 |
(!x y z. {IN . x . (INSERT . y . z)} \/ ~{IN . x . z}) ==>
|
|
1841 |
(!x y z. f . x . (f . y . z) = f . y . (f . x . z)) /\
|
|
1842 |
(!x y z.
|
|
1843 |
ITSET . f . (INSERT . x . y) . z =
|
|
1844 |
ITSET . f . (DELETE . y . x) . (f . x . z) \/
|
|
1845 |
~{less_than . (CARD . y) . v} \/ ~{FINITE . y}) /\ v = CARD . s /\
|
|
1846 |
{FINITE . s} /\ REST . (INSERT . x . s) = t /\
|
|
1847 |
CHOICE . (INSERT . x . s) = y /\ ~{IN . y . t} /\ ~{IN . x . s} /\
|
|
1848 |
INSERT . x . s = INSERT . y . t /\ ~(x = y) /\ ~{IN . x . t} ==> F`},
|
|
1849 |
|
|
1850 |
{name = "prob_44",
|
|
1851 |
comments = [],
|
|
1852 |
goal = `
|
|
1853 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1854 |
(!x y.
|
|
1855 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1856 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1857 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1858 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\
|
|
1859 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\
|
|
1860 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1861 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1862 |
{truth} ==>
|
|
1863 |
(!x y z.
|
|
1864 |
~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/
|
|
1865 |
~{IN . y . c} \/ ~{IN . (gv24939 . y) . (prefix_set . (x))} \/
|
|
1866 |
~{IN . (gv24939 . y) . (prefix_set . y)} \/
|
|
1867 |
~{IN . (gv24940 . z) . (prefix_set . z)} \/
|
|
1868 |
~{IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\
|
|
1869 |
(!x y z.
|
|
1870 |
~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/
|
|
1871 |
~{IN . y . c} \/ {IN . (gv24939 . y) . (prefix_set . (x))} \/
|
|
1872 |
{IN . (gv24939 . y) . (prefix_set . y)} \/
|
|
1873 |
~{IN . (gv24940 . z) . (prefix_set . z)} \/
|
|
1874 |
~{IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\
|
|
1875 |
(!x y z.
|
|
1876 |
~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/
|
|
1877 |
~{IN . y . c} \/ {IN . (gv24939 . y) . (prefix_set . (x))} \/
|
|
1878 |
{IN . (gv24939 . y) . (prefix_set . y)} \/
|
|
1879 |
{IN . (gv24940 . z) . (prefix_set . z)} \/
|
|
1880 |
{IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\
|
|
1881 |
(!x y z.
|
|
1882 |
~{IN . x . (prefix_set . x')} \/ ~{IN . x . (prefix_set . (x))} \/
|
|
1883 |
~{IN . y . c} \/ ~{IN . (gv24939 . y) . (prefix_set . (x))} \/
|
|
1884 |
~{IN . (gv24939 . y) . (prefix_set . y)} \/
|
|
1885 |
{IN . (gv24940 . z) . (prefix_set . z)} \/
|
|
1886 |
{IN . (gv24940 . z) . (prefix_set . x')} \/ ~{IN . z . c}) /\
|
|
1887 |
{IN . x' . c} /\
|
|
1888 |
{IN . (PREIMAGE . ((o) . SND . f) . s) . (events . bern)} /\
|
|
1889 |
(!x y.
|
|
1890 |
f . x =
|
|
1891 |
pair . (FST . (f . (prefix_seq . y))) . (sdrop . (LENGTH . y) . x) \/
|
|
1892 |
~{IN . y . c} \/ ~{IN . x . (prefix_set . y)}) /\
|
|
1893 |
{IN . ((o) . SND . f) .
