TSTP Solution File: GRP029-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP029-1 : 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 : n006.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 01:16:40 EDT 2023
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% 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 : GRP029-1 : 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 : n006.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 : Tue Aug 29 02:26:36 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.19/0.41 Command-line arguments: --flatten
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% 0.19/0.41 % SZS status Unsatisfiable
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% 0.19/0.42 % SZS output start Proof
% 0.19/0.42 Take the following subset of the input axioms:
% 0.19/0.42 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))))).
% 0.19/0.42 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))))).
% 0.19/0.42 fof(left_identity, axiom, ![A]: product(identity, A, A)).
% 0.19/0.42 fof(left_inverse, axiom, ![A2]: product(inverse(A2), A2, identity)).
% 0.19/0.42 fof(prove_there_is_a_right_identity, negated_conjecture, ![A2]: ~product(not_right_identity(A2), A2, not_right_identity(A2))).
% 0.19/0.42 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.19/0.42 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 0.19/0.42
% 0.19/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.42 fresh(y, y, x1...xn) = u
% 0.19/0.42 C => fresh(s, t, x1...xn) = v
% 0.19/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.42 variables of u and v.
% 0.19/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.42 input problem has no model of domain size 1).
% 0.19/0.42
% 0.19/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.42
% 0.19/0.42 Axiom 1 (left_identity): product(identity, X, X) = true2.
% 0.19/0.42 Axiom 2 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.19/0.42 Axiom 3 (left_inverse): product(inverse(X), X, identity) = true2.
% 0.19/0.42 Axiom 4 (associativity1): fresh8(X, X, Y, Z, W) = true2.
% 0.19/0.42 Axiom 5 (associativity2): fresh6(X, X, Y, Z, W) = true2.
% 0.19/0.42 Axiom 6 (total_function1): product(X, Y, multiply(X, Y)) = true2.
% 0.19/0.42 Axiom 7 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.19/0.42 Axiom 8 (associativity1): fresh4(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.19/0.42 Axiom 9 (associativity2): fresh3(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.19/0.42 Axiom 10 (associativity1): fresh7(X, X, Y, Z, W, V, U, T) = fresh8(product(Y, Z, W), true2, Y, U, T).
% 0.19/0.42 Axiom 11 (associativity2): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, W), true2, W, V, T).
% 0.19/0.42 Axiom 12 (total_function2): fresh2(product(X, Y, Z), true2, X, Y, W, Z) = fresh(product(X, Y, W), true2, W, Z).
% 0.19/0.42 Axiom 13 (associativity1): fresh7(product(X, Y, Z), true2, W, V, X, Y, U, Z) = fresh4(product(V, Y, U), true2, W, V, X, U, Z).
% 0.19/0.42 Axiom 14 (associativity2): fresh5(product(X, Y, Z), true2, W, X, V, Y, Z, U) = fresh3(product(W, Z, U), true2, W, X, V, Y, U).
% 0.19/0.42
% 0.19/0.42 Lemma 15: fresh7(X, X, Y, inverse(Z), identity, W, identity, Z) = product(Y, identity, Z).
% 0.19/0.42 Proof:
% 0.19/0.42 fresh7(X, X, Y, inverse(Z), identity, W, identity, Z)
% 0.19/0.42 = { by axiom 10 (associativity1) }
% 0.19/0.42 fresh8(product(Y, inverse(Z), identity), true2, Y, identity, Z)
% 0.19/0.42 = { by axiom 10 (associativity1) R->L }
% 0.19/0.42 fresh7(true2, true2, Y, inverse(Z), identity, Z, identity, Z)
% 0.19/0.42 = { by axiom 1 (left_identity) R->L }
% 0.19/0.42 fresh7(product(identity, Z, Z), true2, Y, inverse(Z), identity, Z, identity, Z)
% 0.19/0.42 = { by axiom 13 (associativity1) }
% 0.19/0.42 fresh4(product(inverse(Z), Z, identity), true2, Y, inverse(Z), identity, identity, Z)
% 0.19/0.42 = { by axiom 3 (left_inverse) }
% 0.19/0.42 fresh4(true2, true2, Y, inverse(Z), identity, identity, Z)
% 0.19/0.42 = { by axiom 8 (associativity1) }
% 0.19/0.42 product(Y, identity, Z)
% 0.19/0.42
% 0.19/0.42 Goal 1 (prove_there_is_a_right_identity): product(not_right_identity(X), X, not_right_identity(X)) = true2.
