TSTP Solution File: TOP049-1 by Toma---0.4
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : TOP049-1 : TPTP v8.1.2. Released v8.1.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 1 05:59:21 EDT 2023
% Result : Unsatisfiable 1.55s 1.86s
% Output : CNFRefutation 1.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : TOP049-1 : TPTP v8.1.2. Released v8.1.0.
% 0.07/0.14 % Command : toma --casc %s
% 0.14/0.35 % Computer : n011.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Aug 26 23:23:09 EDT 2023
% 0.14/0.36 % CPUTime :
% 1.55/1.86 % SZS status Unsatisfiable
% 1.55/1.86 % SZS output start Proof
% 1.55/1.86 original problem:
% 1.55/1.86 axioms:
% 1.55/1.86 product(X, X) = X
% 1.55/1.86 product(product(X, Y), Y) = X
% 1.55/1.86 product(product(X, Y), Z) = product(product(X, Z), product(Y, Z))
% 1.55/1.86 a1() = product(a9(), a7())
% 1.55/1.86 a3() = product(a1(), a2())
% 1.55/1.86 a2() = product(a3(), a4())
% 1.55/1.86 a5() = product(a2(), a10())
% 1.55/1.86 a6() = product(a5(), a4())
% 1.55/1.86 a7() = product(a6(), a1())
% 1.55/1.86 a8() = product(a7(), a4())
% 1.55/1.86 a10() = product(a8(), a9())
% 1.55/1.86 a4() = product(a10(), a3())
% 1.55/1.86 a9() = product(a4(), a8())
% 1.55/1.86 goal:
% 1.55/1.86 tuple(a1(), a9(), a8(), a6(), a7(), a2(), a3(), a4(), a5()) != tuple(a2(), a10(), a9(), a7(), a8(), a3(), a4(), a5(), a6())
% 1.55/1.86 To show the unsatisfiability of the original goal,
% 1.55/1.86 it suffices to show that tuple(a1(), a9(), a8(), a6(), a7(), a2(), a3(), a4(), a5()) = tuple(a2(), a10(), a9(), a7(), a8(), a3(), a4(), a5(), a6()) (skolemized goal) is valid under the axioms.
% 1.55/1.86 Here is an equational proof:
% 1.55/1.86 0: product(X0, X0) = X0.
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 1: product(product(X0, X1), X1) = X0.
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 2: product(product(X0, X1), X2) = product(product(X0, X2), product(X1, X2)).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 3: a1() = product(a9(), a7()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 4: a3() = product(a1(), a2()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 5: a2() = product(a3(), a4()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 6: a5() = product(a2(), a10()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 7: a6() = product(a5(), a4()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 8: a7() = product(a6(), a1()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 9: a8() = product(a7(), a4()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 10: a10() = product(a8(), a9()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 11: a4() = product(a10(), a3()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 12: a9() = product(a4(), a8()).
% 1.55/1.86 Proof: Axiom.
% 1.55/1.86
% 1.55/1.86 13: a1() = product(product(a4(), a7()), product(product(a7(), a4()), a7())).
% 1.55/1.86 Proof: Rewrite equation 3,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [12,9,2].
% 1.55/1.86
% 1.55/1.86 14: a3() = product(product(product(a4(), a7()), product(product(a7(), a4()), a7())), a2()).
% 1.55/1.86 Proof: Rewrite equation 4,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [13].
% 1.55/1.86
% 1.55/1.86 15: a2() = product(product(product(a4(), a7()), product(a7(), a4())), product(a2(), a4())).
% 1.55/1.86 Proof: Rewrite equation 5,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [14,2,2,2,0,2,1,2].
% 1.55/1.86
% 1.55/1.86 16: a5() = product(a2(), product(product(a7(), a4()), product(a4(), product(a7(), a4())))).
% 1.55/1.86 Proof: Rewrite equation 6,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [10,9,12,9].
% 1.55/1.86
% 1.55/1.86 17: a6() = product(product(a2(), a4()), product(a7(), product(a4(), a7()))).
% 1.55/1.86 Proof: Rewrite equation 7,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [16,2,2,1,2,0,1].
% 1.55/1.86
% 1.55/1.86 18: a7() = product(product(product(a2(), a4()), product(a7(), product(a4(), a7()))), product(product(a4(), a7()), product(product(a7(), a4()), a7()))).
% 1.55/1.86 Proof: Rewrite equation 8,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [17,13].
% 1.55/1.86
% 1.55/1.86 19: a10() = product(product(a7(), a4()), product(a4(), product(a7(), a4()))).
% 1.55/1.86 Proof: Rewrite equation 10,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,12,9].
