TSTP Solution File: HEN004-4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : HEN004-4 : 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 : n019.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 01:56:53 EDT 2023
% Result : Unsatisfiable 0.20s 0.39s
% 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
% 0.00/0.12 % Problem : HEN004-4 : 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 : n019.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
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 13:38:43 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.39 Command-line arguments: --no-flatten-goal
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% 0.20/0.39 % SZS status Unsatisfiable
% 0.20/0.39
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 Take the following subset of the input axioms:
% 0.20/0.40 fof(less_equal_and_equal, axiom, ![X, Y]: (~less_equal(X, Y) | (~less_equal(Y, X) | X=Y))).
% 0.20/0.40 fof(prove_x_divide_zero_is_x, negated_conjecture, divide(a, zero)!=a).
% 0.20/0.40 fof(quotient_less_equal2, axiom, ![X2, Y2]: (divide(X2, Y2)!=zero | less_equal(X2, Y2))).
% 0.20/0.40 fof(quotient_property, axiom, ![Z, X2, Y2]: less_equal(divide(divide(X2, Z), divide(Y2, Z)), divide(divide(X2, Y2), Z))).
% 0.20/0.40 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: less_equal(divide(X2, Y2), X2)).
% 0.20/0.40 fof(x_divide_x_is_zero, axiom, ![X2]: divide(X2, X2)=zero).
% 0.20/0.40 fof(zero_divide_anything_is_zero, axiom, ![X2]: divide(zero, X2)=zero).
% 0.20/0.40 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 0.20/0.40
% 0.20/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.40 fresh(y, y, x1...xn) = u
% 0.20/0.40 C => fresh(s, t, x1...xn) = v
% 0.20/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.40 variables of u and v.
% 0.20/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.40 input problem has no model of domain size 1).
% 0.20/0.40
% 0.20/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.40
% 0.20/0.40 Axiom 1 (zero_is_smallest): less_equal(zero, X) = true.
% 0.20/0.40 Axiom 2 (x_divide_x_is_zero): divide(X, X) = zero.
% 0.20/0.40 Axiom 3 (zero_divide_anything_is_zero): divide(zero, X) = zero.
% 0.20/0.40 Axiom 4 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.20/0.40 Axiom 5 (quotient_less_equal2): fresh4(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 6 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.20/0.40 Axiom 7 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.20/0.40 Axiom 8 (quotient_less_equal2): fresh4(divide(X, Y), zero, X, Y) = less_equal(X, Y).
% 0.20/0.40 Axiom 9 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.20/0.40 Axiom 10 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.20/0.40
% 0.20/0.40 Goal 1 (prove_x_divide_zero_is_x): divide(a, zero) = a.
% 0.20/0.40 Proof:
% 0.20/0.40 divide(a, zero)
% 0.20/0.40 = { by axiom 6 (less_equal_and_equal) R->L }
% 0.20/0.40 fresh2(true, true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 5 (quotient_less_equal2) R->L }
% 0.20/0.40 fresh2(fresh4(zero, zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 6 (less_equal_and_equal) R->L }
% 0.20/0.40 fresh2(fresh4(fresh2(true, true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 5 (quotient_less_equal2) R->L }
% 0.20/0.40 fresh2(fresh4(fresh2(fresh4(zero, zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 6 (less_equal_and_equal) R->L }
% 0.20/0.40 fresh2(fresh4(fresh2(fresh4(fresh2(true, true, zero, divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero)))), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 10 (quotient_property) R->L }
% 0.20/0.40 fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), divide(divide(a, zero), divide(a, zero))), true, zero, divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero)))), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 2 (x_divide_x_is_zero) }
% 0.20/0.40 fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), zero), true, zero, divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero)))), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.40 = { by axiom 9 (less_equal_and_equal) }
% 0.20/0.40 fresh2(fresh4(fresh2(fresh4(fresh(less_equal(zero, divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero)))), true, zero, divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero)))), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.41 fresh2(fresh4(fresh2(fresh4(fresh(true, true, zero, divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero)))), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.41 fresh2(fresh4(fresh2(fresh4(divide(divide(a, divide(a, zero)), divide(zero, divide(a, zero))), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 3 (zero_divide_anything_is_zero) }
% 0.20/0.41 fresh2(fresh4(fresh2(fresh4(divide(divide(a, divide(a, zero)), zero), zero, divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 8 (quotient_less_equal2) }
% 0.20/0.41 fresh2(fresh4(fresh2(less_equal(divide(a, divide(a, zero)), zero), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 9 (less_equal_and_equal) }
% 0.20/0.41 fresh2(fresh4(fresh(less_equal(zero, divide(a, divide(a, zero))), true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.41 fresh2(fresh4(fresh(true, true, zero, divide(a, divide(a, zero))), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.41 fresh2(fresh4(divide(a, divide(a, zero)), zero, a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 8 (quotient_less_equal2) }
% 0.20/0.41 fresh2(less_equal(a, divide(a, zero)), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 9 (less_equal_and_equal) }
% 0.20/0.41 fresh(less_equal(divide(a, zero), a), true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 7 (quotient_smaller_than_numerator) }
% 0.20/0.41 fresh(true, true, divide(a, zero), a)
% 0.20/0.41 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.41 a
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41
% 0.20/0.41 RESULT: Unsatisfiable (the axioms are contradictory).
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