|
|
1894 |
(measurable . (events . bern) . (events . bern))} /\
|
|
1895 |
{countable . (range . ((o) . FST . f))} /\
|
|
1896 |
{IN . ((o) . FST . f) . (measurable . (events . bern) . UNIV)} /\
|
|
1897 |
{prefix_cover . c} /\ {IN . s . (events . bern)} /\ {IN . x . c} /\
|
|
1898 |
({IN . x'' . (prefix_set . x)} \/ {IN . x'' . (prefix_set . x')}) /\
|
|
1899 |
(~{IN . x'' . (prefix_set . x)} \/ ~{IN . x'' . (prefix_set . x')}) /\
|
|
1900 |
{IN . x''' . (prefix_set . x)} /\ {IN . x''' . (prefix_set . x')} ==> F`},
|
|
1901 |
|
|
1902 |
{name = "prob_53",
|
|
1903 |
comments = [],
|
|
1904 |
goal = `
|
|
1905 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
1906 |
(!x y.
|
|
1907 |
x = y \/ ~{x . (EXT_POINT . x . y)} \/ ~{y . (EXT_POINT . x . y)}) /\
|
|
1908 |
(!x y. x = y \/ {x . (EXT_POINT . x . y)} \/ {y . (EXT_POINT . x . y)}) /\
|
|
1909 |
(!x y. x = y \/ ~(x . (EXT_POINT . x . y) = y . (EXT_POINT . x . y))) /\
|
|
1910 |
(!x y. ~{x} \/ ~(x = y) \/ {y}) /\
|
|
1911 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x . z = y . v) /\
|
|
1912 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1913 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1914 |
{truth} ==>
|
|
1915 |
(!x y z.
|
|
1916 |
~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/
|
|
1917 |
~{IN . y . c} \/ ~{IN . (gv39280 . y) . (prefix_set . x'')} \/
|
|
1918 |
~{IN . (gv39280 . y) . (prefix_set . y)} \/
|
|
1919 |
~{IN . (gv39280 . z) . (prefix_set . z)} \/
|
|
1920 |
~{IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\
|
|
1921 |
(!x y z.
|
|
1922 |
~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/
|
|
1923 |
~{IN . y . c} \/ {IN . (gv39280 . y) . (prefix_set . x'')} \/
|
|
1924 |
{IN . (gv39280 . y) . (prefix_set . y)} \/
|
|
1925 |
~{IN . (gv39280 . z) . (prefix_set . z)} \/
|
|
1926 |
~{IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\
|
|
1927 |
(!x y z.
|
|
1928 |
~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/
|
|
1929 |
~{IN . y . c} \/ {IN . (gv39280 . y) . (prefix_set . x'')} \/
|
|
1930 |
{IN . (gv39280 . y) . (prefix_set . y)} \/
|
|
1931 |
{IN . (gv39280 . z) . (prefix_set . z)} \/
|
|
1932 |
{IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\
|
|
1933 |
(!x y z.
|
|
1934 |
~{IN . x . (prefix_set . x''')} \/ ~{IN . x . (prefix_set . x'')} \/
|
|
1935 |
~{IN . y . c} \/ ~{IN . (gv39280 . y) . (prefix_set . x'')} \/
|
|
1936 |
~{IN . (gv39280 . y) . (prefix_set . y)} \/
|
|
1937 |
{IN . (gv39280 . z) . (prefix_set . z)} \/
|
|
1938 |
{IN . (gv39280 . z) . (prefix_set . x''')} \/ ~{IN . z . c}) /\
|
|
1939 |
{IN . x''' . c} /\
|
|
1940 |
{IN . (PREIMAGE . ((o) . SND . f) . x') . (events . bern)} /\
|
|
1941 |
{IN . x' . (events . bern)} /\ {prefix_cover . c} /\
|
|
1942 |
{IN . ((o) . FST . f) . (measurable . (events . bern) . UNIV)} /\
|
|
1943 |
{countable . (range . ((o) . FST . f))} /\
|
|
1944 |
{IN . ((o) . SND . f) .
|
|
1945 |
(measurable . (events . bern) . (events . bern))} /\
|
|
1946 |
(!x y.
