39348
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(* ========================================================================= *)
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(* METIS TESTS *)
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(* Copyright (c) 2004 Joe Hurd, distributed under the GNU GPL version 2 *)
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(* ========================================================================= *)
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(* ------------------------------------------------------------------------- *)
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(* Dummy versions of Moscow ML declarations to stop real compilers barfing. *)
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(* ------------------------------------------------------------------------- *)
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(*mlton
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val quotation = ref true;
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val quietdec = ref true;
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val loadPath = ref ([] : string list);
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val load = fn (_ : string) => ();
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*)
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(*polyml
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val quotation = ref true;
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val quietdec = ref true;
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val loadPath = ref ([] : string list);
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val load = fn (_ : string) => ();
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*)
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(* ------------------------------------------------------------------------- *)
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(* Load and open some useful modules *)
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(* ------------------------------------------------------------------------- *)
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val () = loadPath := !loadPath @ ["../bin/mosml"];
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val () = app load ["Options"];
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open Useful;
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val time = Portable.time;
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(* ------------------------------------------------------------------------- *)
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(* Problem data. *)
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(* ------------------------------------------------------------------------- *)
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val ref oldquietdec = quietdec;
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val () = quietdec := true;
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val () = quotation := true;
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use "../src/problems.sml";
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val () = quietdec := oldquietdec;
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(* ------------------------------------------------------------------------- *)
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(* Helper functions. *)
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(* ------------------------------------------------------------------------- *)
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fun partialOrderToString (SOME LESS) = "SOME LESS"
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| partialOrderToString (SOME GREATER) = "SOME GREATER"
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| partialOrderToString (SOME EQUAL) = "SOME EQUAL"
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| partialOrderToString NONE = "NONE";
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fun SAY s =
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print
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("-------------------------------------" ^
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"-------------------------------------\n" ^ s ^ "\n\n");
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fun printval p x = (print (Print.toString p x ^ "\n\n"); x);
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fun mkCl p th = Clause.mk {parameters = p, id = Clause.newId (), thm = th};
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val pvBool = printval Print.ppBool
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and pvPo = printval (Print.ppMap partialOrderToString Print.ppString)
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and pvFm = printval Formula.pp
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and pvFms = printval (Print.ppList Formula.pp)
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and pvThm = printval Thm.pp
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and pvEqn : Rule.equation -> Rule.equation = printval (Print.ppMap snd Thm.pp)
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and pvNet = printval (LiteralNet.pp Print.ppInt)
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and pvRw = printval Rewrite.pp
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and pvU = printval Units.pp
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and pvLits = printval LiteralSet.pp
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and pvCl = printval Clause.pp
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and pvCls = printval (Print.ppList Clause.pp)
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and pvM = printval Model.pp;
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val NV = Name.fromString
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and NF = Name.fromString
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and NR = Name.fromString;
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val V = Term.Var o NV
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and C = (fn c => Term.Fn (NF c, []))
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and T = Term.parse
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and A = Atom.parse
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and L = Literal.parse
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and F = Formula.parse
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and S = Subst.fromList;
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val LS = LiteralSet.fromList o map L;
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val AX = Thm.axiom o LS;
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val CL = mkCl Clause.default o AX;
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val Q = (fn th => (Thm.destUnitEq th, th)) o AX o singleton
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and U = (fn th => (Thm.destUnit th, th)) o AX o singleton;
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fun test_fun eq p r a =
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if eq r a then p a ^ "\n" else
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(print ("\n\n" ^
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"test: should have\n-->" ^ p r ^ "<--\n\n" ^
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"test: actually have\n-->" ^ p a ^ "<--\n\n");
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raise Fail "test: failed a test");
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fun test eq p r a = print (test_fun eq p r a ^ "\n");
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val test_tm = test Term.equal Term.toString o Term.parse;
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val test_fm = test Formula.equal Formula.toString o Formula.parse;
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fun test_id p f a = test p a (f a);
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fun chop_newline s =
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if String.sub (s,0) = #"\n" then String.extract (s,1,NONE) else s;
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fun unquote (QUOTE q) = q
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| unquote (ANTIQUOTE _) = raise Fail "unquote";
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(* ------------------------------------------------------------------------- *)
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val () = SAY "The parser and pretty-printer";
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(* ------------------------------------------------------------------------- *)
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fun prep l = (chop_newline o String.concat o map unquote) l;
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fun mini_print n fm = withRef (Print.lineLength,n) Formula.toString fm;
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fun testlen_pp n q =
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(fn s => test_fun equal I s ((mini_print n o Formula.fromString) s))
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(prep q);
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fun test_pp q = print (testlen_pp 40 q ^ "\n");
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val () = test_pp `3 = f x`;
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val () = test_pp `f x y = y`;
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val () = test_pp `P x y`;
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val () = test_pp `P (f x) y`;
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val () = test_pp `f x = 3`;
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val () = test_pp `!x. P x y`;
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val () = test_pp `!x y. P x y`;
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val () = test_pp `!x y z. P x y z`;
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val () = test_pp `x = y`;
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val () = test_pp `x = 3`;
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val () = test_pp `x + y = y`;
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val () = test_pp `x / y * z = w`;
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val () = test_pp `x * y * z = x * (y * z)`;
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val () = test_pp `!x. ?y. x <= y /\ y <= x`;
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val () = test_pp `?x. !y. x + y = y /\ y <= x`;
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val () = test_pp `p /\ q \/ r /\ p ==> q <=> p`;
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val () = test_pp `p`;
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val () = test_pp `~!x. bool x`;
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val () = test_pp `p ==> !x. bool x`;
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val () = test_pp `p ==> ~!x. bool x`;
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val () = test_pp `~!x. bool x`;
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val () = test_pp `~~!x. bool x`;
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val () = test_pp `hello + there <> everybody`;
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val () = test_pp `!x y. ?z. x < z /\ y < z`;
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val () = test_pp `~(!x. P x) <=> ?y. ~P y`;
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val () = test_pp `?y. x < y ==> !v. ?w. x * v < y * w`;
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val () = test_pp `(<=)`;
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val () = test_pp `(<=) <= b`;
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val () = test_pp `(<=) <= (+)`;
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val () = test_pp `(<=) x`;
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val () = test_pp `(<=) <= (+) x`;
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val () = test_pp `~B (P % ((,) % c_a % v_b))`;
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val () = test_pp `B ((<=) % 0 % (LENGTH % NIL))`;
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val () = test_pp `~(a = b)`;
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val () = test_pp `!x. p x ==> !y. p y`;
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val () = test_pp `(!x. p x) ==> !y. p y`;
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val () = test_pp `!x. ~~x = x`;
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val () = test_pp `x + (y + z) = a`;
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val () = test_pp `(x @ y) @ z = a`;
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val () = test_pp `p ((a @ a) @ a = a)`;
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val () = test_pp `!x y z. (x @ y) @ z = (x @ z) @ y @ z`;
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val () = test_pp `~(!x. q x) /\ p`;
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val () = test_pp `!x. f (~~x) b (~c)`;
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val () = test_pp `p ==> ~(a /\ b)`;
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val () = test_pp `!water. drinks (water)`;
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val () = test_pp `!vat water. drinks ((vat) p x (water))`;
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val () = test_pp `!x y. ~{x < y} /\ T`;
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val () = test_pp `[3]`;
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val () = test_pp `
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!x y z. ?x' y' z'.
