TSTP Solution File: HEN010-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN010-2 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n022.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 : Thu Aug 31 01:57:01 EDT 2023
% Result : Unsatisfiable 6.57s 1.22s
% Output : Proof 6.57s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HEN010-2 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n022.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Aug 24 13:31:10 EDT 2023
% 0.12/0.34 % CPUTime :
% 6.57/1.22 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 6.57/1.22
% 6.57/1.22 % SZS status Unsatisfiable
% 6.57/1.22
% 6.57/1.23 % SZS output start Proof
% 6.57/1.23 Take the following subset of the input axioms:
% 6.57/1.23 fof(closure, axiom, ![X, Y]: quotient(X, Y, divide(X, Y))).
% 6.57/1.23 fof(divisor_existence, axiom, ![Z, X2, Y4]: (~quotient(X2, Y4, Z) | less_equal(Z, X2))).
% 6.57/1.23 fof(identity_divide_a, hypothesis, quotient(identity, a, idQa)).
% 6.57/1.23 fof(identity_divide_idQ_idQa, hypothesis, quotient(idQa, idQ_idQa, idQa_Q__idQ_idQa)).
% 6.57/1.23 fof(identity_divide_idQa, hypothesis, quotient(identity, idQa, idQ_idQa)).
% 6.57/1.23 fof(less_equal_and_equal, axiom, ![X2, Y4]: (~less_equal(X2, Y4) | (~less_equal(Y4, X2) | X2=Y4))).
% 6.57/1.23 fof(one_inversion_equals_three, axiom, ![Y1, Y2, Y3, X2]: (~quotient(identity, X2, Y1) | (~quotient(identity, Y1, Y2) | (~quotient(identity, Y2, Y3) | Y1=Y3)))).
% 6.57/1.23 fof(prove_idQa_equals_idQa_Q__idQ_idQa, negated_conjecture, idQa!=idQa_Q__idQ_idQa).
% 6.57/1.23 fof(quotient_property, axiom, ![V1, V2, V3, V4, V5, X2, Y4, Z2]: (~quotient(X2, Y4, V1) | (~quotient(Y4, Z2, V2) | (~quotient(X2, Z2, V3) | (~quotient(V3, V2, V4) | (~quotient(V1, Z2, V5) | less_equal(V4, V5))))))).
% 6.57/1.23 fof(well_defined, axiom, ![W, X2, Y4, Z2]: (~quotient(X2, Y4, Z2) | (~quotient(X2, Y4, W) | Z2=W))).
% 6.57/1.23 fof(xLEy_implies_zQyLEzQx, axiom, ![W1, W2, X2, Y4, Z2]: (~less_equal(X2, Y4) | (~quotient(Z2, Y4, W1) | (~quotient(Z2, X2, W2) | less_equal(W1, W2))))).
% 6.57/1.23 fof(x_divde_zero_is_x, axiom, ![X2]: quotient(X2, zero, X2)).
% 6.57/1.23 fof(x_divide_x_is_zero, axiom, ![X2]: quotient(X2, X2, zero)).
% 6.57/1.23
% 6.57/1.23 Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.57/1.23 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.57/1.23 We repeatedly replace C & s=t => u=v by the two clauses:
% 6.57/1.23 fresh(y, y, x1...xn) = u
% 6.57/1.23 C => fresh(s, t, x1...xn) = v
% 6.57/1.23 where fresh is a fresh function symbol and x1..xn are the free
% 6.57/1.23 variables of u and v.
% 6.57/1.23 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.57/1.23 input problem has no model of domain size 1).
% 6.57/1.23
% 6.57/1.23 The encoding turns the above axioms into the following unit equations and goals:
% 6.57/1.23
% 6.57/1.23 Axiom 1 (x_divide_x_is_zero): quotient(X, X, zero) = true.
% 6.57/1.23 Axiom 2 (x_divde_zero_is_x): quotient(X, zero, X) = true.
% 6.57/1.23 Axiom 3 (identity_divide_idQa): quotient(identity, idQa, idQ_idQa) = true.
% 6.57/1.23 Axiom 4 (identity_divide_a): quotient(identity, a, idQa) = true.
% 6.57/1.23 Axiom 5 (identity_divide_idQ_idQa): quotient(idQa, idQ_idQa, idQa_Q__idQ_idQa) = true.
