TSTP Solution File: HEN008-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN008-2 : 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 : n007.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 5.11s 1.06s
% Output : Proof 5.67s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : HEN008-2 : TPTP v8.1.2. Released v1.0.0.
% 0.06/0.12 % 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 : n007.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 13:11:55 EDT 2023
% 0.12/0.34 % CPUTime :
% 5.11/1.06 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 5.11/1.06
% 5.11/1.06 % SZS status Unsatisfiable
% 5.11/1.06
% 5.67/1.07 % SZS output start Proof
% 5.67/1.07 Take the following subset of the input axioms:
% 5.67/1.07 fof(aLEb, hypothesis, less_equal(a, b)).
% 5.67/1.07 fof(aQc, hypothesis, quotient(a, c, aQc)).
% 5.67/1.07 fof(bQc, hypothesis, quotient(b, c, bQc)).
% 5.67/1.07 fof(closure, axiom, ![X, Y]: quotient(X, Y, divide(X, Y))).
% 5.67/1.07 fof(prove_aQcLEbQc, negated_conjecture, ~less_equal(aQc, bQc)).
% 5.67/1.07 fof(quotient_less_equal, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | quotient(X2, Y2, zero))).
% 5.67/1.07 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))))))).
% 5.67/1.07 fof(well_defined, axiom, ![W, X2, Y2, Z2]: (~quotient(X2, Y2, Z2) | (~quotient(X2, Y2, W) | Z2=W))).
% 5.67/1.07 fof(xQyLEz_implies_xQzLEy, axiom, ![W1, W2, X2, Y2, Z2]: (~quotient(X2, Y2, W1) | (~less_equal(W1, Z2) | (~quotient(X2, Z2, W2) | less_equal(W2, Y2))))).
% 5.67/1.07 fof(x_divde_zero_is_x, axiom, ![X2]: quotient(X2, zero, X2)).
% 5.67/1.07 fof(zero_divide_anything_is_zero, axiom, ![X2]: quotient(zero, X2, zero)).
% 5.67/1.07
% 5.67/1.07 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.67/1.07 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.67/1.07 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.67/1.07 fresh(y, y, x1...xn) = u
% 5.67/1.07 C => fresh(s, t, x1...xn) = v
% 5.67/1.07 where fresh is a fresh function symbol and x1..xn are the free
% 5.67/1.07 variables of u and v.
% 5.67/1.07 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.67/1.07 input problem has no model of domain size 1).
% 5.67/1.07
% 5.67/1.07 The encoding turns the above axioms into the following unit equations and goals:
% 5.67/1.07
% 5.67/1.07 Axiom 1 (aLEb): less_equal(a, b) = true.
% 5.67/1.07 Axiom 2 (x_divde_zero_is_x): quotient(X, zero, X) = true.
% 5.67/1.07 Axiom 3 (zero_divide_anything_is_zero): quotient(zero, X, zero) = true.
% 5.67/1.07 Axiom 4 (aQc): quotient(a, c, aQc) = true.
% 5.67/1.07 Axiom 5 (bQc): quotient(b, c, bQc) = true.
% 5.67/1.07 Axiom 6 (well_defined): fresh(X, X, Y, Z) = Z.
% 5.67/1.07 Axiom 7 (quotient_property): fresh20(X, X, Y, Z) = true.
% 5.67/1.07 Axiom 8 (xQyLEz_implies_xQzLEy): fresh15(X, X, Y, Z) = true.
% 5.67/1.07 Axiom 9 (quotient_less_equal): fresh9(X, X, Y, Z) = true.
% 5.67/1.07 Axiom 10 (closure): quotient(X, Y, divide(X, Y)) = true.
% 5.67/1.07 Axiom 11 (quotient_less_equal): fresh9(less_equal(X, Y), true, X, Y) = quotient(X, Y, zero).
% 5.67/1.07 Axiom 12 (xQyLEz_implies_xQzLEy): fresh5(X, X, Y, Z, W, V) = less_equal(V, Y).
% 5.67/1.07 Axiom 13 (well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 5.67/1.07 Axiom 14 (quotient_property): fresh18(X, X, Y, Z, W, V, U) = less_equal(V, U).
% 5.67/1.07 Axiom 15 (xQyLEz_implies_xQzLEy): fresh14(X, X, Y, Z, W, V, U) = fresh15(less_equal(W, V), true, Z, U).
% 5.67/1.07 Axiom 16 (quotient_property): fresh19(X, X, Y, Z, W, V, U, T, S) = fresh20(quotient(Y, Z, W), true, T, S).
% 5.67/1.07 Axiom 17 (well_defined): fresh2(quotient(X, Y, Z), true, X, Y, W, Z) = fresh(quotient(X, Y, W), true, W, Z).
% 5.67/1.07 Axiom 18 (quotient_property): fresh17(X, X, Y, Z, W, V, U, T, S, X2) = fresh18(quotient(Y, V, T), true, Y, Z, W, S, X2).
