TSTP Solution File: GRP012-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP012-1 : 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 : 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 01:16:33 EDT 2023
% Result : Unsatisfiable 0.21s 0.45s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : GRP012-1 : TPTP v8.1.2. Released v1.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n028.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 00:51:40 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.21/0.45 Command-line arguments: --no-flatten-goal
% 0.21/0.45
% 0.21/0.45 % SZS status Unsatisfiable
% 0.21/0.45
% 0.21/0.47 % SZS output start Proof
% 0.21/0.47 Take the following subset of the input axioms:
% 0.21/0.47 fof(a_multiply_b_is_c, hypothesis, product(a, b, c)).
% 0.21/0.47 fof(associativity1, axiom, ![X, Y, Z, W, U, V]: (~product(X, Y, U) | (~product(Y, Z, V) | (~product(U, Z, W) | product(X, V, W))))).
% 0.21/0.47 fof(associativity2, axiom, ![X2, Y2, Z2, W2, U2, V2]: (~product(X2, Y2, U2) | (~product(Y2, Z2, V2) | (~product(X2, V2, W2) | product(U2, Z2, W2))))).
% 0.21/0.47 fof(inverse_b_multiply_inverse_a_is_d, hypothesis, product(inverse(b), inverse(a), d)).
% 0.21/0.47 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 0.21/0.47 fof(left_inverse, axiom, ![X2]: product(inverse(X2), X2, identity)).
% 0.21/0.47 fof(prove_c_multiply_d_is_identity, negated_conjecture, ~product(c, d, identity)).
% 0.21/0.47 fof(right_identity, axiom, ![X2]: product(X2, identity, X2)).
% 0.21/0.47 fof(right_inverse, axiom, ![X2]: product(X2, inverse(X2), identity)).
% 0.21/0.47 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.21/0.47 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 0.21/0.47
% 0.21/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.47 fresh(y, y, x1...xn) = u
% 0.21/0.47 C => fresh(s, t, x1...xn) = v
% 0.21/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.47 variables of u and v.
% 0.21/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.47 input problem has no model of domain size 1).
% 0.21/0.47
% 0.21/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.47
% 0.21/0.47 Axiom 1 (right_identity): product(X, identity, X) = true.
% 0.21/0.47 Axiom 2 (a_multiply_b_is_c): product(a, b, c) = true.
% 0.21/0.47 Axiom 3 (left_identity): product(identity, X, X) = true.
% 0.21/0.47 Axiom 4 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.21/0.47 Axiom 5 (right_inverse): product(X, inverse(X), identity) = true.
% 0.21/0.47 Axiom 6 (left_inverse): product(inverse(X), X, identity) = true.
% 0.21/0.47 Axiom 7 (associativity1): fresh8(X, X, Y, Z, W) = true.
% 0.21/0.47 Axiom 8 (associativity2): fresh6(X, X, Y, Z, W) = true.
% 0.21/0.47 Axiom 9 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 0.21/0.47 Axiom 10 (inverse_b_multiply_inverse_a_is_d): product(inverse(b), inverse(a), d) = true.
% 0.21/0.47 Axiom 11 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.21/0.47 Axiom 12 (associativity1): fresh4(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.21/0.47 Axiom 13 (associativity2): fresh3(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.21/0.47 Axiom 14 (associativity1): fresh7(X, X, Y, Z, W, V, U, T) = fresh8(product(Y, Z, W), true, Y, U, T).
% 0.21/0.47 Axiom 15 (associativity2): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, W), true, W, V, T).
% 0.21/0.47 Axiom 16 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.21/0.47 Axiom 17 (associativity1): fresh7(product(X, Y, Z), true, W, V, X, Y, U, Z) = fresh4(product(V, Y, U), true, W, V, X, U, Z).
% 0.21/0.47 Axiom 18 (associativity2): fresh5(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh3(product(W, Z, U), true, W, X, V, Y, U).
% 0.21/0.47
% 0.21/0.47 Lemma 19: fresh(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 0.21/0.47 Proof:
% 0.21/0.47 fresh(product(X, Y, Z), true, Z, multiply(X, Y))
% 0.21/0.47 = { by axiom 16 (total_function2) R->L }
% 0.21/0.47 fresh2(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 0.21/0.47 = { by axiom 9 (total_function1) }
% 0.21/0.47 fresh2(true, true, X, Y, Z, multiply(X, Y))
% 0.21/0.47 = { by axiom 11 (total_function2) }
% 0.21/0.47 Z
% 0.21/0.47
% 0.21/0.47 Lemma 20: multiply(multiply(X, inverse(Y)), Y) = X.