% 6.57/1.23 Axiom 6 (quotient_property): fresh26(X, X, Y, Z) = true.
% 6.57/1.23 Axiom 7 (xLEy_implies_zQyLEzQx): fresh19(X, X, Y, Z) = true.
% 6.57/1.23 Axiom 8 (one_inversion_equals_three): fresh15(X, X, Y, Z) = Z.
% 6.57/1.23 Axiom 9 (divisor_existence): fresh12(X, X, Y, Z) = true.
% 6.57/1.23 Axiom 10 (less_equal_and_equal): fresh5(X, X, Y, Z) = Y.
% 6.57/1.23 Axiom 11 (less_equal_and_equal): fresh4(X, X, Y, Z) = Z.
% 6.57/1.23 Axiom 12 (well_defined): fresh2(X, X, Y, Z) = Z.
% 6.57/1.23 Axiom 13 (closure): quotient(X, Y, divide(X, Y)) = true.
% 6.57/1.23 Axiom 14 (one_inversion_equals_three): fresh(X, X, Y, Z, W) = Z.
% 6.57/1.23 Axiom 15 (xLEy_implies_zQyLEzQx): fresh7(X, X, Y, Z, W, V) = less_equal(W, V).
% 6.57/1.23 Axiom 16 (less_equal_and_equal): fresh5(less_equal(X, Y), true, Y, X) = fresh4(less_equal(Y, X), true, Y, X).
% 6.57/1.23 Axiom 17 (well_defined): fresh3(X, X, Y, Z, W, V) = W.
% 6.57/1.23 Axiom 18 (quotient_property): fresh24(X, X, Y, Z, W, V, U) = less_equal(V, U).
% 6.57/1.23 Axiom 19 (xLEy_implies_zQyLEzQx): fresh18(X, X, Y, Z, W, V, U) = fresh19(less_equal(Y, Z), true, V, U).
% 6.57/1.23 Axiom 20 (one_inversion_equals_three): fresh14(X, X, Y, Z, W, V) = fresh15(quotient(identity, Y, Z), true, Z, V).
% 6.57/1.23 Axiom 21 (divisor_existence): fresh12(quotient(X, Y, Z), true, X, Z) = less_equal(Z, X).
% 6.57/1.23 Axiom 22 (quotient_property): fresh25(X, X, Y, Z, W, V, U, T, S) = fresh26(quotient(Y, Z, W), true, T, S).
% 6.57/1.23 Axiom 23 (one_inversion_equals_three): fresh14(quotient(identity, X, Y), true, Z, W, X, Y) = fresh(quotient(identity, W, X), true, Z, W, Y).
% 6.57/1.23 Axiom 24 (well_defined): fresh3(quotient(X, Y, Z), true, X, Y, W, Z) = fresh2(quotient(X, Y, W), true, W, Z).
% 6.57/1.23 Axiom 25 (quotient_property): fresh23(X, X, Y, Z, W, V, U, T, S, X2) = fresh24(quotient(Y, V, T), true, Y, Z, W, S, X2).
% 6.57/1.23 Axiom 26 (xLEy_implies_zQyLEzQx): fresh18(quotient(X, Y, Z), true, W, Y, X, Z, V) = fresh7(quotient(X, W, V), true, W, Y, Z, V).
% 6.57/1.23 Axiom 27 (quotient_property): fresh22(X, X, Y, Z, W, V, U, T, S, X2) = fresh25(quotient(Z, V, U), true, Y, Z, W, V, T, S, X2).
% 6.57/1.23 Axiom 28 (quotient_property): fresh22(quotient(X, Y, Z), true, W, V, U, T, Y, X, Z, S) = fresh23(quotient(U, T, S), true, W, V, U, T, Y, X, Z, S).
% 6.57/1.23
% 6.57/1.23 Lemma 29: divide(X, X) = zero.
% 6.57/1.23 Proof:
% 6.57/1.23 divide(X, X)
% 6.57/1.23 = { by axiom 17 (well_defined) R->L }
% 6.57/1.23 fresh3(true, true, X, X, divide(X, X), zero)
% 6.57/1.23 = { by axiom 1 (x_divide_x_is_zero) R->L }
% 6.57/1.23 fresh3(quotient(X, X, zero), true, X, X, divide(X, X), zero)
% 6.57/1.23 = { by axiom 24 (well_defined) }
% 6.57/1.23 fresh2(quotient(X, X, divide(X, X)), true, divide(X, X), zero)
% 6.57/1.23 = { by axiom 13 (closure) }
% 6.57/1.23 fresh2(true, true, divide(X, X), zero)
% 6.57/1.23 = { by axiom 12 (well_defined) }
% 6.57/1.23 zero
% 6.57/1.23
% 6.57/1.23 Lemma 30: divide(X, zero) = X.
