TSTP Solution File: GRP012-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP012-3 : 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 : n014.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 01:16:34 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
% 0.00/0.13 % Problem : GRP012-3 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14 % 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 : n014.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 00:01:31 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.44 Command-line arguments: --no-flatten-goal
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% 0.20/0.44 % SZS status Unsatisfiable
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% 0.20/0.45 % SZS output start Proof
% 0.20/0.45 Take the following subset of the input axioms:
% 0.20/0.46 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))))).
% 0.20/0.46 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))))).
% 0.20/0.46 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 0.20/0.46 fof(left_inverse, axiom, ![X2]: product(inverse(X2), X2, identity)).
% 0.20/0.46 fof(prove_inverse_of_product_is_product_of_inverses, negated_conjecture, inverse(multiply(a, b))!=multiply(inverse(b), inverse(a))).
% 0.20/0.46 fof(right_identity, axiom, ![X2]: product(X2, identity, X2)).
% 0.20/0.46 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.20/0.46 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 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 (right_identity): product(X, identity, X) = true.
% 0.20/0.46 Axiom 2 (left_identity): product(identity, X, X) = true.
% 0.20/0.46 Axiom 3 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.20/0.46 Axiom 4 (left_inverse): product(inverse(X), X, identity) = true.
% 0.20/0.46 Axiom 5 (associativity1): fresh8(X, X, Y, Z, W) = true.
% 0.20/0.46 Axiom 6 (associativity2): fresh6(X, X, Y, Z, W) = true.
% 0.20/0.46 Axiom 7 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 0.20/0.46 Axiom 8 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.46 Axiom 9 (associativity1): fresh4(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.20/0.46 Axiom 10 (associativity2): fresh3(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.20/0.46 Axiom 11 (associativity1): fresh7(X, X, Y, Z, W, V, U, T) = fresh8(product(Y, Z, W), true, Y, U, T).
% 0.20/0.46 Axiom 12 (associativity2): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, W), true, W, V, T).
% 0.20/0.46 Axiom 13 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.20/0.46 Axiom 14 (associativity1): fresh7(product(X, Y, Z), true, W, V, X, Y, U, Z) = fresh4(product(V, Y, U), true, W, V, X, U, Z).
% 0.20/0.46 Axiom 15 (associativity2): fresh5(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh3(product(W, Z, U), true, W, X, V, Y, U).
% 0.20/0.46
% 0.20/0.46 Lemma 16: fresh(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 0.20/0.46 Proof:
% 0.20/0.46 fresh(product(X, Y, Z), true, Z, multiply(X, Y))
% 0.20/0.46 = { by axiom 13 (total_function2) R->L }
% 0.20/0.46 fresh2(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 0.20/0.46 = { by axiom 7 (total_function1) }
% 0.20/0.46 fresh2(true, true, X, Y, Z, multiply(X, Y))
% 0.20/0.46 = { by axiom 8 (total_function2) }
% 0.20/0.46 Z
% 0.20/0.46
% 0.20/0.46 Lemma 17: multiply(multiply(X, inverse(Y)), Y) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 multiply(multiply(X, inverse(Y)), Y)
% 0.20/0.46 = { by axiom 3 (total_function2) R->L }
% 0.20/0.46 fresh(true, true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.46 = { by axiom 6 (associativity2) R->L }
% 0.20/0.46 fresh(fresh6(true, true, multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.46 = { by axiom 7 (total_function1) R->L }
% 0.20/0.46 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))
% 0.20/0.46 = { by axiom 12 (associativity2) R->L }
% 0.20/0.46 fresh(fresh5(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.46 = { by axiom 4 (left_inverse) R->L }
% 0.20/0.46 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))
% 0.20/0.46 = { by axiom 15 (associativity2) }
% 0.20/0.46 fresh(fresh3(product(X, identity, X), true, X, inverse(Y), multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.46 = { by axiom 1 (right_identity) }
% 0.20/0.46 fresh(fresh3(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.46 = { by axiom 10 (associativity2) }
% 0.20/0.46 fresh(product(multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.46 = { by lemma 16 }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Lemma 18: inverse(inverse(X)) = X.
