TSTP Solution File: HEN007-2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN007-2 : 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 : n015.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 01:56:57 EDT 2023
% Result : Unsatisfiable 0.20s 0.46s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HEN007-2 : 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
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 13:42:06 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.46 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.46
% 0.20/0.46 % SZS status Unsatisfiable
% 0.20/0.46
% 0.20/0.47 % SZS output start Proof
% 0.20/0.47 Take the following subset of the input axioms:
% 0.20/0.47 fof(closure, axiom, ![X, Y]: quotient(X, Y, divide(X, Y))).
% 0.20/0.47 fof(less_equal_quotient, axiom, ![X2, Y2]: (~quotient(X2, Y2, zero) | less_equal(X2, Y2))).
% 0.20/0.47 fof(prove_zQyLEzQx, negated_conjecture, ~less_equal(zQy, zQx)).
% 0.20/0.47 fof(transitivity_of_less_equal, axiom, ![Z, X2, Y2]: (~less_equal(X2, Y2) | (~less_equal(Y2, Z) | less_equal(X2, Z)))).
% 0.20/0.47 fof(well_defined, axiom, ![W, X2, Y2, Z2]: (~quotient(X2, Y2, Z2) | (~quotient(X2, Y2, W) | Z2=W))).
% 0.20/0.47 fof(xLEy, hypothesis, less_equal(x, y)).
% 0.20/0.47 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))))).
% 0.20/0.47 fof(x_divide_x_is_zero, axiom, ![X2]: quotient(X2, X2, zero)).
% 0.20/0.47 fof(zQx, hypothesis, quotient(z, x, zQx)).
% 0.20/0.47 fof(zQy, hypothesis, quotient(z, y, zQy)).
% 0.20/0.47
% 0.20/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.47 fresh(y, y, x1...xn) = u
% 0.20/0.47 C => fresh(s, t, x1...xn) = v
% 0.20/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.47 variables of u and v.
% 0.20/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.47 input problem has no model of domain size 1).
% 0.20/0.47
% 0.20/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.47
% 0.20/0.47 Axiom 1 (xLEy): less_equal(x, y) = true.
% 0.20/0.47 Axiom 2 (x_divide_x_is_zero): quotient(X, X, zero) = true.
% 0.20/0.47 Axiom 3 (zQx): quotient(z, x, zQx) = true.
% 0.20/0.47 Axiom 4 (zQy): quotient(z, y, zQy) = true.
% 0.20/0.47 Axiom 5 (well_defined): fresh(X, X, Y, Z) = Z.
% 0.20/0.47 Axiom 6 (xQyLEz_implies_xQzLEy): fresh12(X, X, Y, Z) = true.
% 0.20/0.47 Axiom 7 (less_equal_quotient): fresh10(X, X, Y, Z) = true.
% 0.20/0.47 Axiom 8 (transitivity_of_less_equal): fresh6(X, X, Y, Z) = true.
% 0.20/0.47 Axiom 9 (closure): quotient(X, Y, divide(X, Y)) = true.
% 0.20/0.47 Axiom 10 (transitivity_of_less_equal): fresh7(X, X, Y, Z, W) = less_equal(Y, W).
% 0.20/0.47 Axiom 11 (xQyLEz_implies_xQzLEy): fresh5(X, X, Y, Z, W, V) = less_equal(V, Y).
% 0.20/0.47 Axiom 12 (well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.47 Axiom 13 (xQyLEz_implies_xQzLEy): fresh11(X, X, Y, Z, W, V, U) = fresh12(less_equal(W, V), true, Z, U).
% 0.20/0.47 Axiom 14 (less_equal_quotient): fresh10(quotient(X, Y, zero), true, X, Y) = less_equal(X, Y).
% 0.20/0.47 Axiom 15 (transitivity_of_less_equal): fresh7(less_equal(X, Y), true, Z, X, Y) = fresh6(less_equal(Z, X), true, Z, Y).
% 0.20/0.47 Axiom 16 (well_defined): fresh2(quotient(X, Y, Z), true, X, Y, W, Z) = fresh(quotient(X, Y, W), true, W, Z).
% 0.20/0.47 Axiom 17 (xQyLEz_implies_xQzLEy): fresh11(quotient(X, Y, Z), true, X, W, V, Y, Z) = fresh5(quotient(X, W, V), true, W, V, Y, Z).
% 0.20/0.47
% 0.20/0.47 Lemma 18: fresh11(quotient(X, Y, Z), true, X, W, divide(X, W), Y, Z) = less_equal(Z, W).