|
|
1947 |
f . x =
|
|
1948 |
pair . (FST . (f . (prefix_seq . y))) . (sdrop . (LENGTH . y) . x) \/
|
|
1949 |
~{IN . y . c} \/ ~{IN . x . (prefix_set . y)}) /\
|
|
1950 |
{IN . (PREIMAGE . ((o) . FST . f) . x) . (events . bern)} /\
|
|
1951 |
{IN . x'' . c} /\
|
|
1952 |
({IN . x'''' . (prefix_set . x'')} \/
|
|
1953 |
{IN . x'''' . (prefix_set . x''')}) /\
|
|
1954 |
(~{IN . x'''' . (prefix_set . x'')} \/
|
|
1955 |
~{IN . x'''' . (prefix_set . x''')}) /\
|
|
1956 |
{IN . x''''' . (prefix_set . x'')} /\
|
|
1957 |
{IN . x''''' . (prefix_set . x''')} ==> F`},
|
|
1958 |
|
|
1959 |
{name = "prob_extra_22",
|
|
1960 |
comments = [],
|
|
1961 |
goal = `
|
|
1962 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
1963 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
1964 |
{truth} ==>
|
|
1965 |
{P . x} /\ (!x. {real_lte . x . z} \/ ~{P . x}) /\
|
|
1966 |
(!x y z v.
|
|
1967 |
{real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/
|
|
1968 |
~{P . z} \/ ~{real_lte . (gv13960 . v) . v}) /\
|
|
1969 |
(!x y z.
|
|
1970 |
~{real_lt . x . (sup . P)} \/ {P . (gv13960 . x)} \/ ~{P . y} \/
|
|
1971 |
~{real_lte . (gv13960 . z) . z}) /\
|
|
1972 |
(!x y z.
|
|
1973 |
~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv13960 . x)} \/
|
|
1974 |
~{P . y} \/ ~{real_lte . (gv13960 . z) . z}) /\
|
|
1975 |
(!x y z.
|
|
1976 |
~{real_lt . x . (sup . P)} \/ {real_lt . x . (gv13960 . x)} \/
|
|
1977 |
~{P . y} \/ {P . (gv13960 . z)}) /\
|
|
1978 |
(!x y z.
|
|
1979 |
~{real_lt . x . (sup . P)} \/ {P . (gv13960 . x)} \/ ~{P . y} \/
|
|
1980 |
{P . (gv13960 . z)}) /\
|
|
1981 |
(!x y z v.
|
|
1982 |
{real_lt . x . (sup . P)} \/ ~{P . y} \/ ~{real_lt . x . y} \/
|
|
1983 |
~{P . z} \/ {P . (gv13960 . v)}) /\
|
|
1984 |
(!x.
|
|
1985 |
{real_lt . (gv13925 . x) . (gv13926 . x)} \/
|
|
1986 |
{real_lt . (gv13925 . x) . x}) /\
|
|
1987 |
(!x. {P . (gv13926 . x)} \/ {real_lt . (gv13925 . x) . x}) /\
|
|
1988 |
(!x y.
|
|
1989 |
~{real_lt . (gv13925 . x) . y} \/ ~{P . y} \/
|
|
1990 |
~{real_lt . (gv13925 . x) . x}) ==> F`},
|
|
1991 |
|
|
1992 |
{name = "root2_2",
|
|
1993 |
comments = [],
|
|
1994 |
goal = `
|
|
1995 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ x * z = y * v) /\
|
|
1996 |
(!x y. ~(x = y) \/ NUMERAL x = NUMERAL y) /\
|
|
1997 |
(!x y. ~(x = y) \/ BIT2 x = BIT2 y) /\
|
|
1998 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ EXP x z = EXP y v) /\
|
|
1999 |
(!x y z v. ~(x = y) \/ ~(z = v) \/ y < v \/ ~(x < z)) /\
|
|
2000 |
(!x y. ~(x = y) \/ EVEN y \/ ~EVEN x) /\
|
|
2001 |
(!x y z. ~(x = y) \/ ~(x = z) \/ y = z) /\ (!x. x = x) /\
|
|
2002 |
(!x y.
|
|
2003 |
~(EXP (NUMERAL (BIT2 ZERO) * x) (NUMERAL (BIT2 ZERO)) =
|
|
2004 |
NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO))) \/
|
|
2005 |
NUMERAL (BIT2 ZERO) * EXP x (NUMERAL (BIT2 ZERO)) =
|
|
2006 |
EXP y (NUMERAL (BIT2 ZERO))) /\
|
|
2007 |
(!x y.
