TSTP Solution File: RNG039-2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG039-2 : 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 : n021.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 13:58:59 EDT 2023
% Result : Unsatisfiable 86.91s 11.56s
% Output : Proof 86.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : RNG039-2 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n021.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Sun Aug 27 02:39:42 EDT 2023
% 0.21/0.36 % CPUTime :
% 86.91/11.56 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 86.91/11.56
% 86.91/11.56 % SZS status Unsatisfiable
% 86.91/11.56
% 86.91/11.58 % SZS output start Proof
% 86.91/11.58 Take the following subset of the input axioms:
% 86.91/11.58 fof(absorbtion2, axiom, ![A, B]: sum(add(A, B), B, A)).
% 86.91/11.58 fof(add_substitution1, axiom, ![X, Y, W]: (~equalish(X, Y) | equalish(add(X, W), add(Y, W)))).
% 86.91/11.58 fof(addition_is_well_defined, axiom, ![U, V, X2, Y2]: (~sum(X2, Y2, U) | (~sum(X2, Y2, V) | equalish(U, V)))).
% 86.91/11.58 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 86.91/11.58 fof(clause35, axiom, ![A2]: equalish(multiply(A2, A2), A2)).
% 86.91/11.58 fof(clause38, axiom, ![A2, B2]: sum(A2, B2, add(B2, A2))).
% 86.91/11.58 fof(clause44, axiom, ![A2, B2]: product(a, multiply(b, A2), multiply(B2, A2))).
% 86.91/11.58 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 86.91/11.58 fof(multiplication_is_well_defined, axiom, ![V2, X2, Y2, U2]: (~product(X2, Y2, U2) | (~product(X2, Y2, V2) | equalish(U2, V2)))).
% 86.91/11.58 fof(multiplicative_identity1, axiom, ![X2]: product(additive_identity, X2, additive_identity)).
% 86.91/11.58 fof(product_substitution3, axiom, ![Z, X2, Y2, W2]: (~equalish(X2, Y2) | (~product(W2, Z, X2) | product(W2, Z, Y2)))).
% 86.91/11.58 fof(prove_c_equals_d, negated_conjecture, ~equalish(c, d)).
% 86.91/11.58 fof(sum_substitution1, axiom, ![X2, Y2, Z2, W2]: (~equalish(X2, Y2) | (~sum(X2, W2, Z2) | sum(Y2, W2, Z2)))).
% 86.91/11.58 fof(transitivity, axiom, ![X2, Y2, Z2]: (~equalish(X2, Y2) | (~equalish(Y2, Z2) | equalish(X2, Z2)))).
% 86.91/11.58
% 86.91/11.58 Now clausify the problem and encode Horn clauses using encoding 3 of
% 86.91/11.58 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 86.91/11.58 We repeatedly replace C & s=t => u=v by the two clauses:
% 86.91/11.58 fresh(y, y, x1...xn) = u
% 86.91/11.58 C => fresh(s, t, x1...xn) = v
% 86.91/11.58 where fresh is a fresh function symbol and x1..xn are the free
% 86.91/11.58 variables of u and v.
% 86.91/11.58 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 86.91/11.58 input problem has no model of domain size 1).
% 86.91/11.58
% 86.91/11.58 The encoding turns the above axioms into the following unit equations and goals:
% 86.91/11.58
% 86.91/11.58 Axiom 1 (multiplicative_identity1): product(additive_identity, X, additive_identity) = true.
% 86.91/11.58 Axiom 2 (additive_identity2): sum(X, additive_identity, X) = true.
% 86.91/11.58 Axiom 3 (clause35): equalish(multiply(X, X), X) = true.
% 86.91/11.58 Axiom 4 (transitivity): fresh(X, X, Y, Z) = true.
% 86.91/11.58 Axiom 5 (addition_is_well_defined): fresh28(X, X, Y, Z) = true.
% 86.91/11.58 Axiom 6 (multiplication_is_well_defined): fresh21(X, X, Y, Z) = true.
% 86.91/11.58 Axiom 7 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 86.91/11.58 Axiom 8 (clause38): sum(X, Y, add(Y, X)) = true.
