TSTP Solution File: GRP012-2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP012-2 : 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 01:16:33 EDT 2023
% Result : Unsatisfiable 0.20s 0.43s
% 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
% 0.07/0.13 % Problem : GRP012-2 : 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
% 0.13/0.35 % Computer : n021.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % 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 : Tue Aug 29 01:49:44 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.43 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
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% 0.20/0.43 % SZS status Unsatisfiable
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% 0.20/0.44 % SZS output start Proof
% 0.20/0.44 Take the following subset of the input axioms:
% 0.20/0.44 fof(a_multiply_b_is_c, hypothesis, product(a, b, c)).
% 0.20/0.44 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))))).
% 0.20/0.44 fof(inverse_b_multiply_inverse_a_is_d, hypothesis, product(inverse(b), inverse(a), d)).
% 0.20/0.44 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 0.20/0.44 fof(left_inverse, axiom, ![X2]: product(inverse(X2), X2, identity)).
% 0.20/0.44 fof(prove_c_inverse_equals_d, negated_conjecture, inverse(c)!=d).
% 0.20/0.44 fof(right_identity, axiom, ![X2]: product(X2, identity, X2)).
% 0.20/0.44 fof(right_inverse, axiom, ![X2]: product(X2, inverse(X2), identity)).
% 0.20/0.44 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.20/0.44 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 0.20/0.44
% 0.20/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.44 fresh(y, y, x1...xn) = u
% 0.20/0.44 C => fresh(s, t, x1...xn) = v
% 0.20/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.44 variables of u and v.
% 0.20/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.44 input problem has no model of domain size 1).
% 0.20/0.44
% 0.20/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.44
% 0.20/0.44 Axiom 1 (right_identity): product(X, identity, X) = true.
% 0.20/0.44 Axiom 2 (left_identity): product(identity, X, X) = true.
% 0.20/0.44 Axiom 3 (a_multiply_b_is_c): product(a, b, c) = true.
% 0.20/0.44 Axiom 4 (right_inverse): product(X, inverse(X), identity) = true.
% 0.20/0.44 Axiom 5 (left_inverse): product(inverse(X), X, identity) = true.
% 0.20/0.44 Axiom 6 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.20/0.44 Axiom 7 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 0.20/0.44 Axiom 8 (inverse_b_multiply_inverse_a_is_d): product(inverse(b), inverse(a), d) = true.
% 0.20/0.44 Axiom 9 (associativity2): fresh6(X, X, Y, Z, W) = true.
% 0.20/0.44 Axiom 10 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.44 Axiom 11 (associativity2): fresh3(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.20/0.45 Axiom 12 (associativity2): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, W), true, W, V, T).
% 0.20/0.45 Axiom 13 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.20/0.45 Axiom 14 (associativity2): fresh5(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh3(product(W, Z, U), true, W, X, V, Y, U).
% 0.20/0.45
% 0.20/0.45 Lemma 15: fresh5(X, X, Y, Z, multiply(Y, Z), W, V, U) = true.
% 0.20/0.45 Proof:
% 0.20/0.45 fresh5(X, X, Y, Z, multiply(Y, Z), W, V, U)
% 0.20/0.45 = { by axiom 12 (associativity2) }
% 0.20/0.45 fresh6(product(Y, Z, multiply(Y, Z)), true, multiply(Y, Z), W, U)
% 0.20/0.45 = { by axiom 7 (total_function1) }
% 0.20/0.45 fresh6(true, true, multiply(Y, Z), W, U)
% 0.20/0.45 = { by axiom 9 (associativity2) }
% 0.20/0.45 true
% 0.20/0.45
% 0.20/0.45 Lemma 16: fresh(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 0.20/0.45 Proof:
% 0.20/0.45 fresh(product(X, Y, Z), true, Z, multiply(X, Y))
% 0.20/0.45 = { by axiom 13 (total_function2) R->L }
% 0.20/0.45 fresh2(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 0.20/0.45 = { by axiom 7 (total_function1) }
% 0.20/0.45 fresh2(true, true, X, Y, Z, multiply(X, Y))
% 0.20/0.45 = { by axiom 10 (total_function2) }
% 0.20/0.45 Z
% 0.20/0.45
% 0.20/0.45 Lemma 17: multiply(multiply(X, Y), inverse(Y)) = X.
