TSTP Solution File: HEN008-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN008-1 : 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 : 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 : Thu Aug 31 01:56:58 EDT 2023
% Result : Unsatisfiable 0.20s 0.50s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : HEN008-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13 % 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 : n011.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 13:37:52 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.50 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.50
% 0.20/0.50 % SZS status Unsatisfiable
% 0.20/0.50
% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 Take the following subset of the input axioms:
% 0.20/0.51 fof(aLEb, hypothesis, less_equal(a, b)).
% 0.20/0.51 fof(aQc, hypothesis, quotient(a, c, aQc)).
% 0.20/0.51 fof(bQc, hypothesis, quotient(b, c, bQc)).
% 0.20/0.51 fof(closure, axiom, ![X, Y]: quotient(X, Y, divide(X, Y))).
% 0.20/0.51 fof(less_equal_and_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | (~less_equal(Y2, X2) | X2=Y2))).
% 0.20/0.51 fof(less_equal_quotient, axiom, ![X2, Y2]: (~quotient(X2, Y2, zero) | less_equal(X2, Y2))).
% 0.20/0.51 fof(prove_aQcLEbQc, negated_conjecture, ~less_equal(aQc, bQc)).
% 0.20/0.51 fof(quotient_less_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | quotient(X2, Y2, zero))).
% 0.20/0.51 fof(quotient_property, axiom, ![Z, V1, V2, V3, V4, V5, X2, Y2]: (~quotient(X2, Y2, V1) | (~quotient(Y2, Z, V2) | (~quotient(X2, Z, V3) | (~quotient(V3, V2, V4) | (~quotient(V1, Z, V5) | less_equal(V4, V5))))))).
% 0.20/0.51 fof(well_defined, axiom, ![W, X2, Y2, Z2]: (~quotient(X2, Y2, Z2) | (~quotient(X2, Y2, W) | Z2=W))).
% 0.20/0.51 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (zero_is_smallest): less_equal(zero, X) = true.
% 0.20/0.51 Axiom 2 (aLEb): less_equal(a, b) = true.
% 0.20/0.51 Axiom 3 (aQc): quotient(a, c, aQc) = true.
% 0.20/0.51 Axiom 4 (bQc): quotient(b, c, bQc) = true.
% 0.20/0.51 Axiom 5 (well_defined): fresh(X, X, Y, Z) = Z.
% 0.20/0.51 Axiom 6 (quotient_property): fresh12(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 7 (less_equal_quotient): fresh7(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 8 (quotient_less_equal): fresh5(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 9 (less_equal_and_equal): fresh4(X, X, Y, Z) = Y.
% 0.20/0.51 Axiom 10 (less_equal_and_equal): fresh3(X, X, Y, Z) = Z.
% 0.20/0.51 Axiom 11 (closure): quotient(X, Y, divide(X, Y)) = true.
% 0.20/0.51 Axiom 12 (quotient_less_equal): fresh5(less_equal(X, Y), true, X, Y) = quotient(X, Y, zero).
% 0.20/0.51 Axiom 13 (less_equal_and_equal): fresh4(less_equal(X, Y), true, Y, X) = fresh3(less_equal(Y, X), true, Y, X).
% 0.20/0.51 Axiom 14 (well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.51 Axiom 15 (quotient_property): fresh10(X, X, Y, Z, W, V, U) = less_equal(V, U).
% 0.20/0.51 Axiom 16 (less_equal_quotient): fresh7(quotient(X, Y, zero), true, X, Y) = less_equal(X, Y).
% 0.20/0.51 Axiom 17 (quotient_property): fresh11(X, X, Y, Z, W, V, U, T, S) = fresh12(quotient(Y, Z, W), true, T, S).
% 0.20/0.51 Axiom 18 (well_defined): fresh2(quotient(X, Y, Z), true, X, Y, W, Z) = fresh(quotient(X, Y, W), true, W, Z).
% 0.20/0.51 Axiom 19 (quotient_property): fresh9(X, X, Y, Z, W, V, U, T, S, X2) = fresh10(quotient(Y, V, T), true, Y, Z, W, S, X2).