% 86.91/11.58 Axiom 9 (absorbtion2): sum(add(X, Y), Y, X) = true.
% 86.91/11.58 Axiom 10 (add_substitution1): fresh30(X, X, Y, Z, W) = true.
% 86.91/11.58 Axiom 11 (product_substitution3): fresh13(X, X, Y, Z, W) = true.
% 86.91/11.58 Axiom 12 (sum_substitution1): fresh7(X, X, Y, Z, W) = true.
% 86.91/11.58 Axiom 13 (transitivity): fresh2(X, X, Y, Z, W) = equalish(Y, W).
% 86.91/11.58 Axiom 14 (addition_is_well_defined): fresh29(X, X, Y, Z, W, V) = equalish(W, V).
% 86.91/11.58 Axiom 15 (multiplication_is_well_defined): fresh22(X, X, Y, Z, W, V) = equalish(W, V).
% 86.91/11.58 Axiom 16 (product_substitution3): fresh14(X, X, Y, Z, W, V) = product(W, V, Z).
% 86.91/11.58 Axiom 17 (sum_substitution1): fresh8(X, X, Y, Z, W, V) = sum(Z, W, V).
% 86.91/11.58 Axiom 18 (clause44): product(a, multiply(b, X), multiply(Y, X)) = true.
% 86.91/11.58 Axiom 19 (add_substitution1): fresh30(equalish(X, Y), true, X, Y, Z) = equalish(add(X, Z), add(Y, Z)).
% 86.91/11.58 Axiom 20 (transitivity): fresh2(equalish(X, Y), true, Z, X, Y) = fresh(equalish(Z, X), true, Z, Y).
% 86.91/11.58 Axiom 21 (addition_is_well_defined): fresh29(sum(X, Y, Z), true, X, Y, W, Z) = fresh28(sum(X, Y, W), true, W, Z).
% 86.91/11.58 Axiom 22 (multiplication_is_well_defined): fresh22(product(X, Y, Z), true, X, Y, W, Z) = fresh21(product(X, Y, W), true, W, Z).
% 86.91/11.58 Axiom 23 (product_substitution3): fresh14(product(X, Y, Z), true, Z, W, X, Y) = fresh13(equalish(Z, W), true, W, X, Y).
% 86.91/11.58 Axiom 24 (sum_substitution1): fresh8(sum(X, Y, Z), true, X, W, Y, Z) = fresh7(equalish(X, W), true, W, Y, Z).
% 86.91/11.58
% 86.91/11.58 Lemma 25: sum(X, X, Y) = true.
% 86.91/11.58 Proof:
% 86.91/11.58 sum(X, X, Y)
% 86.91/11.58 = { by axiom 17 (sum_substitution1) R->L }
% 86.91/11.58 fresh8(true, true, add(Y, X), X, X, Y)
% 86.91/11.58 = { by axiom 9 (absorbtion2) R->L }
% 86.91/11.58 fresh8(sum(add(Y, X), X, Y), true, add(Y, X), X, X, Y)
% 86.91/11.58 = { by axiom 24 (sum_substitution1) }
% 86.91/11.58 fresh7(equalish(add(Y, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 13 (transitivity) R->L }
% 86.91/11.58 fresh7(fresh2(true, true, add(Y, X), add(additive_identity, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 5 (addition_is_well_defined) R->L }
% 86.91/11.58 fresh7(fresh2(fresh28(true, true, add(additive_identity, X), X), true, add(Y, X), add(additive_identity, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 8 (clause38) R->L }
% 86.91/11.58 fresh7(fresh2(fresh28(sum(X, additive_identity, add(additive_identity, X)), true, add(additive_identity, X), X), true, add(Y, X), add(additive_identity, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 21 (addition_is_well_defined) R->L }
% 86.91/11.58 fresh7(fresh2(fresh29(sum(X, additive_identity, X), true, X, additive_identity, add(additive_identity, X), X), true, add(Y, X), add(additive_identity, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 2 (additive_identity2) }
% 86.91/11.58 fresh7(fresh2(fresh29(true, true, X, additive_identity, add(additive_identity, X), X), true, add(Y, X), add(additive_identity, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 14 (addition_is_well_defined) }
% 86.91/11.58 fresh7(fresh2(equalish(add(additive_identity, X), X), true, add(Y, X), add(additive_identity, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 20 (transitivity) }
% 86.91/11.58 fresh7(fresh(equalish(add(Y, X), add(additive_identity, X)), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 19 (add_substitution1) R->L }
% 86.91/11.58 fresh7(fresh(fresh30(equalish(Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 15 (multiplication_is_well_defined) R->L }
% 86.91/11.58 fresh7(fresh(fresh30(fresh22(true, true, additive_identity, Y, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 1 (multiplicative_identity1) R->L }
% 86.91/11.58 fresh7(fresh(fresh30(fresh22(product(additive_identity, Y, additive_identity), true, additive_identity, Y, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.58 = { by axiom 22 (multiplication_is_well_defined) }
