TSTP Solution File: HEN009-5 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN009-5 : TPTP v8.1.2. Bugfixed v1.2.1.
% 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 : n028.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 01:57:00 EDT 2023
% Result : Unsatisfiable 0.20s 0.44s
% 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 : HEN009-5 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.18/0.34 % Computer : n028.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Thu Aug 24 13:39:20 EDT 2023
% 0.18/0.35 % CPUTime :
% 0.20/0.44 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.44
% 0.20/0.44 % SZS status Unsatisfiable
% 0.20/0.45
% 0.20/0.46 % SZS output start Proof
% 0.20/0.46 Take the following subset of the input axioms:
% 0.20/0.46 fof(divide_and_equal, axiom, ![X, Y]: (divide(X, Y)!=zero | (divide(Y, X)!=zero | X=Y))).
% 0.20/0.46 fof(identity_is_largest, axiom, ![X2]: divide(X2, identity)=zero).
% 0.20/0.46 fof(property_of_divide1, axiom, ![Z, X2, Y2]: (divide(divide(X2, Y2), Z)!=zero | divide(divide(X2, Z), Y2)=zero)).
% 0.20/0.46 fof(property_of_divide2, axiom, ![X2, Y2, Z2]: (divide(X2, Y2)!=zero | divide(divide(Z2, Y2), divide(Z2, X2))=zero)).
% 0.20/0.46 fof(prove_this, negated_conjecture, divide(identity, a)!=divide(identity, divide(identity, divide(identity, a)))).
% 0.20/0.46 fof(quotient_property, axiom, ![X2, Y2, Z2]: divide(divide(divide(X2, Z2), divide(Y2, Z2)), divide(divide(X2, Y2), Z2))=zero).
% 0.20/0.46 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: divide(divide(X2, Y2), X2)=zero).
% 0.20/0.46 fof(zero_is_smallest, axiom, ![X2]: divide(zero, X2)=zero).
% 0.20/0.46
% 0.20/0.46 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.46 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.46 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.46 fresh(y, y, x1...xn) = u
% 0.20/0.46 C => fresh(s, t, x1...xn) = v
% 0.20/0.46 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.46 variables of u and v.
% 0.20/0.46 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.46 input problem has no model of domain size 1).
% 0.20/0.46
% 0.20/0.46 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.46
% 0.20/0.46 Axiom 1 (identity_is_largest): divide(X, identity) = zero.
% 0.20/0.46 Axiom 2 (zero_is_smallest): divide(zero, X) = zero.
% 0.20/0.46 Axiom 3 (quotient_smaller_than_numerator): divide(divide(X, Y), X) = zero.
% 0.20/0.46 Axiom 4 (divide_and_equal): fresh(X, X, Y, Z) = Z.
% 0.20/0.46 Axiom 5 (divide_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.20/0.46 Axiom 6 (property_of_divide2): fresh6(X, X, Y, Z, W) = zero.
% 0.20/0.46 Axiom 7 (property_of_divide1): fresh5(X, X, Y, Z, W) = zero.
% 0.20/0.46 Axiom 8 (divide_and_equal): fresh2(divide(X, Y), zero, Y, X) = fresh(divide(Y, X), zero, Y, X).
% 0.20/0.46 Axiom 9 (property_of_divide2): fresh6(divide(X, Y), zero, X, Y, Z) = divide(divide(Z, Y), divide(Z, X)).
% 0.20/0.46 Axiom 10 (property_of_divide1): fresh5(divide(divide(X, Y), Z), zero, X, Y, Z) = divide(divide(X, Z), Y).
% 0.20/0.46 Axiom 11 (quotient_property): divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = zero.
% 0.20/0.46
% 0.20/0.46 Lemma 12: fresh(divide(X, zero), zero, X, zero) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 fresh(divide(X, zero), zero, X, zero)
% 0.20/0.46 = { by axiom 8 (divide_and_equal) R->L }
% 0.20/0.46 fresh2(divide(zero, X), zero, X, zero)
% 0.20/0.46 = { by axiom 2 (zero_is_smallest) }
% 0.20/0.46 fresh2(zero, zero, X, zero)
% 0.20/0.46 = { by axiom 5 (divide_and_equal) }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Lemma 13: divide(divide(X, Y), divide(identity, Y)) = zero.