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P x y z ==> P x' y' z'`;
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val () = test_pp `
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(!x. P x ==> !x. Q x) /\
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((!x. Q x \/ R x) ==> ?x. Q x /\ R x) /\
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((?x. R x) ==> !x. L x ==> M x) ==>
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!x. P x /\ L x ==> M x`;
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val () = test_pp `
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!x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
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x12 x13 x14 x15 x16 x17 x18 x19 x20
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x21 x22 x23 x24 x25 x26 x27 x28 x29
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x30 x31 x32. ?y0 y1 y2 y3 y4 y5 y6 y7.
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P`;
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val () = test_pp `
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!x x x x x x x x x x x x x x x x x x x x
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x x x x x x x x x x. ?y y y y y y y y
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y y y y y y y y y y y.
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P (x, y) /\ P (x, y) /\ P (x, y) /\
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P (x, y) /\ P (x, y) /\ P (x, y) /\
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P (x, y) /\ P (x, y) /\ P (x, y) /\
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P (x, y) /\ P (x, y) /\ P (x, y) /\
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P (x, y) /\ P (x, y) /\
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~~~~~~~~~~~~~f
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(f (f (f x y) (f x y))
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(f (f x y) (f x y)))
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(f (f (f x y) (f x y))
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(f (f x y) (f x y)))`;
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val () = test_pp `
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(!x.
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extremely__long__predicate__name) /\
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F`;
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val () = test_pp `
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(!x. x = x) /\
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(!x y. ~(x = y) \/ y = x) /\
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(!x y z.
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~(x = y) \/ ~(y = z) \/ x = z) /\
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(!x y z. b . x . y . z = x . (y . z)) /\
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(!x y. t . x . y = y . x) /\
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(!x y z. ~(x = y) \/ x . z = y . z) /\
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(!x y z. ~(x = y) \/ z . x = z . y) ==>
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~(b . (b . (t . b) . b) . t . x . y .
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z = y . (x . z)) ==> F`;
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(* ------------------------------------------------------------------------- *)
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val () = SAY "Substitution";
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(* ------------------------------------------------------------------------- *)
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val () =
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test Name.equal Name.toString (NV"x")
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(Term.variantPrime (NameSet.fromList [NV"y",NV"z" ]) (NV"x"));
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val () =
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test Name.equal Name.toString (NV"x'")
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(Term.variantPrime (NameSet.fromList [NV"x",NV"y" ]) (NV"x"));
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val () =
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test Name.equal Name.toString (NV"x''")
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(Term.variantPrime (NameSet.fromList [NV"x",NV"x'"]) (NV"x"));
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val () =
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test Name.equal Name.toString (NV"x")
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(Term.variantNum (NameSet.fromList [NV"y",NV"z"]) (NV"x"));
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val () =
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test Name.equal Name.toString (NV"x0")
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(Term.variantNum (NameSet.fromList [NV"x",NV"y"]) (NV"x"));
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val () =
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test Name.equal Name.toString (NV"x1")
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(Term.variantNum (NameSet.fromList [NV"x",NV"x0"]) (NV"x"));
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val () =
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test_fm
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`!x. x = $z`
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(Formula.subst (S [(NV"y", V"z")]) (F`!x. x = $y`));
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val () =
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test_fm
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`!x'. x' = $x`
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(Formula.subst (S [(NV"y", V"x")]) (F`!x. x = $y`));
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val () =
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test_fm
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`!x' x''. x' = $x ==> x' = x''`
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(Formula.subst (S [(NV"y", V"x")])
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(F`!x x'. x = $y ==> x = x'`));
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(* ------------------------------------------------------------------------- *)
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val () = SAY "Unification";
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(* ------------------------------------------------------------------------- *)