% 0.20/0.47 Proof:
% 0.20/0.47 fresh11(quotient(X, Y, Z), true, X, W, divide(X, W), Y, Z)
% 0.20/0.47 = { by axiom 17 (xQyLEz_implies_xQzLEy) }
% 0.20/0.47 fresh5(quotient(X, W, divide(X, W)), true, W, divide(X, W), Y, Z)
% 0.20/0.47 = { by axiom 9 (closure) }
% 0.20/0.47 fresh5(true, true, W, divide(X, W), Y, Z)
% 0.20/0.47 = { by axiom 11 (xQyLEz_implies_xQzLEy) }
% 0.20/0.47 less_equal(Z, W)
% 0.20/0.47
% 0.20/0.47 Goal 1 (prove_zQyLEzQx): less_equal(zQy, zQx) = true.
% 0.20/0.47 Proof:
% 0.20/0.47 less_equal(zQy, zQx)
% 0.20/0.47 = { by axiom 5 (well_defined) R->L }
% 0.20/0.47 less_equal(zQy, fresh(true, true, divide(z, x), zQx))
% 0.20/0.47 = { by axiom 9 (closure) R->L }
% 0.20/0.47 less_equal(zQy, fresh(quotient(z, x, divide(z, x)), true, divide(z, x), zQx))
% 0.20/0.47 = { by axiom 16 (well_defined) R->L }
% 0.20/0.47 less_equal(zQy, fresh2(quotient(z, x, zQx), true, z, x, divide(z, x), zQx))
% 0.20/0.47 = { by axiom 3 (zQx) }
% 0.20/0.47 less_equal(zQy, fresh2(true, true, z, x, divide(z, x), zQx))
% 0.20/0.47 = { by axiom 12 (well_defined) }
% 0.20/0.47 less_equal(zQy, divide(z, x))
% 0.20/0.47 = { by lemma 18 R->L }
% 0.20/0.47 fresh11(quotient(z, y, zQy), true, z, divide(z, x), divide(z, divide(z, x)), y, zQy)
% 0.20/0.47 = { by axiom 4 (zQy) }
% 0.20/0.47 fresh11(true, true, z, divide(z, x), divide(z, divide(z, x)), y, zQy)
% 0.20/0.47 = { by axiom 13 (xQyLEz_implies_xQzLEy) }
% 0.20/0.47 fresh12(less_equal(divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 10 (transitivity_of_less_equal) R->L }
% 0.20/0.47 fresh12(fresh7(true, true, divide(z, divide(z, x)), x, y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 1 (xLEy) R->L }
% 0.20/0.47 fresh12(fresh7(less_equal(x, y), true, divide(z, divide(z, x)), x, y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 15 (transitivity_of_less_equal) }
% 0.20/0.47 fresh12(fresh6(less_equal(divide(z, divide(z, x)), x), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by lemma 18 R->L }
% 0.20/0.47 fresh12(fresh6(fresh11(quotient(z, divide(z, x), divide(z, divide(z, x))), true, z, x, divide(z, x), divide(z, x), divide(z, divide(z, x))), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 9 (closure) }
% 0.20/0.47 fresh12(fresh6(fresh11(true, true, z, x, divide(z, x), divide(z, x), divide(z, divide(z, x))), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 13 (xQyLEz_implies_xQzLEy) }
% 0.20/0.47 fresh12(fresh6(fresh12(less_equal(divide(z, x), divide(z, x)), true, x, divide(z, divide(z, x))), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 14 (less_equal_quotient) R->L }
% 0.20/0.47 fresh12(fresh6(fresh12(fresh10(quotient(divide(z, x), divide(z, x), zero), true, divide(z, x), divide(z, x)), true, x, divide(z, divide(z, x))), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 2 (x_divide_x_is_zero) }
% 0.20/0.47 fresh12(fresh6(fresh12(fresh10(true, true, divide(z, x), divide(z, x)), true, x, divide(z, divide(z, x))), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 7 (less_equal_quotient) }
% 0.20/0.47 fresh12(fresh6(fresh12(true, true, x, divide(z, divide(z, x))), true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 6 (xQyLEz_implies_xQzLEy) }
% 0.20/0.47 fresh12(fresh6(true, true, divide(z, divide(z, x)), y), true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 8 (transitivity_of_less_equal) }
% 0.20/0.47 fresh12(true, true, divide(z, x), zQy)
% 0.20/0.47 = { by axiom 6 (xQyLEz_implies_xQzLEy) }
% 0.20/0.47 true
% 0.20/0.47 % SZS output end Proof
% 0.20/0.47
% 0.20/0.47 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------