% 0.20/0.46 Proof:
% 0.20/0.46 divide(divide(X, Y), divide(identity, Y))
% 0.20/0.46 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.46 fresh(zero, zero, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.20/0.46 = { by axiom 2 (zero_is_smallest) R->L }
% 0.20/0.46 fresh(divide(zero, divide(divide(X, Y), divide(identity, Y))), zero, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.20/0.46 = { by axiom 8 (divide_and_equal) R->L }
% 0.20/0.46 fresh2(divide(divide(divide(X, Y), divide(identity, Y)), zero), zero, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.20/0.46 = { by axiom 2 (zero_is_smallest) R->L }
% 0.20/0.46 fresh2(divide(divide(divide(X, Y), divide(identity, Y)), divide(zero, Y)), zero, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.20/0.46 = { by axiom 1 (identity_is_largest) R->L }
% 0.20/0.46 fresh2(divide(divide(divide(X, Y), divide(identity, Y)), divide(divide(X, identity), Y)), zero, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.20/0.46 = { by axiom 11 (quotient_property) }
% 0.20/0.46 fresh2(zero, zero, zero, divide(divide(X, Y), divide(identity, Y)))
% 0.20/0.46 = { by axiom 5 (divide_and_equal) }
% 0.20/0.46 zero
% 0.20/0.46
% 0.20/0.46 Goal 1 (prove_this): divide(identity, a) = divide(identity, divide(identity, divide(identity, a))).
% 0.20/0.46 Proof:
% 0.20/0.46 divide(identity, a)
% 0.20/0.46 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.46 fresh(zero, zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.46 = { by axiom 7 (property_of_divide1) R->L }
% 0.20/0.46 fresh(fresh5(zero, zero, identity, divide(identity, a), divide(identity, divide(identity, a))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.46 = { by lemma 13 R->L }
% 0.20/0.46 fresh(fresh5(divide(divide(identity, divide(identity, a)), divide(identity, divide(identity, a))), zero, identity, divide(identity, a), divide(identity, divide(identity, a))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.46 = { by axiom 10 (property_of_divide1) }
% 0.20/0.47 fresh(divide(divide(identity, divide(identity, divide(identity, a))), divide(identity, a)), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 8 (divide_and_equal) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, divide(identity, divide(identity, a)))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh(zero, zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by lemma 13 R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh(divide(divide(a, divide(identity, a)), divide(identity, divide(identity, a))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 8 (divide_and_equal) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh(zero, zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 3 (quotient_smaller_than_numerator) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh(divide(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 8 (divide_and_equal) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by lemma 12 R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(fresh(divide(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 10 (property_of_divide1) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(fresh(fresh5(divide(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero, divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 9 (property_of_divide2) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(fresh(fresh5(fresh6(divide(zero, zero), zero, zero, zero, divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 2 (zero_is_smallest) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(fresh(fresh5(fresh6(zero, zero, zero, zero, divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 6 (property_of_divide2) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(fresh(fresh5(zero, zero, divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 7 (property_of_divide1) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(fresh(zero, zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero)), zero), zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 4 (divide_and_equal) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(fresh2(zero, zero, divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a)))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 5 (divide_and_equal) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), zero), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 4 (divide_and_equal) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), fresh(zero, zero, divide(divide(identity, a), divide(identity, a)), zero)), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 7 (property_of_divide1) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), fresh(fresh5(zero, zero, divide(identity, a), zero, divide(identity, a)), zero, divide(divide(identity, a), divide(identity, a)), zero)), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 3 (quotient_smaller_than_numerator) R->L }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), fresh(fresh5(divide(divide(divide(identity, a), zero), divide(identity, a)), zero, divide(identity, a), zero, divide(identity, a)), zero, divide(divide(identity, a), divide(identity, a)), zero)), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 10 (property_of_divide1) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), fresh(divide(divide(divide(identity, a), divide(identity, a)), zero), zero, divide(divide(identity, a), divide(identity, a)), zero)), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by lemma 12 }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(divide(divide(divide(identity, divide(identity, a)), divide(a, divide(identity, a))), divide(divide(identity, a), divide(identity, a))), zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 11 (quotient_property) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, fresh2(zero, zero, divide(a, divide(identity, a)), divide(identity, divide(identity, a))))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 5 (divide_and_equal) }
% 0.20/0.47 fresh2(divide(divide(identity, a), divide(identity, divide(a, divide(identity, a)))), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 9 (property_of_divide2) R->L }
% 0.20/0.47 fresh2(fresh6(divide(divide(a, divide(identity, a)), a), zero, divide(a, divide(identity, a)), a, identity), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 3 (quotient_smaller_than_numerator) }
% 0.20/0.47 fresh2(fresh6(zero, zero, divide(a, divide(identity, a)), a, identity), zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 6 (property_of_divide2) }
% 0.20/0.47 fresh2(zero, zero, divide(identity, divide(identity, divide(identity, a))), divide(identity, a))
% 0.20/0.47 = { by axiom 5 (divide_and_equal) }
% 0.20/0.47 divide(identity, divide(identity, divide(identity, a)))
% 0.20/0.47 % SZS output end Proof
% 0.20/0.47
% 0.20/0.47 RESULT: Unsatisfiable (the axioms are contradictory).
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