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fun unify_and_apply tm1 tm2 =
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Subst.subst (Subst.unify Subst.empty tm1 tm2) tm1;
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val () = test_tm `c` (unify_and_apply (V"x") (C"c"));
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val () = test_tm `c` (unify_and_apply (C"c") (V"x"));
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val () =
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test_tm
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`f c`
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(unify_and_apply
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(Term.Fn (NF"f", [V"x"]))
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(Term.Fn (NF"f", [C"c"])));
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val () = test_tm `f 0 0 0` (unify_and_apply (T`f 0 $x $x`) (T`f $y $y $z`));
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fun f x y = (printval Subst.pp (Atom.unify Subst.empty x y); ());
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val () = f (NR"P", [V"x"]) (NR"P", [V"x"]);
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val () = f (NR"P", [V"x"]) (NR"P", [C"c"]);
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val () = f (A`P c_x`) (A`P $x`);
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val () = f (A`q $x (f $x)`) (A`q $y $z`);
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(* ------------------------------------------------------------------------- *)
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val () = SAY "The logical kernel";
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(* ------------------------------------------------------------------------- *)
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val th0 = AX [`p`,`q`];
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val th1 = AX [`~p`,`r`];
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val th2 = Thm.resolve (L`p`) th0 th1;
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val _ = printval Proof.pp (Proof.proof th2);
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val th0 = Rule.relationCongruence Atom.eqRelation;
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val th1 =
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Thm.subst (S [(NV"y0",T`$x`),(NV"y1",T`$y`),(NV"x1",T`$z`),(NV"x0",T`$x`)]) th0;
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val th2 = Thm.resolve (L`$x = $x`) Rule.reflexivity th1;
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val th3 = Rule.symNeq (L`~($z = $y)`) th2;
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val _ = printval Proof.pp (Proof.proof th3);
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(* Testing the elimination of redundancies in proofs *)
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val th0 = Rule.reflexivity;
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val th1 = Thm.subst (S [(NV"x", Term.Fn (NF"f", [V"y"]))]) th0;
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val th2 = Thm.subst (S [(NV"y", C"c")]) th1;
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val _ = printval Proof.pp (Proof.proof th2);
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(* ------------------------------------------------------------------------- *)
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val () = SAY "Derived rules of inference";
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(* ------------------------------------------------------------------------- *)
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val th0 = pvThm (AX [`$x = a`, `f a = $x`, `~(a = b)`, `a = $x`,
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`$x = a`, `a = $x`, `~(b = a)`]);
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val th1 = pvThm (Rule.removeSym th0);
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val th2 = pvThm (Rule.symEq (L`a = $x`) th0);
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val th3 = pvThm (Rule.symEq (L`f a = $x`) th0);
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val th5 = pvThm (Rule.symNeq (L`~(a = b)`) th0);
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(* Testing the rewrConv conversion *)
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val (x_y as (x,y), eqTh) = pvEqn (Q`e * (i $z * $z) = e`);
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val tm = Term.Fn (NF"f",[x]);
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|
384 |
val path : int list = [0];
|
|
385 |
val reflTh = Thm.refl tm;
|
|
386 |
val reflLit = Thm.destUnit reflTh;
|
|
387 |
val th = Thm.equality reflLit (1 :: path) y;
|
|
388 |
val th = Thm.resolve reflLit reflTh th;
|
|
389 |
val th = pvThm (try (Thm.resolve (Literal.mkEq x_y) eqTh) th);
|
|
390 |
|
|
391 |
(* ------------------------------------------------------------------------- *)
|
|
392 |
val () = SAY "Discrimination nets for literals";
|
|
393 |
(* ------------------------------------------------------------------------- *)
|
|
394 |
|
|
395 |
val n = pvNet (LiteralNet.new {fifo = true});
|
|
396 |
val n = pvNet (LiteralNet.insert n (L`P (f c $x a)`, 1));
|
|
397 |
val n = pvNet (LiteralNet.insert n (L`P (f c $y a)`, 2));
|
|
398 |
val n = pvNet (LiteralNet.insert n (L`P (f c a a)`, 3));
|
|
399 |
val n = pvNet (LiteralNet.insert n (L`P (f c b a)`, 4));
|
|
400 |
val n = pvNet (LiteralNet.insert n (L`~Q`, 5));
|
|
401 |
val n = pvNet (LiteralNet.insert n (L`~Q`, 6));
|
|
402 |
val n = pvNet (LiteralNet.insert n (L`~Q`, 7));
|
|
403 |
val n = pvNet (LiteralNet.insert n (L`~Q`, 8));
|
|
404 |
|
|
405 |
(* ------------------------------------------------------------------------- *)
|
|
406 |
val () = SAY "The Knuth-Bendix ordering on terms";
|
|
407 |
(* ------------------------------------------------------------------------- *)
|
|
408 |
|
|
409 |
val kboOrder = KnuthBendixOrder.default;
|
|
410 |
val kboCmp = KnuthBendixOrder.compare kboOrder;
|
|
411 |
|
|
412 |
val x = pvPo (kboCmp (T`f a`, T`g b`));
|
|
413 |
val x = pvPo (kboCmp (T`f a b`, T`g b`));
|
|
414 |
val x = pvPo (kboCmp (T`f $x`, T`g a`));
|
|
415 |
val x = pvPo (kboCmp (T`f a $x`, T`g $x`));
|
|
416 |
val x = pvPo (kboCmp (T`f $x`, T`g $x`));
|
|
417 |
val x = pvPo (kboCmp (T`f $x`, T`f $x`));
|
|
418 |
val x = pvPo (kboCmp (T`$x + $y`, T`$x + $x`));
|
|
419 |
val x = pvPo (kboCmp (T`$x + $y + $x`, T`$y + $x + $x`));
|
|
420 |
val x = pvPo (kboCmp (T`$x + $y + $x`, T`$y * $x + $x`));
|
|
421 |
val x = pvPo (kboCmp (T`a`, T`$x`));
|
|
422 |
val x = pvPo (kboCmp (T`f a`, T`$x`));
|
|
423 |