% 0.19/0.42 The goal is true when:
% 0.19/0.42 X = identity
% 0.19/0.42
% 0.19/0.42 Proof:
% 0.19/0.42 product(not_right_identity(identity), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 7 (total_function2) R->L }
% 0.19/0.42 product(fresh2(true2, true2, inverse(inverse(not_right_identity(identity))), identity, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 6 (total_function1) R->L }
% 0.19/0.42 product(fresh2(product(inverse(inverse(not_right_identity(identity))), identity, multiply(inverse(inverse(not_right_identity(identity))), identity)), true2, inverse(inverse(not_right_identity(identity))), identity, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 12 (total_function2) }
% 0.19/0.42 product(fresh(product(inverse(inverse(not_right_identity(identity))), identity, not_right_identity(identity)), true2, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by lemma 15 R->L }
% 0.19/0.42 product(fresh(fresh7(Z, Z, inverse(inverse(not_right_identity(identity))), inverse(not_right_identity(identity)), identity, W, identity, not_right_identity(identity)), true2, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 10 (associativity1) }
% 0.19/0.42 product(fresh(fresh8(product(inverse(inverse(not_right_identity(identity))), inverse(not_right_identity(identity)), identity), true2, inverse(inverse(not_right_identity(identity))), identity, not_right_identity(identity)), true2, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 3 (left_inverse) }
% 0.19/0.42 product(fresh(fresh8(true2, true2, inverse(inverse(not_right_identity(identity))), identity, not_right_identity(identity)), true2, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 4 (associativity1) }
% 0.19/0.42 product(fresh(true2, true2, not_right_identity(identity), multiply(inverse(inverse(not_right_identity(identity))), identity)), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 2 (total_function2) }
% 0.19/0.42 product(multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.42 = { by lemma 15 R->L }
% 0.19/0.42 fresh7(X, X, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), identity, Y, identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 10 (associativity1) }
% 0.19/0.42 fresh8(product(multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.42 = { by axiom 9 (associativity2) R->L }
% 0.19/0.43 fresh8(fresh3(true2, true2, inverse(inverse(not_right_identity(identity))), identity, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 3 (left_inverse) R->L }
% 0.19/0.43 fresh8(fresh3(product(inverse(inverse(not_right_identity(identity))), inverse(not_right_identity(identity)), identity), true2, inverse(inverse(not_right_identity(identity))), identity, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 14 (associativity2) R->L }
% 0.19/0.43 fresh8(fresh5(product(identity, inverse(not_right_identity(identity)), inverse(not_right_identity(identity))), true2, inverse(inverse(not_right_identity(identity))), identity, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 1 (left_identity) }
% 0.19/0.43 fresh8(fresh5(true2, true2, inverse(inverse(not_right_identity(identity))), identity, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 11 (associativity2) }
% 0.19/0.43 fresh8(fresh6(product(inverse(inverse(not_right_identity(identity))), identity, multiply(inverse(inverse(not_right_identity(identity))), identity)), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 6 (total_function1) }
% 0.19/0.43 fresh8(fresh6(true2, true2, multiply(inverse(inverse(not_right_identity(identity))), identity), inverse(not_right_identity(identity)), identity), true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 5 (associativity2) }
% 0.19/0.43 fresh8(true2, true2, multiply(inverse(inverse(not_right_identity(identity))), identity), identity, not_right_identity(identity))
% 0.19/0.43 = { by axiom 4 (associativity1) }
% 0.19/0.43 true2
% 0.19/0.43 % SZS output end Proof
% 0.19/0.43
% 0.19/0.43 RESULT: Unsatisfiable (the axioms are contradictory).
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