% 0.20/0.46 Proof:
% 0.20/0.46 inverse(inverse(X))
% 0.20/0.46 = { by lemma 17 R->L }
% 0.20/0.46 multiply(multiply(inverse(inverse(X)), inverse(X)), X)
% 0.20/0.46 = { by axiom 8 (total_function2) R->L }
% 0.20/0.46 multiply(fresh2(true, true, inverse(inverse(X)), inverse(X), multiply(inverse(inverse(X)), inverse(X)), identity), X)
% 0.20/0.46 = { by axiom 4 (left_inverse) R->L }
% 0.20/0.46 multiply(fresh2(product(inverse(inverse(X)), inverse(X), identity), true, inverse(inverse(X)), inverse(X), multiply(inverse(inverse(X)), inverse(X)), identity), X)
% 0.20/0.46 = { by axiom 13 (total_function2) }
% 0.20/0.46 multiply(fresh(product(inverse(inverse(X)), inverse(X), multiply(inverse(inverse(X)), inverse(X))), true, multiply(inverse(inverse(X)), inverse(X)), identity), X)
% 0.20/0.46 = { by axiom 7 (total_function1) }
% 0.20/0.46 multiply(fresh(true, true, multiply(inverse(inverse(X)), inverse(X)), identity), X)
% 0.20/0.46 = { by axiom 3 (total_function2) }
% 0.20/0.46 multiply(identity, X)
% 0.20/0.46 = { by axiom 8 (total_function2) R->L }
% 0.20/0.46 fresh2(true, true, identity, X, multiply(identity, X), X)
% 0.20/0.46 = { by axiom 2 (left_identity) R->L }
% 0.20/0.46 fresh2(product(identity, X, X), true, identity, X, multiply(identity, X), X)
% 0.20/0.46 = { by axiom 13 (total_function2) }
% 0.20/0.46 fresh(product(identity, X, multiply(identity, X)), true, multiply(identity, X), X)
% 0.20/0.46 = { by axiom 7 (total_function1) }
% 0.20/0.46 fresh(true, true, multiply(identity, X), X)
% 0.20/0.46 = { by axiom 3 (total_function2) }
% 0.20/0.46 X
% 0.20/0.46
% 0.20/0.46 Goal 1 (prove_inverse_of_product_is_product_of_inverses): inverse(multiply(a, b)) = multiply(inverse(b), inverse(a)).
% 0.20/0.46 Proof:
% 0.20/0.46 inverse(multiply(a, b))
% 0.20/0.46 = { by lemma 17 R->L }
% 0.20/0.46 multiply(multiply(inverse(multiply(a, b)), inverse(inverse(a))), inverse(a))
% 0.20/0.46 = { by lemma 18 }
% 0.20/0.46 multiply(multiply(inverse(multiply(a, b)), a), inverse(a))
% 0.20/0.46 = { by lemma 18 R->L }
% 0.20/0.46 multiply(multiply(inverse(multiply(a, inverse(inverse(b)))), a), inverse(a))
% 0.20/0.46 = { by lemma 17 R->L }
% 0.20/0.46 multiply(multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b))), inverse(a))
% 0.20/0.46 = { by axiom 3 (total_function2) R->L }
% 0.20/0.46 multiply(fresh(true, true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 5 (associativity1) R->L }
% 0.20/0.46 multiply(fresh(fresh8(true, true, inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 4 (left_inverse) R->L }
% 0.20/0.46 multiply(fresh(fresh8(product(inverse(multiply(a, inverse(inverse(b)))), multiply(a, inverse(inverse(b))), identity), true, inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 11 (associativity1) R->L }
% 0.20/0.46 multiply(fresh(fresh7(true, true, inverse(multiply(a, inverse(inverse(b)))), multiply(a, inverse(inverse(b))), identity, inverse(b), multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 2 (left_identity) R->L }
% 0.20/0.46 multiply(fresh(fresh7(product(identity, inverse(b), inverse(b)), true, inverse(multiply(a, inverse(inverse(b)))), multiply(a, inverse(inverse(b))), identity, inverse(b), multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 14 (associativity1) }
% 0.20/0.46 multiply(fresh(fresh4(product(multiply(a, inverse(inverse(b))), inverse(b), multiply(multiply(a, inverse(inverse(b))), inverse(b))), true, inverse(multiply(a, inverse(inverse(b)))), multiply(a, inverse(inverse(b))), identity, multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 7 (total_function1) }
% 0.20/0.46 multiply(fresh(fresh4(true, true, inverse(multiply(a, inverse(inverse(b)))), multiply(a, inverse(inverse(b))), identity, multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by axiom 9 (associativity1) }
% 0.20/0.46 multiply(fresh(product(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)), inverse(b)), true, inverse(b), multiply(inverse(multiply(a, inverse(inverse(b)))), multiply(multiply(a, inverse(inverse(b))), inverse(b)))), inverse(a))
% 0.20/0.46 = { by lemma 16 }
% 0.20/0.46 multiply(inverse(b), inverse(a))
% 0.20/0.46 % SZS output end Proof
% 0.20/0.46
% 0.20/0.46 RESULT: Unsatisfiable (the axioms are contradictory).
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