|
|
2008 |
EXP (NUMERAL (BIT2 ZERO) * x) (NUMERAL (BIT2 ZERO)) =
|
|
2009 |
NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO)) \/
|
|
2010 |
~(NUMERAL (BIT2 ZERO) * EXP x (NUMERAL (BIT2 ZERO)) =
|
|
2011 |
EXP y (NUMERAL (BIT2 ZERO)))) /\
|
|
2012 |
(!x. ~EVEN x \/ x = NUMERAL (BIT2 ZERO) * gv4671 x) /\
|
|
2013 |
(!x y. EVEN x \/ ~(x = NUMERAL (BIT2 ZERO) * y)) /\
|
|
2014 |
(!x y. ~EVEN (x * y) \/ EVEN x \/ EVEN y) /\
|
|
2015 |
(!x y. EVEN (x * y) \/ ~EVEN x) /\ (!x y. EVEN (x * y) \/ ~EVEN y) /\
|
|
2016 |
(!x. EXP x (NUMERAL (BIT2 ZERO)) = x * x) /\
|
|
2017 |
(!x. EVEN (NUMERAL (BIT2 ZERO) * x)) ==>
|
|
2018 |
EXP (NUMERAL (BIT2 ZERO) * k) (NUMERAL (BIT2 ZERO)) =
|
|
2019 |
NUMERAL (BIT2 ZERO) * EXP n (NUMERAL (BIT2 ZERO)) /\
|
|
2020 |
(!x y.
|
|
2021 |
~(EXP x (NUMERAL (BIT2 ZERO)) =
|
|
2022 |
NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO))) \/ x = 0 \/
|
|
2023 |
~(x < NUMERAL (BIT2 ZERO) * k)) /\
|
|
2024 |
(!x y.
|
|
2025 |
~(EXP x (NUMERAL (BIT2 ZERO)) =
|
|
2026 |
NUMERAL (BIT2 ZERO) * EXP y (NUMERAL (BIT2 ZERO))) \/ y = 0 \/
|
|
2027 |
~(x < NUMERAL (BIT2 ZERO) * k)) /\
|
|
2028 |
(!x. ~(n = NUMERAL (BIT2 ZERO) * x)) ==> F`},
|
|
2029 |
|
|
2030 |
{name = "TermRewriting_13",
|
|
2031 |
comments = [],
|
|
2032 |
goal = `
|
|
2033 |
~{existential . (K . falsity)} /\ {existential . (K . truth)} /\
|
|
2034 |
~{universal . (K . falsity)} /\ {universal . (K . truth)} /\ ~{falsity} /\
|
|
2035 |
{truth} /\
|
|
2036 |
(!x y z v.
|
|
2037 |
~{RTC . x . y . z} \/ {RTC . x . v . z} \/ ~{RTC . x . v . y}) ==>
|
|
2038 |
{WCR . R} /\ {SN . R} /\
|
|
2039 |
(!x y z.
|
|
2040 |
~{RTC . R . x . y} \/ ~{RTC . R . x . z} \/
|
|
2041 |
{RTC . R . y . (gv1300 . x . z . y)} \/ ~{TC . R . (x) . x}) /\
|
|
2042 |
(!x y z.
|
|
2043 |
~{RTC . R . x . y} \/ ~{RTC . R . x . z} \/
|
|
2044 |
{RTC . R . z . (gv1300 . x . z . y)} \/ ~{TC . R . (x) . x}) /\
|
|
2045 |
{RTC . R . x . y} /\ {RTC . R . x . z} /\ {R . x . y1} /\
|
|
2046 |
{RTC . R . y1 . y} /\ {R . x . z1} /\ {RTC . R . z1 . z} /\
|
|
2047 |
{RTC . R . y1 . x0} /\ {RTC . R . z1 . x0} /\ {TC . R . x . y1} /\
|
|
2048 |
{TC . R . x . z1} /\ {RTC . R . y . y2} /\ {RTC . R . x0 . y2} /\
|
|
2049 |
{RTC . R . z . z2} /\ {RTC . R . x0 . z2} /\ {TC . R . x . x0} /\
|
|
2050 |
(!x. ~{RTC . R . y . x} \/ ~{RTC . R . z . x}) ==> F`}
|
|
2051 |
|
|
2052 |
];
|