% 0.20/0.51 Axiom 20 (quotient_property): fresh8(X, X, Y, Z, W, V, U, T, S, X2) = fresh11(quotient(Z, V, U), true, Y, Z, W, V, T, S, X2).
% 0.20/0.51 Axiom 21 (quotient_property): fresh8(quotient(X, Y, Z), true, W, V, U, T, Y, X, Z, S) = fresh9(quotient(U, T, S), true, W, V, U, T, Y, X, Z, S).
% 0.20/0.51
% 0.20/0.51 Goal 1 (prove_aQcLEbQc): less_equal(aQc, bQc) = true.
% 0.20/0.51 Proof:
% 0.20/0.51 less_equal(aQc, bQc)
% 0.20/0.51 = { by axiom 16 (less_equal_quotient) R->L }
% 0.20/0.51 fresh7(quotient(aQc, bQc, zero), true, aQc, bQc)
% 0.20/0.51 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.20/0.51 fresh7(quotient(aQc, bQc, fresh3(true, true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.51 = { by axiom 6 (quotient_property) R->L }
% 0.20/0.51 fresh7(quotient(aQc, bQc, fresh3(fresh12(true, true, divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.51 = { by axiom 8 (quotient_less_equal) R->L }
% 0.20/0.51 fresh7(quotient(aQc, bQc, fresh3(fresh12(fresh5(true, true, a, b), true, divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.51 = { by axiom 2 (aLEb) R->L }
% 0.20/0.51 fresh7(quotient(aQc, bQc, fresh3(fresh12(fresh5(less_equal(a, b), true, a, b), true, divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.51 = { by axiom 12 (quotient_less_equal) }
% 0.20/0.51 fresh7(quotient(aQc, bQc, fresh3(fresh12(quotient(a, b, zero), true, divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.51 = { by axiom 17 (quotient_property) R->L }
% 0.20/0.51 fresh7(quotient(aQc, bQc, fresh3(fresh11(true, true, a, b, zero, c, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 4 (bQc) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh11(quotient(b, c, bQc), true, a, b, zero, c, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 20 (quotient_property) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh8(true, true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 11 (closure) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh8(quotient(divide(a, c), bQc, divide(divide(a, c), bQc)), true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 21 (quotient_property) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh9(quotient(zero, c, zero), true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 12 (quotient_less_equal) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh9(fresh5(less_equal(zero, c), true, zero, c), true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh9(fresh5(true, true, zero, c), true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 8 (quotient_less_equal) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh9(true, true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 19 (quotient_property) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh10(quotient(a, c, divide(a, c)), true, a, b, zero, divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 11 (closure) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(fresh10(true, true, a, b, zero, divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 15 (quotient_property) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(less_equal(divide(divide(a, c), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 14 (well_defined) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(less_equal(divide(fresh2(true, true, a, c, divide(a, c), aQc), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 3 (aQc) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(less_equal(divide(fresh2(quotient(a, c, aQc), true, a, c, divide(a, c), aQc), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 18 (well_defined) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(less_equal(divide(fresh(quotient(a, c, divide(a, c)), true, divide(a, c), aQc), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 11 (closure) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(less_equal(divide(fresh(true, true, divide(a, c), aQc), bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 5 (well_defined) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh3(less_equal(divide(aQc, bQc), zero), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 13 (less_equal_and_equal) R->L }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh4(less_equal(zero, divide(aQc, bQc)), true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, fresh4(true, true, divide(aQc, bQc), zero)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 9 (less_equal_and_equal) }
% 0.20/0.52 fresh7(quotient(aQc, bQc, divide(aQc, bQc)), true, aQc, bQc)
% 0.20/0.52 = { by axiom 11 (closure) }
% 0.20/0.52 fresh7(true, true, aQc, bQc)
% 0.20/0.52 = { by axiom 7 (less_equal_quotient) }
% 0.20/0.52 true
% 0.20/0.52 % SZS output end Proof
% 0.20/0.52
% 0.20/0.52 RESULT: Unsatisfiable (the axioms are contradictory).
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