% 1.55/1.86
% 1.55/1.86 20: a4() = product(product(product(a7(), a4()), product(a4(), product(a7(), a4()))), product(product(product(a4(), a7()), product(product(a7(), a4()), a7())), a2())).
% 1.55/1.86 Proof: Rewrite equation 11,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [19,14].
% 1.55/1.86
% 1.55/1.86 21: a9() = product(a4(), product(a7(), a4())).
% 1.55/1.86 Proof: Rewrite equation 12,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9].
% 1.55/1.86
% 1.55/1.86 22: product(product(X3, X1), X3) = product(X3, product(X1, X3)).
% 1.55/1.86 Proof: A critical pair between equations 2 and 0.
% 1.55/1.86
% 1.55/1.86 23: product(product(product(X3, X4), X1), X4) = product(X3, product(X1, X4)).
% 1.55/1.86 Proof: A critical pair between equations 2 and 1.
% 1.55/1.86
% 1.55/1.86 28: product(product(product(X3, X4), X2), product(product(X5, X4), X2)) = product(product(product(X3, X5), X4), X2).
% 1.55/1.86 Proof: A critical pair between equations 2 and 2.
% 1.55/1.86
% 1.55/1.86 30: product(product(product(product(a4(), a7()), product(a7(), a4())), X2), product(product(a2(), a4()), X2)) = product(a2(), X2).
% 1.55/1.86 Proof: A critical pair between equations 2 and 15.
% 1.55/1.86
% 1.55/1.86 35: product(product(product(product(a4(), a7()), a8()), X2), product(product(a2(), a4()), X2)) = product(a2(), X2).
% 1.55/1.86 Proof: Rewrite equation 30,
% 1.55/1.86 lhs with equations [9]
% 1.55/1.86 rhs with equations [].
% 1.55/1.86
% 1.55/1.86 37: a9() = product(a4(), a8()).
% 1.55/1.86 Proof: Rewrite equation 21,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9].
% 1.55/1.86
% 1.55/1.86 38: a4() = product(product(a8(), product(a4(), a8())), product(product(product(a4(), a7()), product(a8(), a7())), a2())).
% 1.55/1.86 Proof: Rewrite equation 20,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,9,9].
% 1.55/1.86
% 1.55/1.86 39: a10() = product(a8(), product(a4(), a8())).
% 1.55/1.86 Proof: Rewrite equation 19,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,9].
% 1.55/1.86
% 1.55/1.86 40: a7() = product(a6(), product(product(a4(), a7()), product(a8(), a7()))).
% 1.55/1.86 Proof: Rewrite equation 18,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [17,9].
% 1.55/1.86
% 1.55/1.86 41: a5() = product(a2(), product(a8(), product(a4(), a8()))).
% 1.55/1.86 Proof: Rewrite equation 16,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,9].
% 1.55/1.86
% 1.55/1.86 42: a2() = product(product(product(a4(), a7()), a8()), product(a2(), a4())).
% 1.55/1.86 Proof: Rewrite equation 15,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9].
% 1.55/1.86
% 1.55/1.86 43: a3() = product(product(product(a4(), a7()), product(a8(), a7())), a2()).
% 1.55/1.86 Proof: Rewrite equation 14,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9].
% 1.55/1.86
% 1.55/1.86 44: a1() = product(product(a4(), a7()), product(a8(), a7())).
% 1.55/1.86 Proof: Rewrite equation 13,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9].
% 1.55/1.86
% 1.55/1.86 53: a6() = product(a7(), product(product(a4(), a7()), product(a8(), a7()))).
% 1.55/1.86 Proof: A critical pair between equations 1 and 40.
% 1.55/1.86
% 1.55/1.86 54: product(a2(), a4()) = product(a6(), product(a7(), product(a4(), a7()))).
% 1.55/1.86 Proof: A critical pair between equations 1 and 17.
% 1.55/1.86
% 1.55/1.86 55: product(a2(), a4()) = product(product(product(product(a4(), a7()), a8()), a4()), a2()).
% 1.55/1.86 Proof: A critical pair between equations 35 and 1.
% 1.55/1.86
% 1.55/1.86 58: product(a7(), a4()) = product(product(product(a4(), a7()), product(a7(), product(a4(), a7()))), a2()).
% 1.55/1.86 Proof: Rewrite equation 55,
% 1.55/1.86 lhs with equations [54,53,9,22,2,1,2,1,22,1]
% 1.55/1.86 rhs with equations [9,2,22,1,2,22].
% 1.55/1.86
% 1.55/1.86 59: product(a2(), a4()) = product(a7(), a4()).
% 1.55/1.86 Proof: Rewrite equation 54,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [53,9,22,2,1,2,1,22,1].