val x = pvPo (kboCmp (T`f $x (f $y $z)`, T`f (f $x $y) $z`));
|
|
424 |
val x = pvPo (kboCmp (T`f (g $x a)`, T`f (h a $x)`));
|
|
425 |
val x = pvPo (kboCmp (T`f (g a)`, T`f (h $x)`));
|
|
426 |
val x = pvPo (kboCmp (T`f (h a)`, T`f (g $x)`));
|
|
427 |
val x = pvPo (kboCmp (T`f $y`, T`f (g a b c)`));
|
|
428 |
val x = pvPo (kboCmp (T`$x * $y + $x * $z`, T`$x * ($y + $z)`));
|
|
429 |
|
|
430 |
(* ------------------------------------------------------------------------- *)
|
|
431 |
val () = SAY "Rewriting";
|
|
432 |
(* ------------------------------------------------------------------------- *)
|
|
433 |
|
|
434 |
val eqnsToRw = Rewrite.addList (Rewrite.new kboCmp) o enumerate;
|
|
435 |
|
|
436 |
val eqns = [Q`e * $x = $x`, Q`$x * e = $x`, Q`i $x * $x = e`, Q`$x * i $x = e`];
|
|
437 |
val ax = pvThm (AX [`e * (i $z * $z) = i e * e`]);
|
|
438 |
val th = pvThm (Rewrite.orderedRewrite kboCmp eqns ax);
|
|
439 |
|
|
440 |
val rw = pvRw (eqnsToRw eqns);
|
|
441 |
val th = pvThm (snd (try (Rewrite.rewriteConv rw kboCmp) (T`e * e`)));
|
|
442 |
val th = pvThm (snd (try (Rewrite.rewriteConv rw kboCmp) (T`e * (i $z * $z)`)));
|
|
443 |
val th = pvThm (try (Rewrite.rewriteRule rw kboCmp) ax);
|
|
444 |
|
|
445 |
(* Bug check: in this one a literal goes missing, due to the Resolve in Subst *)
|
|
446 |
val eqns = [Q`f a = a`];
|
|
447 |
val ax = pvThm (AX [`~(g (f a) = b)`, `~(f a = a)`]);
|
|
448 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp eqns) ax);
|
|
449 |
|
|
450 |
(* Bug check: term paths were not being reversed before use *)
|
|
451 |
val eqns = [Q`a = f a`];
|
|
452 |
val ax = pvThm (AX [`a <= f (f a)`]);
|
|
453 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp eqns) ax);
|
|
454 |
|
|
455 |
(* Bug check: Equality used to complain if the literal didn't exist *)
|
|
456 |
val eqns = [Q`b = f a`];
|
|
457 |
val ax = pvThm (AX [`~(f a = f a)`]);
|
|
458 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp eqns) ax);
|
|
459 |
|
|
460 |
(* Testing the rewriting with disequalities in the same clause *)
|
|
461 |
val ax = pvThm (AX [`~(a = b)`, `P a`, `P b`]);
|
|
462 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp []) ax);
|
|
463 |
|
|
464 |
val ax = pvThm (AX [`~(f $x = $x)`, `P (f a)`, `P (f $x)`, `P (f $y)`]);
|
|
465 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp []) ax);
|
|
466 |
|
|
467 |
val ax = pvThm
|
|
468 |
(AX [`~(f (f (f (f (f $x)))) = $x)`,
|
|
469 |
`~(f (f (f (f (f (f (f (f $x))))))) = $x)`,
|
|
470 |
`P (f $x)`]);
|
|
471 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp []) ax);
|
|
472 |
|
|
473 |
(* Symmetry should yield a tautology on ground clauses *)
|
|
474 |
val ax = pvThm (AX [`~(a = b)`, `b = a`]);
|
|
475 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp []) ax);
|
|
476 |
|
|
477 |
(* Transitivity should yield a tautology on ground clauses *)
|
|
478 |
val ax = pvThm (AX [`~(a = b)`, `~(b = c)`, `a = c`]);
|
|
479 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp []) ax);
|
|
480 |
|
|
481 |
(* Extended transitivity should yield a tautology on ground clauses *)
|
|
482 |
val ax = pvThm (AX [`~(a = b)`, `~(b = c)`, `~(c = d)`, `a = d`]);
|
|
483 |
val th = pvThm (try (Rewrite.orderedRewrite kboCmp []) ax);
|
|
484 |
|
|
485 |
(* ------------------------------------------------------------------------- *)
|
|
486 |
val () = SAY "Unit cache";
|
|
487 |
(* ------------------------------------------------------------------------- *)
|
|
488 |
|
|
489 |
val u = pvU (Units.add Units.empty (U`~p $x`));
|
|
490 |
val u = pvU (Units.add u (U`a = b`));
|
|
491 |
val _ = pvThm (Units.reduce u (AX [`p 0`,`~(b = a)`]));
|
|
492 |
|
|
493 |
(* ------------------------------------------------------------------------- *)
|
|
494 |
val () = SAY "Negation normal form";
|
|
495 |
(* ------------------------------------------------------------------------- *)
|
|
496 |
|
|
497 |
val nnf = Normalize.nnf;
|
|
498 |
|
|
499 |
val _ = pvFm (nnf (F`p /\ ~p`));
|
|
500 |
val _ = pvFm (nnf (F`(!x. P x) ==> ((?y. Q y) <=> (?z. P z /\ Q z))`));
|
|
501 |
val _ = pvFm (nnf (F`~(~(p <=> q) <=> r) <=> ~(p <=> ~(q <=> r))`));
|
|
502 |
|
|
503 |
(* ------------------------------------------------------------------------- *)
|
|
504 |
val () = SAY "Conjunctive normal form";
|
|
505 |
(* ------------------------------------------------------------------------- *)
|
|
506 |
|
|
507 |
local
|
|
508 |
fun clauseToFormula cl =
|
|
509 |
Formula.listMkDisj (LiteralSet.transform Literal.toFormula cl);
|
|
510 |
in
|
|
511 |
fun clausesToFormula cls = Formula.listMkConj (map clauseToFormula cls);
|
|
512 |
end;
|
|
513 |
|
|
514 |
val cnf' = pvFm o clausesToFormula o Normalize.cnf o F;
|
|
515 |
|
|
516 |
val cnf = pvFm o clausesToFormula o Normalize.cnf o
|
|
517 |
Formula.Not o Formula.generalize o F;
|
|
518 |
|
|
519 |
val _ = cnf `p \/ ~p`;
|
|
520 |
val _ = cnf `~((p /\ (q \/ r /\ s)) /\ (~p \/ ~q \/ ~s))`;
|
|
521 |
val _ = cnf `~((p /\ (q \/ r /\ s)) /\ (~p \/ ~q \/ ~s) /\ (p \/ ~p))`;
|
|
522 |
val _ = cnf `~(~(p <=> q) <=> r) <=> ~(p <=> ~(q <=> r))`;
|
|
523 |
val _ = cnf `((p <=> q) <=> r) <=> (p <=> (q <=> r))`;
|
|
524 |
val _ = cnf `~(!x. ?y. x < y ==> !v. ?w. x * v < y * w)`;
|
|
525 |
val _ = cnf `~(!x. P x ==> (?y z. Q y \/ ~(?z. P z /\ Q z)))`;
|
|
526 |
val _ = cnf `~(?x y. x + y = 2)`;
|
|
527 |
val _ = cnf' `(!x. p x) \/ (!y. r $x y)`;
|
|
528 |
|
|
529 |
val _ = cnf
|
|
530 |
`(!x. P x ==> (!x. Q x)) /\ ((!x. Q x \/ R x) ==> (?x. Q x /\ R x)) /\
|
|
531 |
((?x. R x) ==> (!x. L x ==> M x)) ==> (!x. P x /\ L x ==> M x)`;
|
|
532 |
|
|
533 |
(* ------------------------------------------------------------------------- *)
|
|
534 |
val () = SAY "Finite models";
|
|
535 |
(* ------------------------------------------------------------------------- *)
|
|
536 |
|
|
537 |
fun checkModelClause M cl =
|
|
538 |
let
|
|
539 |
val randomSamples = 100
|
|
540 |
|
|
541 |
fun addRandomSample {T,F} =
|
|
542 |
let
|
|
543 |
val {T = T', F = F'} = Model.checkClause {maxChecks = SOME 1} M cl
|
|
544 |
in
|
|
545 |
{T = T + T', F = F + F'}
|
|
546 |
end
|
|
547 |
|
|
548 |
val {T,F} = funpow randomSamples addRandomSample {T = 0, F = 0}
|
|
549 |
val rx = Real.fromInt T / Real.fromInt (T + F)
|
|
550 |
|
|
551 |
val {T,F} = Model.checkClause {maxChecks = NONE} M cl
|
|
552 |
val ry = Real.fromInt T / Real.fromInt (T + F)
|
|
553 |
in
|
|
554 |
[Formula.toString (LiteralSet.disjoin cl),
|
|
555 |
" | random sampling = " ^ percentToString rx,
|
|
556 |
" | exhaustive = " ^ percentToString ry]
|
|
557 |
end;
|
|
558 |
|
|
559 |
local
|
|
560 |
val format =
|
|
561 |
