TSTP Solution File: GRP047-2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP047-2 : 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 : n022.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 01:16:46 EDT 2023
% Result : Unsatisfiable 5.65s 1.13s
% Output : Proof 5.93s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP047-2 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13 % 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 : n022.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 : Mon Aug 28 21:11:39 EDT 2023
% 0.13/0.34 % CPUTime :
% 5.65/1.13 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 5.65/1.13
% 5.65/1.13 % SZS status Unsatisfiable
% 5.65/1.13
% 5.93/1.14 % SZS output start Proof
% 5.93/1.14 Take the following subset of the input axioms:
% 5.93/1.14 fof(a_equals_b, hypothesis, equalish(a, b)).
% 5.93/1.14 fof(associativity2, axiom, ![X, Y, Z, W, U, V]: (~product(X, Y, U) | (~product(Y, Z, V) | (~product(X, V, W) | product(U, Z, W))))).
% 5.93/1.14 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 5.93/1.14 fof(product_substitution3, axiom, ![X2, Y2, Z2, W2]: (~equalish(X2, Y2) | (~product(W2, Z2, X2) | product(W2, Z2, Y2)))).
% 5.93/1.14 fof(prove_multiply_substitution2, negated_conjecture, ~equalish(multiply(c, a), multiply(c, b))).
% 5.93/1.14 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 5.93/1.14 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | equalish(Z2, W2)))).
% 5.93/1.14
% 5.93/1.14 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.93/1.14 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.93/1.14 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.93/1.14 fresh(y, y, x1...xn) = u
% 5.93/1.14 C => fresh(s, t, x1...xn) = v
% 5.93/1.14 where fresh is a fresh function symbol and x1..xn are the free
% 5.93/1.14 variables of u and v.
% 5.93/1.14 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.93/1.14 input problem has no model of domain size 1).
% 5.93/1.14
% 5.93/1.14 The encoding turns the above axioms into the following unit equations and goals:
% 5.93/1.14
% 5.93/1.14 Axiom 1 (a_equals_b): equalish(a, b) = true.
% 5.93/1.14 Axiom 2 (left_identity): product(identity, X, X) = true.
% 5.93/1.14 Axiom 3 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 5.93/1.14 Axiom 4 (total_function2): fresh(X, X, Y, Z) = true.
% 5.93/1.14 Axiom 5 (associativity2): fresh8(X, X, Y, Z, W) = true.
% 5.93/1.14 Axiom 6 (product_substitution3): fresh3(X, X, Y, Z, W) = true.
% 5.93/1.14 Axiom 7 (product_substitution3): fresh4(X, X, Y, Z, W, V) = product(W, V, Z).
% 5.93/1.14 Axiom 8 (total_function2): fresh2(X, X, Y, Z, W, V) = equalish(W, V).
% 5.93/1.14 Axiom 9 (associativity2): fresh5(X, X, Y, Z, W, V, U) = product(W, V, U).
% 5.93/1.14 Axiom 10 (product_substitution3): fresh4(equalish(X, Y), true, X, Y, Z, W) = fresh3(product(Z, W, X), true, Y, Z, W).
% 5.93/1.14 Axiom 11 (associativity2): fresh7(X, X, Y, Z, W, V, U, T) = fresh8(product(Y, Z, W), true, W, V, T).
% 5.93/1.14 Axiom 12 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 5.93/1.14 Axiom 13 (associativity2): fresh7(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh5(product(W, Z, U), true, W, X, V, Y, U).
% 5.93/1.14
% 5.93/1.14 Lemma 14: fresh4(equalish(X, Y), true, X, Y, identity, X) = true.
% 5.93/1.14 Proof:
% 5.93/1.14 fresh4(equalish(X, Y), true, X, Y, identity, X)
% 5.93/1.14 = { by axiom 10 (product_substitution3) }
% 5.93/1.14 fresh3(product(identity, X, X), true, Y, identity, X)
% 5.93/1.14 = { by axiom 2 (left_identity) }
% 5.93/1.14 fresh3(true, true, Y, identity, X)
% 5.93/1.14 = { by axiom 6 (product_substitution3) }
% 5.93/1.14 true
% 5.93/1.14
% 5.93/1.14 Lemma 15: fresh7(X, X, Y, Z, multiply(Y, Z), W, V, U) = true.