% 1.55/1.86
% 1.55/1.86 60: a6() = product(a7(), product(product(a4(), a7()), product(a7(), product(a4(), a7())))).
% 1.55/1.86 Proof: Rewrite equation 53,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,22].
% 1.55/1.86
% 1.55/1.86 61: a1() = product(product(a4(), a7()), product(a7(), product(a4(), a7()))).
% 1.55/1.86 Proof: Rewrite equation 44,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,22].
% 1.55/1.86
% 1.55/1.86 62: a3() = product(a7(), a4()).
% 1.55/1.86 Proof: Rewrite equation 43,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,22,58].
% 1.55/1.86
% 1.55/1.86 63: a2() = product(a4(), a7()).
% 1.55/1.86 Proof: Rewrite equation 42,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,59,1].
% 1.55/1.86
% 1.55/1.86 64: a5() = product(product(a4(), a7()), product(product(a7(), a4()), product(a4(), product(a7(), a4())))).
% 1.55/1.86 Proof: Rewrite equation 41,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [63,9,9].
% 1.55/1.86
% 1.55/1.86 66: a10() = product(product(a7(), a4()), product(a4(), product(a7(), a4()))).
% 1.55/1.86 Proof: Rewrite equation 39,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,9].
% 1.55/1.86
% 1.55/1.86 67: a4() = product(a7(), product(a4(), a7())).
% 1.55/1.86 Proof: Rewrite equation 38,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9,9,9,22,63,22,1,1,23,22,1].
% 1.55/1.86
% 1.55/1.86 68: a9() = product(a4(), product(a7(), a4())).
% 1.55/1.86 Proof: Rewrite equation 37,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [9].
% 1.55/1.86
% 1.55/1.86 70: product(a4(), X2) = product(product(a4(), a7()), X2).
% 1.55/1.86 Proof: Rewrite equation 35,
% 1.55/1.86 lhs with equations [9,63,22,28,22,1]
% 1.55/1.86 rhs with equations [63].
% 1.55/1.86
% 1.55/1.86 74: a4() = product(a4(), a7()).
% 1.55/1.86 Proof: A critical pair between equations 1 and 70.
% 1.55/1.86
% 1.55/1.86 76: a7() = product(a4(), product(a4(), a7())).
% 1.55/1.86 Proof: A critical pair between equations 1 and 67.
% 1.55/1.86
% 1.55/1.86 85: product(a7(), product(product(a4(), a7()), a7())) = product(a4(), a7()).
% 1.55/1.86 Proof: A critical pair between equations 22 and 67.
% 1.55/1.86
% 1.55/1.86 86: a8() = a4().
% 1.55/1.86 Proof: Rewrite equation 85,
% 1.55/1.86 lhs with equations [74,74,9]
% 1.55/1.86 rhs with equations [74].
% 1.55/1.86
% 1.55/1.86 87: a7() = a4().
% 1.55/1.86 Proof: Rewrite equation 76,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [74,0].
% 1.55/1.86
% 1.55/1.86 88: a9() = a4().
% 1.55/1.86 Proof: Rewrite equation 68,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,0,0].
% 1.55/1.86
% 1.55/1.86 89: a10() = a4().
% 1.55/1.86 Proof: Rewrite equation 66,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,0,87,0,0,0].
% 1.55/1.86
% 1.55/1.86 90: a5() = a4().
% 1.55/1.86 Proof: Rewrite equation 64,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,0,87,0,87,0,0,0,0].
% 1.55/1.86
% 1.55/1.86 91: a2() = a4().
% 1.55/1.86 Proof: Rewrite equation 63,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,0].
% 1.55/1.86
% 1.55/1.86 92: a3() = a4().
% 1.55/1.86 Proof: Rewrite equation 62,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,0].
% 1.55/1.86
% 1.55/1.86 93: a1() = a4().
% 1.55/1.86 Proof: Rewrite equation 61,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,0,87,87,0,0,0].
% 1.55/1.86
% 1.55/1.86 94: a6() = a4().
% 1.55/1.86 Proof: Rewrite equation 60,
% 1.55/1.86 lhs with equations []
% 1.55/1.86 rhs with equations [87,87,0,87,87,0,0,0,0].
% 1.55/1.86
% 1.55/1.86 95: tuple(a1(), a9(), a8(), a6(), a7(), a2(), a3(), a4(), a5()) = tuple(a2(), a10(), a9(), a7(), a8(), a3(), a4(), a5(), a6()).
% 1.55/1.86 Proof: Rewrite lhs with equations [93,88,86,94,87,91,92,90]
% 1.55/1.86 rhs with equations [91,89,88,87,86,92,90,94].
% 1.55/1.86
% 1.55/1.86 % SZS output end Proof
%------------------------------------------------------------------------------