% 86.91/11.58 fresh7(fresh(fresh30(fresh21(product(additive_identity, Y, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 16 (product_substitution3) R->L }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh14(true, true, multiply(additive_identity, Y), Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 7 (closure_of_multiplication) R->L }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh14(product(additive_identity, Y, multiply(additive_identity, Y)), true, multiply(additive_identity, Y), Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 23 (product_substitution3) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(equalish(multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 13 (transitivity) R->L }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh2(true, true, multiply(additive_identity, Y), multiply(Y, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 3 (clause35) R->L }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh2(equalish(multiply(Y, Y), Y), true, multiply(additive_identity, Y), multiply(Y, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 20 (transitivity) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh(equalish(multiply(additive_identity, Y), multiply(Y, Y)), true, multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 15 (multiplication_is_well_defined) R->L }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh(fresh22(true, true, a, multiply(b, Y), multiply(additive_identity, Y), multiply(Y, Y)), true, multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 18 (clause44) R->L }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh(fresh22(product(a, multiply(b, Y), multiply(Y, Y)), true, a, multiply(b, Y), multiply(additive_identity, Y), multiply(Y, Y)), true, multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 22 (multiplication_is_well_defined) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh(fresh21(product(a, multiply(b, Y), multiply(additive_identity, Y)), true, multiply(additive_identity, Y), multiply(Y, Y)), true, multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 18 (clause44) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh(fresh21(true, true, multiply(additive_identity, Y), multiply(Y, Y)), true, multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 6 (multiplication_is_well_defined) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(fresh(true, true, multiply(additive_identity, Y), Y), true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 4 (transitivity) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(fresh13(true, true, Y, additive_identity, Y), true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 11 (product_substitution3) }
% 86.91/11.59 fresh7(fresh(fresh30(fresh21(true, true, Y, additive_identity), true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 6 (multiplication_is_well_defined) }
% 86.91/11.59 fresh7(fresh(fresh30(true, true, Y, additive_identity, X), true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 10 (add_substitution1) }
% 86.91/11.59 fresh7(fresh(true, true, add(Y, X), X), true, X, X, Y)
% 86.91/11.59 = { by axiom 4 (transitivity) }
% 86.91/11.59 fresh7(true, true, X, X, Y)
% 86.91/11.59 = { by axiom 12 (sum_substitution1) }
% 86.91/11.59 true
% 86.91/11.59
% 86.91/11.59 Goal 1 (prove_c_equals_d): equalish(c, d) = true.
% 86.91/11.59 Proof:
% 86.91/11.59 equalish(c, d)
% 86.91/11.59 = { by axiom 14 (addition_is_well_defined) R->L }
% 86.91/11.59 fresh29(true, true, X, X, c, d)
% 86.91/11.59 = { by lemma 25 R->L }
% 86.91/11.59 fresh29(sum(X, X, d), true, X, X, c, d)
% 86.91/11.59 = { by axiom 21 (addition_is_well_defined) }
% 86.91/11.59 fresh28(sum(X, X, c), true, c, d)
% 86.91/11.59 = { by lemma 25 }
% 86.91/11.59 fresh28(true, true, c, d)
% 86.91/11.59 = { by axiom 5 (addition_is_well_defined) }
% 86.91/11.59 true
% 86.91/11.59 % SZS output end Proof
% 86.91/11.59
% 86.91/11.59 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------