% 6.57/1.23 Proof:
% 6.57/1.23 divide(X, zero)
% 6.57/1.23 = { by axiom 17 (well_defined) R->L }
% 6.57/1.23 fresh3(true, true, X, zero, divide(X, zero), X)
% 6.57/1.23 = { by axiom 2 (x_divde_zero_is_x) R->L }
% 6.57/1.23 fresh3(quotient(X, zero, X), true, X, zero, divide(X, zero), X)
% 6.57/1.23 = { by axiom 24 (well_defined) }
% 6.57/1.23 fresh2(quotient(X, zero, divide(X, zero)), true, divide(X, zero), X)
% 6.57/1.23 = { by axiom 13 (closure) }
% 6.57/1.23 fresh2(true, true, divide(X, zero), X)
% 6.57/1.23 = { by axiom 12 (well_defined) }
% 6.57/1.23 X
% 6.57/1.23
% 6.57/1.23 Lemma 31: divide(identity, idQ_idQa) = idQa.
% 6.57/1.23 Proof:
% 6.57/1.23 divide(identity, idQ_idQa)
% 6.57/1.23 = { by axiom 8 (one_inversion_equals_three) R->L }
% 6.57/1.23 fresh15(true, true, idQa, divide(identity, idQ_idQa))
% 6.57/1.23 = { by axiom 4 (identity_divide_a) R->L }
% 6.57/1.23 fresh15(quotient(identity, a, idQa), true, idQa, divide(identity, idQ_idQa))
% 6.57/1.23 = { by axiom 20 (one_inversion_equals_three) R->L }
% 6.57/1.23 fresh14(true, true, a, idQa, idQ_idQa, divide(identity, idQ_idQa))
% 6.57/1.23 = { by axiom 13 (closure) R->L }
% 6.57/1.23 fresh14(quotient(identity, idQ_idQa, divide(identity, idQ_idQa)), true, a, idQa, idQ_idQa, divide(identity, idQ_idQa))
% 6.57/1.23 = { by axiom 23 (one_inversion_equals_three) }
% 6.57/1.23 fresh(quotient(identity, idQa, idQ_idQa), true, a, idQa, divide(identity, idQ_idQa))
% 6.57/1.23 = { by axiom 3 (identity_divide_idQa) }
% 6.57/1.23 fresh(true, true, a, idQa, divide(identity, idQ_idQa))
% 6.57/1.23 = { by axiom 14 (one_inversion_equals_three) }
% 6.57/1.23 idQa
% 6.57/1.23
% 6.57/1.23 Goal 1 (prove_idQa_equals_idQa_Q__idQ_idQa): idQa = idQa_Q__idQ_idQa.