% 5.93/1.14 Proof:
% 5.93/1.14 fresh7(X, X, Y, Z, multiply(Y, Z), W, V, U)
% 5.93/1.14 = { by axiom 11 (associativity2) }
% 5.93/1.14 fresh8(product(Y, Z, multiply(Y, Z)), true, multiply(Y, Z), W, U)
% 5.93/1.14 = { by axiom 3 (total_function1) }
% 5.93/1.14 fresh8(true, true, multiply(Y, Z), W, U)
% 5.93/1.14 = { by axiom 5 (associativity2) }
% 5.93/1.14 true
% 5.93/1.14
% 5.93/1.14 Lemma 16: fresh7(product(X, Y, Z), true, W, X, V, Y, Z, multiply(W, Z)) = product(V, Y, multiply(W, Z)).
% 5.93/1.14 Proof:
% 5.93/1.14 fresh7(product(X, Y, Z), true, W, X, V, Y, Z, multiply(W, Z))
% 5.93/1.14 = { by axiom 13 (associativity2) }
% 5.93/1.14 fresh5(product(W, Z, multiply(W, Z)), true, W, X, V, Y, multiply(W, Z))
% 5.93/1.14 = { by axiom 3 (total_function1) }
% 5.93/1.14 fresh5(true, true, W, X, V, Y, multiply(W, Z))
% 5.93/1.14 = { by axiom 9 (associativity2) }
% 5.93/1.14 product(V, Y, multiply(W, Z))
% 5.93/1.14
% 5.93/1.14 Goal 1 (prove_multiply_substitution2): equalish(multiply(c, a), multiply(c, b)) = true.
% 5.93/1.14 Proof:
% 5.93/1.14 equalish(multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 8 (total_function2) R->L }
% 5.93/1.14 fresh2(true, true, identity, multiply(c, b), multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 2 (left_identity) R->L }
% 5.93/1.14 fresh2(product(identity, multiply(c, b), multiply(c, b)), true, identity, multiply(c, b), multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 12 (total_function2) }
% 5.93/1.14 fresh(product(identity, multiply(c, b), multiply(c, a)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 7 (product_substitution3) R->L }
% 5.93/1.14 fresh(fresh4(true, true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 4 (total_function2) R->L }
% 5.93/1.14 fresh(fresh4(fresh(true, true, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by lemma 15 R->L }
% 5.93/1.14 fresh(fresh4(fresh(fresh7(true, true, c, identity, multiply(c, identity), a, b, multiply(c, b)), true, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by lemma 14 R->L }
% 5.93/1.14 fresh(fresh4(fresh(fresh7(fresh4(equalish(a, b), true, a, b, identity, a), true, c, identity, multiply(c, identity), a, b, multiply(c, b)), true, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 1 (a_equals_b) }
% 5.93/1.14 fresh(fresh4(fresh(fresh7(fresh4(true, true, a, b, identity, a), true, c, identity, multiply(c, identity), a, b, multiply(c, b)), true, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 7 (product_substitution3) }
% 5.93/1.14 fresh(fresh4(fresh(fresh7(product(identity, a, b), true, c, identity, multiply(c, identity), a, b, multiply(c, b)), true, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by lemma 16 }
% 5.93/1.14 fresh(fresh4(fresh(product(multiply(c, identity), a, multiply(c, b)), true, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 12 (total_function2) R->L }
% 5.93/1.14 fresh(fresh4(fresh2(product(multiply(c, identity), a, multiply(c, a)), true, multiply(c, identity), a, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by lemma 16 R->L }
% 5.93/1.14 fresh(fresh4(fresh2(fresh7(product(identity, a, a), true, c, identity, multiply(c, identity), a, a, multiply(c, a)), true, multiply(c, identity), a, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 2 (left_identity) }
% 5.93/1.14 fresh(fresh4(fresh2(fresh7(true, true, c, identity, multiply(c, identity), a, a, multiply(c, a)), true, multiply(c, identity), a, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by lemma 15 }
% 5.93/1.14 fresh(fresh4(fresh2(true, true, multiply(c, identity), a, multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 8 (total_function2) }
% 5.93/1.14 fresh(fresh4(equalish(multiply(c, b), multiply(c, a)), true, multiply(c, b), multiply(c, a), identity, multiply(c, b)), true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by lemma 14 }
% 5.93/1.14 fresh(true, true, multiply(c, a), multiply(c, b))
% 5.93/1.14 = { by axiom 4 (total_function2) }
% 5.93/1.14 true
% 5.93/1.14 % SZS output end Proof
% 5.93/1.14
% 5.93/1.14 RESULT: Unsatisfiable (the axioms are contradictory).
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