[{leftAlign = true, padChar = #" "},
|
|
562 |
{leftAlign = true, padChar = #" "},
|
|
563 |
{leftAlign = true, padChar = #" "}];
|
|
564 |
in
|
|
565 |
fun checkModel M cls =
|
|
566 |
let
|
|
567 |
val table = map (checkModelClause M) cls
|
|
568 |
|
|
569 |
val rows = alignTable format table
|
|
570 |
|
|
571 |
val () = print (join "\n" rows ^ "\n\n")
|
|
572 |
in
|
|
573 |
()
|
|
574 |
end;
|
|
575 |
end;
|
|
576 |
|
|
577 |
fun perturbModel M cls n =
|
|
578 |
let
|
|
579 |
val N = {size = Model.size M}
|
|
580 |
|
|
581 |
fun perturbClause (fv,cl) =
|
|
582 |
let
|
|
583 |
val V = Model.randomValuation N fv
|
|
584 |
in
|
|
585 |
if Model.interpretClause M V cl then ()
|
|
586 |
else Model.perturbClause M V cl
|
|
587 |
end
|
|
588 |
|
|
589 |
val cls = map (fn cl => (LiteralSet.freeVars cl, cl)) cls
|
|
590 |
|
|
591 |
fun perturbClauses () = app perturbClause cls
|
|
592 |
|
|
593 |
val () = funpow n perturbClauses ()
|
|
594 |
in
|
|
595 |
M
|
|
596 |
end;
|
|
597 |
|
|
598 |
val groupAxioms =
|
|
599 |
[LS[`0 + $x = $x`],
|
|
600 |
LS[`~$x + $x = 0`],
|
|
601 |
LS[`$x + $y + $z = $x + ($y + $z)`]];
|
|
602 |
|
|
603 |
val groupThms =
|
|
604 |
[LS[`$x + 0 = $x`],
|
|
605 |
LS[`$x + ~$x = 0`],
|
|
606 |
LS[`~~$x = $x`]];
|
|
607 |
|
|
608 |
fun newM fixed = Model.new {size = 8, fixed = fixed};
|
|
609 |
val M = pvM (newM Model.basicFixed);
|
|
610 |
val () = checkModel M (groupAxioms @ groupThms);
|
|
611 |
val M = pvM (perturbModel M groupAxioms 1000);
|
|
612 |
val () = checkModel M (groupAxioms @ groupThms);
|
|
613 |
val M = pvM (newM (Model.unionFixed Model.modularFixed Model.basicFixed));
|
|
614 |
val () = checkModel M (groupAxioms @ groupThms);
|
|
615 |
|
|
616 |
(* ------------------------------------------------------------------------- *)
|
|
617 |
val () = SAY "Checking the standard model";
|
|
618 |
(* ------------------------------------------------------------------------- *)
|
|
619 |
|
|
620 |
fun ppPercentClause (r,cl) =
|
|
621 |
let
|
|
622 |
val ind = 6
|
|
623 |
|
|
624 |
val p = percentToString r
|
|
625 |
|
|
626 |
val fm = LiteralSet.disjoin cl
|
|
627 |
in
|
|
628 |
Print.blockProgram Print.Consistent ind
|
|
629 |
[Print.addString p,
|
|
630 |
Print.addString (nChars #" " (ind - size p)),
|
|
631 |
Formula.pp fm]
|
|
632 |
end;
|
|
633 |
|
|
634 |
val standardModel = Model.new Model.default;
|
|
635 |
|
|
636 |
fun checkStandardModelClause cl =
|
|
637 |
let
|
|
638 |
val {T,F} = Model.checkClause {maxChecks = SOME 1000} standardModel cl
|
|
639 |
val r = Real.fromInt T / Real.fromInt (T + F)
|
|
640 |
in
|
|
641 |
(r,cl)
|
|
642 |
end;
|
|
643 |
|
|
644 |
val pvPCl = printval ppPercentClause
|
|
645 |
|
|
646 |
(* Equality *)
|
|
647 |
|
|
648 |
val cl = LS[`$x = $x`];
|
|
649 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
650 |
val cl = LS[`~($x = $y)`,`$y = $x`];
|
|
651 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
652 |
val cl = LS[`~($x = $y)`,`~($y = $z)`,`$x = $z`];
|
|
653 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
654 |
|
|
655 |
(* Projections *)
|
|
656 |
|
|
657 |
val cl = LS[`project1 $x1 = $x1`];
|
|
658 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
659 |
val cl = LS[`project1 $x1 $x2 = $x1`];
|
|
660 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
661 |
val cl = LS[`project2 $x1 $x2 = $x2`];
|
|
662 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
663 |
val cl = LS[`project1 $x1 $x2 $x3 = $x1`];
|
|
664 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
665 |
val cl = LS[`project2 $x1 $x2 $x3 = $x2`];
|
|
666 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
667 |
val cl = LS[`project3 $x1 $x2 $x3 = $x3`];
|
|
668 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
669 |
val cl = LS[`project1 $x1 $x2 $x3 $x4 = $x1`];
|
|
670 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
671 |
val cl = LS[`project2 $x1 $x2 $x3 $x4 = $x2`];
|
|
672 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
673 |
val cl = LS[`project3 $x1 $x2 $x3 $x4 = $x3`];
|
|
674 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
675 |
val cl = LS[`project4 $x1 $x2 $x3 $x4 = $x4`];
|
|
676 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
677 |
val cl = LS[`project1 $x1 $x2 $x3 $x4 $x5 = $x1`];
|
|
678 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
679 |
val cl = LS[`project2 $x1 $x2 $x3 $x4 $x5 = $x2`];
|
|
680 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
681 |
val cl = LS[`project3 $x1 $x2 $x3 $x4 $x5 = $x3`];
|
|
682 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
683 |
val cl = LS[`project4 $x1 $x2 $x3 $x4 $x5 = $x4`];
|
|
684 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
685 |
val cl = LS[`project5 $x1 $x2 $x3 $x4 $x5 = $x5`];
|
|
686 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
687 |
val cl = LS[`project1 $x1 $x2 $x3 $x4 $x5 $x6 = $x1`];
|
|
688 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
689 |
val cl = LS[`project2 $x1 $x2 $x3 $x4 $x5 $x6 = $x2`];
|
|
690 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
691 |
val cl = LS[`project3 $x1 $x2 $x3 $x4 $x5 $x6 = $x3`];
|
|
692 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
693 |
val cl = LS[`project4 $x1 $x2 $x3 $x4 $x5 $x6 = $x4`];
|
|
694 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
695 |
val cl = LS[`project5 $x1 $x2 $x3 $x4 $x5 $x6 = $x5`];
|
|
696 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
697 |
val cl = LS[`project6 $x1 $x2 $x3 $x4 $x5 $x6 = $x6`];
|
|
698 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
699 |
val cl = LS[`project1 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x1`];
|
|
700 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
701 |
val cl = LS[`project2 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x2`];
|
|
702 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
703 |
val cl = LS[`project3 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x3`];
|
|
704 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
705 |
val cl = LS[`project4 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x4`];
|
|
706 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
707 |
val cl = LS[`project5 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x5`];
|
|
708 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
709 |
val cl = LS[`project6 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x6`];
|
|
710 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
711 |
val cl = LS[`project7 $x1 $x2 $x3 $x4 $x5 $x6 $x7 = $x7`];
|
|
712 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
713 |
val cl = LS[`project1 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x1`];
|
|