% 6.57/1.23 Proof:
% 6.57/1.23 idQa
% 6.57/1.23 = { by lemma 30 R->L }
% 6.57/1.23 divide(idQa, zero)
% 6.57/1.23 = { by lemma 29 R->L }
% 6.57/1.23 divide(idQa, divide(idQ_idQa, idQ_idQa))
% 6.57/1.23 = { by axiom 11 (less_equal_and_equal) R->L }
% 6.57/1.23 fresh4(true, true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 7 (xLEy_implies_zQyLEzQx) R->L }
% 6.57/1.23 fresh4(fresh19(true, true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 9 (divisor_existence) R->L }
% 6.57/1.23 fresh4(fresh19(fresh12(true, true, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 13 (closure) R->L }
% 6.57/1.23 fresh4(fresh19(fresh12(quotient(idQ_idQa, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, idQ_idQa, divide(idQ_idQa, idQ_idQa)), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 21 (divisor_existence) }
% 6.57/1.23 fresh4(fresh19(less_equal(divide(idQ_idQa, idQ_idQa), idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 19 (xLEy_implies_zQyLEzQx) R->L }
% 6.57/1.23 fresh4(fresh18(true, true, divide(idQ_idQa, idQ_idQa), idQ_idQa, idQa, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 5 (identity_divide_idQ_idQa) R->L }
% 6.57/1.23 fresh4(fresh18(quotient(idQa, idQ_idQa, idQa_Q__idQ_idQa), true, divide(idQ_idQa, idQ_idQa), idQ_idQa, idQa, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 26 (xLEy_implies_zQyLEzQx) }
% 6.57/1.23 fresh4(fresh7(quotient(idQa, divide(idQ_idQa, idQ_idQa), divide(idQa, divide(idQ_idQa, idQ_idQa))), true, divide(idQ_idQa, idQ_idQa), idQ_idQa, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 13 (closure) }
% 6.57/1.23 fresh4(fresh7(true, true, divide(idQ_idQa, idQ_idQa), idQ_idQa, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 15 (xLEy_implies_zQyLEzQx) }
% 6.57/1.23 fresh4(less_equal(idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa))), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 16 (less_equal_and_equal) R->L }
% 6.57/1.23 fresh5(less_equal(divide(idQa, divide(idQ_idQa, idQ_idQa)), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by lemma 29 }
% 6.57/1.23 fresh5(less_equal(divide(idQa, zero), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by lemma 30 }
% 6.57/1.23 fresh5(less_equal(idQa, idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by lemma 31 R->L }
% 6.57/1.23 fresh5(less_equal(divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 18 (quotient_property) R->L }
% 6.57/1.23 fresh5(fresh24(true, true, identity, idQ_idQa, idQa, divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 13 (closure) R->L }
% 6.57/1.23 fresh5(fresh24(quotient(identity, idQ_idQa, divide(identity, idQ_idQa)), true, identity, idQ_idQa, idQa, divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 25 (quotient_property) R->L }
% 6.57/1.23 fresh5(fresh23(true, true, identity, idQ_idQa, idQa, idQ_idQa, zero, divide(identity, idQ_idQa), divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 5 (identity_divide_idQ_idQa) R->L }
% 6.57/1.23 fresh5(fresh23(quotient(idQa, idQ_idQa, idQa_Q__idQ_idQa), true, identity, idQ_idQa, idQa, idQ_idQa, zero, divide(identity, idQ_idQa), divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 28 (quotient_property) R->L }
% 6.57/1.23 fresh5(fresh22(quotient(divide(identity, idQ_idQa), zero, divide(identity, idQ_idQa)), true, identity, idQ_idQa, idQa, idQ_idQa, zero, divide(identity, idQ_idQa), divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 2 (x_divde_zero_is_x) }
% 6.57/1.23 fresh5(fresh22(true, true, identity, idQ_idQa, idQa, idQ_idQa, zero, divide(identity, idQ_idQa), divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 27 (quotient_property) }
% 6.57/1.23 fresh5(fresh25(quotient(idQ_idQa, idQ_idQa, zero), true, identity, idQ_idQa, idQa, idQ_idQa, divide(identity, idQ_idQa), divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 1 (x_divide_x_is_zero) }
% 6.57/1.23 fresh5(fresh25(true, true, identity, idQ_idQa, idQa, idQ_idQa, divide(identity, idQ_idQa), divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 22 (quotient_property) }
% 6.57/1.23 fresh5(fresh26(quotient(identity, idQ_idQa, idQa), true, divide(identity, idQ_idQa), idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by lemma 31 }
% 6.57/1.23 fresh5(fresh26(quotient(identity, idQ_idQa, idQa), true, idQa, idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by lemma 31 R->L }
% 6.57/1.23 fresh5(fresh26(quotient(identity, idQ_idQa, divide(identity, idQ_idQa)), true, idQa, idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 13 (closure) }
% 6.57/1.23 fresh5(fresh26(true, true, idQa, idQa_Q__idQ_idQa), true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 6 (quotient_property) }
% 6.57/1.23 fresh5(true, true, idQa_Q__idQ_idQa, divide(idQa, divide(idQ_idQa, idQ_idQa)))
% 6.57/1.23 = { by axiom 10 (less_equal_and_equal) }
% 6.57/1.23 idQa_Q__idQ_idQa
% 6.57/1.23 % SZS output end Proof
% 6.57/1.23
% 6.57/1.23 RESULT: Unsatisfiable (the axioms are contradictory).
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