% 0.21/0.47 Proof:
% 0.21/0.47 multiply(multiply(X, inverse(Y)), Y)
% 0.21/0.47 = { by axiom 4 (total_function2) R->L }
% 0.21/0.47 fresh(true, true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.47 = { by axiom 8 (associativity2) R->L }
% 0.21/0.47 fresh(fresh6(true, true, multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.47 = { by axiom 9 (total_function1) R->L }
% 0.21/0.47 fresh(fresh6(product(X, inverse(Y), multiply(X, inverse(Y))), true, multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.47 = { by axiom 15 (associativity2) R->L }
% 0.21/0.47 fresh(fresh5(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.47 = { by axiom 6 (left_inverse) R->L }
% 0.21/0.47 fresh(fresh5(product(inverse(Y), Y, identity), true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.47 = { by axiom 18 (associativity2) }
% 0.21/0.48 fresh(fresh3(product(X, identity, X), true, X, inverse(Y), multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.48 = { by axiom 1 (right_identity) }
% 0.21/0.48 fresh(fresh3(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.48 = { by axiom 13 (associativity2) }
% 0.21/0.48 fresh(product(multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.21/0.48 = { by lemma 19 }
% 0.21/0.48 X
% 0.21/0.48
% 0.21/0.48 Lemma 21: fresh7(X, X, inverse(Y), Y, identity, Z, W, V) = true.
% 0.21/0.48 Proof:
% 0.21/0.48 fresh7(X, X, inverse(Y), Y, identity, Z, W, V)
% 0.21/0.48 = { by axiom 14 (associativity1) }
% 0.21/0.48 fresh8(product(inverse(Y), Y, identity), true, inverse(Y), W, V)
% 0.21/0.48 = { by axiom 6 (left_inverse) }
% 0.21/0.48 fresh8(true, true, inverse(Y), W, V)
% 0.21/0.48 = { by axiom 7 (associativity1) }
% 0.21/0.48 true
% 0.21/0.48
% 0.21/0.48 Lemma 22: fresh4(product(X, Y, Z), true, W, X, identity, Z, Y) = fresh7(V, V, W, X, identity, U, Z, Y).
% 0.21/0.48 Proof:
% 0.21/0.48 fresh4(product(X, Y, Z), true, W, X, identity, Z, Y)
% 0.21/0.48 = { by axiom 17 (associativity1) R->L }
% 0.21/0.48 fresh7(product(identity, Y, Y), true, W, X, identity, Y, Z, Y)
% 0.21/0.48 = { by axiom 3 (left_identity) }
% 0.21/0.48 fresh7(true, true, W, X, identity, Y, Z, Y)
% 0.21/0.48 = { by axiom 14 (associativity1) }
% 0.21/0.48 fresh8(product(W, X, identity), true, W, Z, Y)
% 0.21/0.48 = { by axiom 14 (associativity1) R->L }
% 0.21/0.48 fresh7(V, V, W, X, identity, U, Z, Y)
% 0.21/0.48
% 0.21/0.48 Goal 1 (prove_c_multiply_d_is_identity): product(c, d, identity) = true.