% 5.67/1.07 Axiom 19 (xQyLEz_implies_xQzLEy): fresh14(quotient(X, Y, Z), true, X, W, V, Y, Z) = fresh5(quotient(X, W, V), true, W, V, Y, Z).
% 5.67/1.07 Axiom 20 (quotient_property): fresh16(X, X, Y, Z, W, V, U, T, S, X2) = fresh19(quotient(Z, V, U), true, Y, Z, W, V, T, S, X2).
% 5.67/1.07 Axiom 21 (quotient_property): fresh16(quotient(X, Y, Z), true, W, V, U, T, Y, X, Z, S) = fresh17(quotient(U, T, S), true, W, V, U, T, Y, X, Z, S).
% 5.67/1.07
% 5.67/1.07 Goal 1 (prove_aQcLEbQc): less_equal(aQc, bQc) = true.
% 5.67/1.07 Proof:
% 5.67/1.07 less_equal(aQc, bQc)
% 5.67/1.07 = { by axiom 12 (xQyLEz_implies_xQzLEy) R->L }
% 5.67/1.07 fresh5(true, true, bQc, divide(aQc, bQc), zero, aQc)
% 5.67/1.07 = { by axiom 10 (closure) R->L }
% 5.67/1.07 fresh5(quotient(aQc, bQc, divide(aQc, bQc)), true, bQc, divide(aQc, bQc), zero, aQc)
% 5.67/1.07 = { by axiom 19 (xQyLEz_implies_xQzLEy) R->L }
% 5.67/1.07 fresh14(quotient(aQc, zero, aQc), true, aQc, bQc, divide(aQc, bQc), zero, aQc)
% 5.67/1.07 = { by axiom 2 (x_divde_zero_is_x) }
% 5.67/1.07 fresh14(true, true, aQc, bQc, divide(aQc, bQc), zero, aQc)
% 5.67/1.07 = { by axiom 15 (xQyLEz_implies_xQzLEy) }
% 5.67/1.07 fresh15(less_equal(divide(aQc, bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 6 (well_defined) R->L }
% 5.67/1.07 fresh15(less_equal(divide(fresh(true, true, divide(a, c), aQc), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 10 (closure) R->L }
% 5.67/1.07 fresh15(less_equal(divide(fresh(quotient(a, c, divide(a, c)), true, divide(a, c), aQc), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 17 (well_defined) R->L }
% 5.67/1.07 fresh15(less_equal(divide(fresh2(quotient(a, c, aQc), true, a, c, divide(a, c), aQc), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 4 (aQc) }
% 5.67/1.07 fresh15(less_equal(divide(fresh2(true, true, a, c, divide(a, c), aQc), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 13 (well_defined) }
% 5.67/1.07 fresh15(less_equal(divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 14 (quotient_property) R->L }
% 5.67/1.07 fresh15(fresh18(true, true, a, b, zero, divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 10 (closure) R->L }
% 5.67/1.07 fresh15(fresh18(quotient(a, c, divide(a, c)), true, a, b, zero, divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 18 (quotient_property) R->L }
% 5.67/1.07 fresh15(fresh17(true, true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 3 (zero_divide_anything_is_zero) R->L }
% 5.67/1.07 fresh15(fresh17(quotient(zero, c, zero), true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 21 (quotient_property) R->L }
% 5.67/1.07 fresh15(fresh16(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, bQc, aQc)
% 5.67/1.07 = { by axiom 10 (closure) }
% 5.67/1.07 fresh15(fresh16(true, true, a, b, zero, c, bQc, divide(a, c), divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 20 (quotient_property) }
% 5.67/1.07 fresh15(fresh19(quotient(b, c, bQc), true, a, b, zero, c, divide(a, c), divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 5 (bQc) }
% 5.67/1.07 fresh15(fresh19(true, true, a, b, zero, c, divide(a, c), divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 16 (quotient_property) }
% 5.67/1.07 fresh15(fresh20(quotient(a, b, zero), true, divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 11 (quotient_less_equal) R->L }
% 5.67/1.07 fresh15(fresh20(fresh9(less_equal(a, b), true, a, b), true, divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 1 (aLEb) }
% 5.67/1.07 fresh15(fresh20(fresh9(true, true, a, b), true, divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 9 (quotient_less_equal) }
% 5.67/1.07 fresh15(fresh20(true, true, divide(divide(a, c), bQc), zero), true, bQc, aQc)
% 5.67/1.07 = { by axiom 7 (quotient_property) }
% 5.67/1.07 fresh15(true, true, bQc, aQc)
% 5.67/1.07 = { by axiom 8 (xQyLEz_implies_xQzLEy) }
% 5.67/1.07 true
% 5.67/1.07 % SZS output end Proof
% 5.67/1.07
% 5.67/1.07 RESULT: Unsatisfiable (the axioms are contradictory).
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