% 0.20/0.45 Proof:
% 0.20/0.45 multiply(multiply(X, Y), inverse(Y))
% 0.20/0.45 = { by axiom 6 (total_function2) R->L }
% 0.20/0.45 fresh(true, true, X, multiply(multiply(X, Y), inverse(Y)))
% 0.20/0.45 = { by lemma 15 R->L }
% 0.20/0.45 fresh(fresh5(true, true, X, Y, multiply(X, Y), inverse(Y), identity, X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 0.20/0.45 = { by axiom 4 (right_inverse) R->L }
% 0.20/0.45 fresh(fresh5(product(Y, inverse(Y), identity), true, X, Y, multiply(X, Y), inverse(Y), identity, X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 0.20/0.45 = { by axiom 14 (associativity2) }
% 0.20/0.45 fresh(fresh3(product(X, identity, X), true, X, Y, multiply(X, Y), inverse(Y), X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 0.20/0.45 = { by axiom 1 (right_identity) }
% 0.20/0.45 fresh(fresh3(true, true, X, Y, multiply(X, Y), inverse(Y), X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 0.20/0.45 = { by axiom 11 (associativity2) }
% 0.20/0.45 fresh(product(multiply(X, Y), inverse(Y), X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 0.20/0.45 = { by lemma 16 }
% 0.20/0.45 X
% 0.20/0.45
% 0.20/0.45 Goal 1 (prove_c_inverse_equals_d): inverse(c) = d.
% 0.20/0.45 Proof:
% 0.20/0.45 inverse(c)
% 0.20/0.45 = { by lemma 17 R->L }
% 0.20/0.45 multiply(multiply(inverse(c), a), inverse(a))
% 0.20/0.45 = { by lemma 17 R->L }
% 0.20/0.45 multiply(multiply(multiply(multiply(inverse(c), a), b), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 6 (total_function2) R->L }
% 0.20/0.45 multiply(multiply(fresh(true, true, identity, multiply(multiply(inverse(c), a), b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by lemma 15 R->L }
% 0.20/0.45 multiply(multiply(fresh(fresh5(true, true, inverse(c), a, multiply(inverse(c), a), b, c, identity), true, identity, multiply(multiply(inverse(c), a), b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 3 (a_multiply_b_is_c) R->L }
% 0.20/0.45 multiply(multiply(fresh(fresh5(product(a, b, c), true, inverse(c), a, multiply(inverse(c), a), b, c, identity), true, identity, multiply(multiply(inverse(c), a), b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 14 (associativity2) }
% 0.20/0.45 multiply(multiply(fresh(fresh3(product(inverse(c), c, identity), true, inverse(c), a, multiply(inverse(c), a), b, identity), true, identity, multiply(multiply(inverse(c), a), b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 5 (left_inverse) }
% 0.20/0.45 multiply(multiply(fresh(fresh3(true, true, inverse(c), a, multiply(inverse(c), a), b, identity), true, identity, multiply(multiply(inverse(c), a), b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 11 (associativity2) }
% 0.20/0.45 multiply(multiply(fresh(product(multiply(inverse(c), a), b, identity), true, identity, multiply(multiply(inverse(c), a), b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by lemma 16 }
% 0.20/0.45 multiply(multiply(identity, inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 10 (total_function2) R->L }
% 0.20/0.45 multiply(fresh2(true, true, identity, inverse(b), multiply(identity, inverse(b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 2 (left_identity) R->L }
% 0.20/0.45 multiply(fresh2(product(identity, inverse(b), inverse(b)), true, identity, inverse(b), multiply(identity, inverse(b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 13 (total_function2) }
% 0.20/0.45 multiply(fresh(product(identity, inverse(b), multiply(identity, inverse(b))), true, multiply(identity, inverse(b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 7 (total_function1) }
% 0.20/0.45 multiply(fresh(true, true, multiply(identity, inverse(b)), inverse(b)), inverse(a))
% 0.20/0.45 = { by axiom 6 (total_function2) }
% 0.20/0.45 multiply(inverse(b), inverse(a))
% 0.20/0.45 = { by axiom 6 (total_function2) R->L }
% 0.20/0.45 fresh(true, true, d, multiply(inverse(b), inverse(a)))
% 0.20/0.45 = { by axiom 8 (inverse_b_multiply_inverse_a_is_d) R->L }
% 0.20/0.45 fresh(product(inverse(b), inverse(a), d), true, d, multiply(inverse(b), inverse(a)))
% 0.20/0.45 = { by lemma 16 }
% 0.20/0.45 d
% 0.20/0.45 % SZS output end Proof
% 0.20/0.45
% 0.20/0.45 RESULT: Unsatisfiable (the axioms are contradictory).
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