% 0.21/0.48 Proof:
% 0.21/0.48 product(c, d, identity)
% 0.21/0.48 = { by lemma 19 R->L }
% 0.21/0.48 product(c, fresh(product(inverse(b), inverse(a), d), true, d, multiply(inverse(b), inverse(a))), identity)
% 0.21/0.48 = { by axiom 10 (inverse_b_multiply_inverse_a_is_d) }
% 0.21/0.48 product(c, fresh(true, true, d, multiply(inverse(b), inverse(a))), identity)
% 0.21/0.48 = { by axiom 4 (total_function2) }
% 0.21/0.48 product(c, multiply(inverse(b), inverse(a)), identity)
% 0.21/0.48 = { by lemma 20 R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), inverse(inverse(c))), inverse(c))), identity)
% 0.21/0.48 = { by lemma 20 R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), multiply(multiply(inverse(inverse(c)), inverse(c)), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 11 (total_function2) R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), multiply(fresh2(true, true, inverse(inverse(c)), inverse(c), multiply(inverse(inverse(c)), inverse(c)), identity), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 6 (left_inverse) R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), multiply(fresh2(product(inverse(inverse(c)), inverse(c), identity), true, inverse(inverse(c)), inverse(c), multiply(inverse(inverse(c)), inverse(c)), identity), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 16 (total_function2) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), multiply(fresh(product(inverse(inverse(c)), inverse(c), multiply(inverse(inverse(c)), inverse(c))), true, multiply(inverse(inverse(c)), inverse(c)), identity), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 9 (total_function1) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), multiply(fresh(true, true, multiply(inverse(inverse(c)), inverse(c)), identity), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 4 (total_function2) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), multiply(identity, c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 11 (total_function2) R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), fresh2(true, true, identity, c, multiply(identity, c), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 3 (left_identity) R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), fresh2(product(identity, c, c), true, identity, c, multiply(identity, c), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 16 (total_function2) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), fresh(product(identity, c, multiply(identity, c)), true, multiply(identity, c), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 9 (total_function1) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), fresh(true, true, multiply(identity, c), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 4 (total_function2) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(multiply(inverse(a), c), inverse(c))), identity)
% 0.21/0.48 = { by axiom 4 (total_function2) R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(fresh(true, true, b, multiply(inverse(a), c)), inverse(c))), identity)
% 0.21/0.48 = { by lemma 21 R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(fresh(fresh7(X, X, inverse(a), a, identity, Y, c, b), true, b, multiply(inverse(a), c)), inverse(c))), identity)
% 0.21/0.48 = { by lemma 22 R->L }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(fresh(fresh4(product(a, b, c), true, inverse(a), a, identity, c, b), true, b, multiply(inverse(a), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 2 (a_multiply_b_is_c) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(fresh(fresh4(true, true, inverse(a), a, identity, c, b), true, b, multiply(inverse(a), c)), inverse(c))), identity)
% 0.21/0.48 = { by axiom 12 (associativity1) }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(fresh(product(inverse(a), c, b), true, b, multiply(inverse(a), c)), inverse(c))), identity)
% 0.21/0.48 = { by lemma 19 }
% 0.21/0.48 product(c, multiply(inverse(b), multiply(b, inverse(c))), identity)
% 0.21/0.48 = { by axiom 4 (total_function2) R->L }
% 0.21/0.48 product(c, fresh(true, true, inverse(c), multiply(inverse(b), multiply(b, inverse(c)))), identity)
% 0.21/0.48 = { by lemma 21 R->L }
% 0.21/0.48 product(c, fresh(fresh7(Z, Z, inverse(b), b, identity, W, multiply(b, inverse(c)), inverse(c)), true, inverse(c), multiply(inverse(b), multiply(b, inverse(c)))), identity)
% 0.21/0.48 = { by lemma 22 R->L }
% 0.21/0.48 product(c, fresh(fresh4(product(b, inverse(c), multiply(b, inverse(c))), true, inverse(b), b, identity, multiply(b, inverse(c)), inverse(c)), true, inverse(c), multiply(inverse(b), multiply(b, inverse(c)))), identity)
% 0.21/0.48 = { by axiom 9 (total_function1) }
% 0.21/0.48 product(c, fresh(fresh4(true, true, inverse(b), b, identity, multiply(b, inverse(c)), inverse(c)), true, inverse(c), multiply(inverse(b), multiply(b, inverse(c)))), identity)
% 0.21/0.48 = { by axiom 12 (associativity1) }
% 0.21/0.48 product(c, fresh(product(inverse(b), multiply(b, inverse(c)), inverse(c)), true, inverse(c), multiply(inverse(b), multiply(b, inverse(c)))), identity)
% 0.21/0.48 = { by lemma 19 }
% 0.21/0.48 product(c, inverse(c), identity)
% 0.21/0.48 = { by axiom 5 (right_inverse) }
% 0.21/0.48 true
% 0.21/0.48 % SZS output end Proof
% 0.21/0.48
% 0.21/0.48 RESULT: Unsatisfiable (the axioms are contradictory).
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