714 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
715 |
val cl = LS[`project2 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x2`];
|
|
716 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
717 |
val cl = LS[`project3 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x3`];
|
|
718 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
719 |
val cl = LS[`project4 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x4`];
|
|
720 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
721 |
val cl = LS[`project5 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x5`];
|
|
722 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
723 |
val cl = LS[`project6 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x6`];
|
|
724 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
725 |
val cl = LS[`project7 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x7`];
|
|
726 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
727 |
val cl = LS[`project8 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 = $x8`];
|
|
728 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
729 |
val cl = LS[`project1 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x1`];
|
|
730 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
731 |
val cl = LS[`project2 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x2`];
|
|
732 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
733 |
val cl = LS[`project3 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x3`];
|
|
734 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
735 |
val cl = LS[`project4 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x4`];
|
|
736 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
737 |
val cl = LS[`project5 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x5`];
|
|
738 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
739 |
val cl = LS[`project6 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x6`];
|
|
740 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
741 |
val cl = LS[`project7 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x7`];
|
|
742 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
743 |
val cl = LS[`project8 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x8`];
|
|
744 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
745 |
val cl = LS[`project9 $x1 $x2 $x3 $x4 $x5 $x6 $x7 $x8 $x9 = $x9`];
|
|
746 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
747 |
|
|
748 |
(* Arithmetic *)
|
|
749 |
|
|
750 |
(* Zero *)
|
|
751 |
val cl = LS[`~isZero $x`,`$x = 0`];
|
|
752 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
753 |
val cl = LS[`isZero $x`,`~($x = 0)`];
|
|
754 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
755 |
|
|
756 |
(* Positive numerals *)
|
|
757 |
val cl = LS[`0 + 1 = 1`];
|
|
758 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
759 |
val cl = LS[`1 + 1 = 2`];
|
|
760 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
761 |
val cl = LS[`2 + 1 = 3`];
|
|
762 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
763 |
val cl = LS[`3 + 1 = 4`];
|
|
764 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
765 |
val cl = LS[`4 + 1 = 5`];
|
|
766 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
767 |
val cl = LS[`5 + 1 = 6`];
|
|
768 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
769 |
val cl = LS[`6 + 1 = 7`];
|
|
770 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
771 |
val cl = LS[`7 + 1 = 8`];
|
|
772 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
773 |
val cl = LS[`8 + 1 = 9`];
|
|
774 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
775 |
val cl = LS[`9 + 1 = 10`];
|
|
776 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
777 |
|
|
778 |
(* Negative numerals *)
|
|
779 |
val cl = LS[`~1 = negative1`];
|
|
780 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
781 |
val cl = LS[`~2 = negative2`];
|
|
782 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
783 |
val cl = LS[`~3 = negative3`];
|
|
784 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
785 |
val cl = LS[`~4 = negative4`];
|
|
786 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
787 |
val cl = LS[`~5 = negative5`];
|
|
788 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
789 |
val cl = LS[`~6 = negative6`];
|
|
790 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
791 |
val cl = LS[`~7 = negative7`];
|
|
792 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
793 |
val cl = LS[`~8 = negative8`];
|
|
794 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
795 |
val cl = LS[`~9 = negative9`];
|
|
796 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
797 |
val cl = LS[`~10 = negative10`];
|
|
798 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
799 |
|
|
800 |
(* Addition *)
|
|
801 |
val cl = LS[`0 + $x = $x`];
|
|
802 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
803 |
val cl = LS[`$x + $y = $y + $x`];
|
|
804 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
805 |
val cl = LS[`$x + ($y + $z) = ($x + $y) + $z`];
|
|
806 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
807 |
|
|
808 |
(* Negation *)
|
|
809 |
val cl = LS[`~$x + $x = 0`];
|
|
810 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
811 |
val cl = LS[`~~$x = $x`];
|
|
812 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
813 |
|
|
814 |
(* Subtraction *)
|
|
815 |
val cl = LS[`$x - $y = $x + ~$y`];
|
|
816 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
817 |
|
|
818 |
(* Successor *)
|
|
819 |
val cl = LS[`suc $x = $x + 1`];
|
|
820 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
821 |
|
|
822 |
(* Predecessor *)
|
|
823 |
val cl = LS[`pre $x = $x - 1`];
|
|
824 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
825 |
|
|
826 |
(* Ordering *)
|
|
827 |
val cl = LS[`$x <= $x`];
|
|
828 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
829 |
val cl = LS[`~($x <= $y)`,`~($y <= $z)`,`$x <= $z`];
|
|
830 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
831 |
val cl = LS[`~($x <= $y)`,`~($y <= $x)`,`$x = $y`];
|
|
832 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
833 |
val cl = LS[`0 <= $x`];
|
|
834 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
835 |
val cl = LS[`~($x >= $y)`,`$y <= $x`];
|
|
836 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
837 |
val cl = LS[`$x >= $y`,`~($y <= $x)`];
|
|
838 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
839 |
val cl = LS[`$x > $y`,`$x <= $y`];
|
|
840 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
841 |
val cl = LS[`~($x > $y)`,`~($x <= $y)`];
|
|
842 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
843 |
val cl = LS[`$x < $y`,`$y <= $x`];
|
|
844 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
845 |
val cl = LS[`~($x < $y)`,`~($y <= $x)`];
|
|
846 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
847 |
val cl = LS[`$x = 0`,`~($x <= $y)`,`~$y <= ~$x`];
|
|
848 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
849 |
|
|
850 |
(* Multiplication *)
|
|
851 |
val cl = LS[`1 * $x = $x`];
|
|
852 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
853 |
val cl = LS[`0 * $x = 0`];
|
|
854 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
855 |
val cl = LS[`$x * $y = $y * $x`];
|
|
856 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
857 |
val cl = LS[`$x * ($y * $z) = ($x * $y) * $z`];
|
|
858 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
859 |
val cl = LS[`$x * ($y + $z) = ($x * $y) + ($x * $z)`];
|
|
860 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
861 |
val cl = LS[`$x * ~$y = ~($x * $y)`];
|
|
862 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
863 |
|
|
864 |
(* Division *)
|
|
865 |
val cl = LS[`$y = 0`,`$x mod $y < $y`];
|
|
866 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
867 |
val cl = LS[`$y * ($x div $y) + $x mod $y = $x`];
|
|
868 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
869 |
|
|
870 |
(* Exponentiation *)
|
|
871 |
val cl = LS[`exp $x 0 = 1`];
|
|
872 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
873 |
val cl = LS[`$y = 0`,`exp $x $y = $x * exp $x (pre $y)`];
|
|
874 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
875 |
|
|
876 |
(* Divides *)
|
|
877 |
val cl = LS[`divides $x $x`];
|
|
878 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
879 |
val cl = LS[`~(divides $x $y)`,`~(divides $y $z)`,`divides $x $z`];
|
|
880 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
881 |
val cl = LS[`~(divides $x $y)`,`~(divides $y $x)`,`$x = $y`];
|
|
882 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
883 |
val cl = LS[`divides 1 $x`];
|
|
884 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
885 |
val cl = LS[`divides $x 0`];
|
|
886 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
887 |
|
|
888 |
(* Even and odd *)
|
|
889 |
val cl = LS[`even 0`];
|
|
890 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
891 |
val cl = LS[`$x = 0`,`~(even (pre $x))`,`odd $x`];
|
|
892 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
893 |
val cl = LS[`$x = 0`,`~(odd (pre $x))`,`even $x`];
|
|
894 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
895 |
|
|
896 |
(* Sets *)
|
|
897 |
|
|
898 |
(* The empty set *)
|
|
899 |
val cl = LS[`~member $x empty`];
|
|
900 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
901 |
|
|
902 |
(* The universal set *)
|
|
903 |
val cl = LS[`member $x universe`];
|
|
904 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
905 |
|
|
906 |
(* Complement *)
|
|
907 |
val cl = LS[`member $x $y`,`member $x (complement $y)`];
|
|
908 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
909 |
val cl = LS[`~(member $x $y)`,`~member $x (complement $y)`];
|
|
910 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
911 |
val cl = LS[`complement (complement $x) = $x`];
|
|
912 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
913 |
val cl = LS[`complement empty = universe`];
|
|
914 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
915 |
val cl = LS[`complement universe = empty`];
|
|
916 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
917 |
|
|
918 |
(* The subset relation *)
|
|
919 |
val cl = LS[`subset $x $x`];
|
|
920 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
921 |
val cl = LS[`~subset $x $y`,`~subset $y $z`,`subset $x $z`];
|
|
922 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
923 |
val cl = LS[`~subset $x $y`,`~subset $y $x`,`$x = $y`];
|
|
924 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
925 |
val cl = LS[`subset empty $x`];
|
|
926 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
927 |
val cl = LS[`subset $x universe`];
|
|
928 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
929 |
val cl = LS[`~subset $x $y`,`subset (complement $y) (complement $x)`];
|
|
930 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
931 |
val cl = LS[`~member $x $y`,`~subset $y $z`,`member $x $z`];
|
|
932 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
933 |
|
|
934 |
(* Union *)
|
|
935 |
val cl = LS[`union $x $y = union $y $x`];
|
|
936 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
937 |
val cl = LS[`union $x (union $y $z) = union (union $x $y) $z`];
|
|
938 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
939 |
val cl = LS[`union empty $x = $x`];
|
|
940 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
941 |
val cl = LS[`union universe $x = universe`];
|
|
942 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
943 |
val cl = LS[`subset $x (union $x $y)`];
|
|
944 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
945 |
val cl = LS[`~member $x (union $y $z)`,`member $x $y`,`member $x $z`];
|
|
946 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
947 |
|
|
948 |
(* Intersection *)
|
|
949 |
val cl = LS[`intersect $x $y =
|
|
950 |
complement (union (complement $x) (complement $y))`];
|
|
951 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
952 |
val cl = LS[`subset (intersect $x $y) $x`];
|
|
953 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
954 |
|
|
955 |
(* Difference *)
|
|
956 |
val cl = LS[`difference $x $y = intersect $x (complement $y)`];
|
|
957 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
958 |
|
|
959 |
(* Symmetric difference *)
|
|
960 |
val cl = LS[`symmetricDifference $x $y =
|
|
961 |
union (difference $x $y) (difference $y $x)`];
|
|
962 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
963 |
|
|
964 |
(* Insert *)
|
|
965 |
val cl = LS[`member $x (insert $x $y)`];
|
|
966 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
967 |
|
|
968 |
(* Singleton *)
|
|
969 |
val cl = LS[`singleton $x = (insert $x empty)`];
|
|
970 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
971 |
|
|
972 |
(* Cardinality *)
|
|
973 |
val cl = LS[`card empty = 0`];
|
|
974 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
975 |
val cl = LS[`member $x $y`,`card (insert $x $y) = suc (card $y)`];
|
|
976 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
977 |
|
|
978 |
(* Lists *)
|
|
979 |
|
|
980 |
(* Nil *)
|
|
981 |
val cl = LS[`null nil`];
|
|
982 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
983 |
val cl = LS[`~null $x`, `$x = nil`];
|
|
984 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
985 |
|
|
986 |
(* Cons *)
|
|
987 |
val cl = LS[`~(nil = $x :: $y)`];
|
|
988 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
989 |
|
|
990 |
(* Append *)
|
|
991 |
val cl = LS[`$x @ ($y @ $z) = ($x @ $y) @ $z`];
|
|
992 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
993 |
val cl = LS[`nil @ $x = $x`];
|
|
994 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
995 |
val cl = LS[`$x @ nil = $x`];
|
|
996 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
997 |
|
|
998 |
(* Length *)
|
|
999 |
val cl = LS[`length nil = 0`];
|
|
1000 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
1001 |
val cl = LS[`length ($x :: $y) >= length $y`];
|
|
1002 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
1003 |
val cl = LS[`length ($x @ $y) >= length $x`];
|
|
1004 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
1005 |
val cl = LS[`length ($x @ $y) >= length $y`];
|
|
1006 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
1007 |
|
|
1008 |
(* Tail *)
|
|
1009 |
val cl = LS[`null $x`,`suc (length (tail $x)) = length $x`];
|
|
1010 |
val _ = pvPCl (checkStandardModelClause cl);
|
|
1011 |
|
|
1012 |
(* ------------------------------------------------------------------------- *)
|
|
1013 |
val () = SAY "Clauses";
|
|
1014 |
(* ------------------------------------------------------------------------- *)
|
|
1015 |
|
|
1016 |
val cl = pvCl (CL[`P $x`,`P $y`]);
|
|
1017 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1018 |
val _ = pvCls (Clause.factor cl);
|
|
1019 |
val cl = pvCl (CL[`P $x`,`~P (f $x)`]);
|
|
1020 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1021 |
val cl = pvCl (CL[`$x = $y`,`f $y = f $x`]);
|
|
1022 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1023 |
val cl = pvCl (CL[`$x = f $y`,`f $x = $y`]);
|
|
1024 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1025 |
val cl = pvCl (CL[`s = a`,`s = b`,`h b c`]);
|
|
1026 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1027 |
val cl = pvCl (CL[`a = a`,`a = b`,`h b c`]);
|
|
1028 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1029 |
|
|
1030 |
(* Test cases contributed by Larry Paulson *)
|
|
1031 |
|
|
1032 |
local
|
|
1033 |
val lnFnName = Name.fromString "ln"
|
|
1034 |
and expFnName = Name.fromString "exp"
|
|
1035 |
and divFnName = Name.fromString "/"
|
|
1036 |
|
|
1037 |
val leRelName = Name.fromString "<";
|
|
1038 |
|
|
1039 |
fun weight na =
|
|
1040 |
case na of
|
|
1041 |
(n,1) =>
|
|
1042 |
if Name.equal n lnFnName then 500
|
|
1043 |
else if Name.equal n expFnName then 500
|
|
1044 |
else 1
|
|
1045 |
| (n,2) =>
|
|
1046 |
if Name.equal n divFnName then 50
|
|
1047 |
else if Name.equal n leRelName then 20
|
|
1048 |
else 1
|
|
1049 |
| _ => 1;
|
|
1050 |
|
|
1051 |
val ordering =
|
|
1052 |
{weight = weight, precedence = #precedence KnuthBendixOrder.default};
|
|
1053 |
|
|
1054 |
val clauseParameters =
|
|
1055 |
{ordering = ordering,
|
|
1056 |
orderLiterals = Clause.UnsignedLiteralOrder,
|
|
1057 |
orderTerms = true};
|
|
1058 |
in
|
|
1059 |
val LcpCL = mkCl clauseParameters o AX;
|
|
1060 |
end;
|
|
1061 |
|
|
1062 |
val cl = pvCl (LcpCL[`~($y <= (2 + (2 * $x + pow $x 2)) / 2)`, `~(0 <= $x)`,
|
|
1063 |
`$y <= exp $x`]);
|
|
1064 |
val _ = pvLits (Clause.largestLiterals cl);
|
|
1065 |
|
|
1066 |
(* ------------------------------------------------------------------------- *)
|
|
1067 |
val () = SAY "Syntax checking the problem sets";
|
|
1068 |
(* ------------------------------------------------------------------------- *)
|
|
1069 |
|
|
1070 |
local
|
|
1071 |
fun same n = raise Fail ("Two goals called " ^ n);
|
|
1072 |
|
|
1073 |
fun dup n n' =
|
|
1074 |
raise Fail ("Goal " ^ n' ^ " is probable duplicate of " ^ n);
|
|
1075 |
|
|
1076 |
fun quot fm =
|
|
1077 |
let
|
|
1078 |
fun f (v,s) = Subst.insert s (v,V"_")
|
|
1079 |
|
|
1080 |
val sub = NameSet.foldl f Subst.empty (Formula.freeVars fm)
|
|
1081 |
in
|
|
1082 |
Formula.subst sub fm
|
|
1083 |
end;
|
|
1084 |
|
|
1085 |
val quot_clauses =
|
|
1086 |
Formula.listMkConj o sort Formula.compare o
|
|
1087 |
map (quot o snd o Formula.stripForall) o Formula.stripConj;
|
|
1088 |
|
|
1089 |
fun quotient (Formula.Imp (a, Formula.Imp (b, Formula.False))) =
|
|
1090 |
Formula.Imp (quot_clauses a, Formula.Imp (quot_clauses b, Formula.False))
|
|
1091 |
| quotient fm = fm;
|
|
1092 |
|
|
1093 |
fun check ({name,goal,...}, acc) =
|
|
1094 |
let
|
|
1095 |
val g = prep goal
|
|
1096 |
val p =
|
|
1097 |
Formula.fromString g
|
|
1098 |
handle Parse.NoParse =>
|
|
1099 |
raise Error ("failed to parse problem " ^ name)
|
|
1100 |
|
|
1101 |
val () =
|
|
1102 |
case List.find (fn (n,_) => n = name) acc of NONE => ()
|
|
1103 |
| SOME _ => same name
|
|
1104 |
|
|
1105 |
val () =
|
|
1106 |
case List.find (fn (_,x) => Formula.equal x p) acc of NONE => ()
|
|
1107 |
| SOME (n,_) => dup n name
|
|
1108 |
|
|
1109 |
val _ =
|
|
1110 |
test_fun equal I g (mini_print (!Print.lineLength) p)
|
|
1111 |
handle e => (print ("Error in problem " ^ name ^ "\n\n"); raise e)
|
|
1112 |
in
|
|
1113 |
(name,p) :: acc
|
|
1114 |
end;
|
|
1115 |
in
|
|
1116 |
fun check_syntax (p : problem list) =
|
|
1117 |
(foldl check [] p; print "ok\n\n");
|
|
1118 |
end;
|
|
1119 |
|
|
1120 |
val () = check_syntax problems;
|
|
1121 |
|
|
1122 |
(* ------------------------------------------------------------------------- *)
|
|
1123 |
val () = SAY "Parsing TPTP problems";
|
|
1124 |
(* ------------------------------------------------------------------------- *)
|
|
1125 |
|
|
1126 |
fun tptp f =
|
|
1127 |
let
|
|
1128 |
val () = print ("parsing " ^ f ^ "... ")
|
|
1129 |
val filename = "tptp/" ^ f ^ ".tptp"
|
|
1130 |
val mapping = Tptp.defaultMapping
|
|
1131 |
val goal = Tptp.goal (Tptp.read {filename = filename, mapping = mapping})
|
|
1132 |
val () = print "ok\n"
|
|
1133 |
in
|
|
1134 |
pvFm goal
|
|
1135 |
end;
|
|
1136 |
|
|
1137 |
val _ = tptp "PUZ001-1";
|
|
1138 |
val _ = tptp "NUMBERED_FORMULAS";
|
|
1139 |
val _ = tptp "DEFINED_TERMS";
|
|
1140 |
val _ = tptp "SYSTEM_TERMS";
|
|
1141 |
val _ = tptp "QUOTED_TERMS";
|
|
1142 |
val _ = tptp "QUOTED_TERMS_IDENTITY";
|
|
1143 |
val _ = tptp "QUOTED_TERMS_DIFFERENT";
|
|
1144 |
val _ = tptp "QUOTED_TERMS_SPECIAL";
|
|
1145 |
val _ = tptp "RENAMING_VARIABLES";
|
|
1146 |
val _ = tptp "MIXED_PROBLEM";
|
|
1147 |
val _ = tptp "BLOCK_COMMENTS";
|
|
1148 |
|
|
1149 |
(* ------------------------------------------------------------------------- *)
|
|
1150 |
val () = SAY "The TPTP finite model";
|
|
1151 |
(* ------------------------------------------------------------------------- *)
|
|
1152 |
|
|
1153 |
val _ = printval (Tptp.ppFixedMap Tptp.defaultMapping) Tptp.defaultFixedMap;
|